CONTENTS About the Aufhor Prefoce
iv
xiii
Guided Tour
xvii
PanrOur Modeling,Computers, ond ErrorAnolysis I l. I Motivofion I 1.2 Port Orgonizotion
2
CHAPTER I Mothemqticol Modeling, Numericol Methods, ond Problem Solving 4 I . l A S i m p l eM o t h e m o t i c M o lo d e l 5 1.2 Conservotion Lowsin Engineering ond Science 12 1.3 NumericolMethodsCoveredin ThisBook l3 P r o b l e m s1 7
CHAPTER2 MATLAB Fundomeniqls 2.1 2.2 2.3 2.4 2.5 2.6 2.7 ^
20
TheMATIABEnvironmenl21 A s s i g n m e n t2 2 Mothemoticol Operotions 27 U s eo f B u i l t l nF u n c t i o n s3 0 Groohics 33 OtherResources36 CoseStudy:Explorotory DotoAnolysis 37
l l
rroblems JY
CHAPTER3 Progromming with MATTAB 42 3.1 M-Files43 3.2 InputOutput 47
CONTENTS
vl
3 . 3 S h u c t u r ePdr o g r o m m i n g5 l 3 . 4 N e s t i n go n d I n d e n t o t i o n6 3 g u n c t i o nt os M - F i l e s 6 6 3 . 5 P o s s i nF 3.6 CoseStudy:BungeeJumperVelocity 71 Problems 75
4 CHAPTER Roundoff qnd Truncotion Errors 79 4.1 ErrorsB0 4.2 RoundoffErrors 84 4.3 Truncotion Errors 92 ,l03 4 . 4 T o t oN l u m e r i c oEl r r o r 4 . 5 B l u n d e r sM, o d e lE r r o r so, n d D o t oU n c e r t o i n t y1 0 8 ,l09 Problems
.|11 PnnrTwo Roots ondOptimizotion 2.1 Overview tll 2.2 Port Orgonizotion
I 12
CHAPTER 5 Roois: Brocketing Methods
I l4
5 . 1 R o o t si n E n g i n e e r i nogn d S c i e n c e I l 5 5 . 2 G r o p h i c oM l ethods I l6 g e t h o d so n d I n i t i oG 5 . 3 B r o c k e t i nM l u e s s e s1 1 7 5 . 4 B i s e c t i o n1 2 2 5 . 5 F o l s eP o s i t i o n 1 2 8 5.6 CoseStudv:Greenhouse Gosesond Roinwoter 132 ,l35 Problems
CHAPTER 6 Roots: Open Methods
| 39
6 . 1 S i m p l eF i x e d - P o ilnt ei r o t i o n 1 4 0 6 . 2 N e w t o n - R o p h s o1n4 4 6 . 3 S e c o nM t ethods 149 6.4 MATLAB F u n c t i o nf :z e r o r 5 1 6 . 5 P o l v n o m l o l s1 5 4 : i p eF r i c t i o n 1 5 7 6 . 6 C o s eS t u d yP Problems 162
CONTENTS
Yrt
7 CHAPTER Optimizofion
166
7.1 Introduciion ond Bockground 167 7 . 2 O n e - D i m e n s i o n o l O p t i m i z o t i1o7n0 7 . 3 M u l t i d i m e n s i o n o l O p t i m i z o t i1o7n9 7 . 4 C o s eS t u d yE : q u i l i b r i uomn d M i n i m u mP o t e n t i oEln e r g y l 8 l P r o b l e m s1 8 3
,|89 PrnrTxnrr LineorSystems 3.1 Overview 189 3.2 Port Orgonizotion l9l
CHAPTER 8 Lineqr Algebroic Equofions ond Motrices
| 93
8.1 MotrixAlgebroOverview 194 with MATLAB 203 8.2 SolvingLineorAlgebroicEquotions ond Voltogesin Circuits 205 8.3 CoseStudy:Currents Problems 209
9 CHAPTER Gouss Eliminotion 212 9 . 1 S o l v i n gS m o l N l u m b e r so f E q u o t i osn 2 1 3 9 . 2 N o i v eG o u s sE l i m i n o t i o n2 1 8 9.3 Pivoting225 9 . 4 T r i d i o g o n oSly s t e m s2 2 7 9.5 CoseStudy:Model of o HeotedRod 229 Problems 233
IO CHAPTER [U Foctorizotion
236
l 0 . l O v e r v i e wo f l U F o c t o r i z o t i o n2 3 7 1 0 . 2 G o u s sE l i m i n o t i oons l U F o c t o r i z o t i o n2 3 8 1 0 . 3 C h o l e s kF y o c t o r i z o t i o n2 4 4 'l0.4 MATLABLeftDivision 246 Problems 247
CONIENTS
vIt!
CHAPTERI I Motrix lnverse ond Condition
249
1 i .l TheMotrix lnverse 249 I I .2 ErrorAnolysisond SystemCondition 253 I 1 . 3 C o s eS t u d yI:n d o o rA i r P o l l u t i o n2 5 8 Problems 261
CHAPTER I2 Iterotive Methods
264
l2.l LineorSystems: Gouss-Seidel264 ,l2.2 N o n l i n e oS r y s t e m s2 7 0 1 2 . 3 C o s eS t u d vC : h e m i c oRl e o c t i o n s2 7 7 P r o b l e m s2 7 9
PtrnrFoun CurveFitting 281 4.1 Overview 281 4.2 Port Orgonizotion
283
CHAPTER I3 Lineor Regression 284 1 3 . I S t o t i s t i cRse v i e w 2 8 6 1 3 . 2 L i n e o Lr e o s t - S q u oR r eesg r e s s i o n2 9 2 ,l3.3 L i n e o r i z o t i o nf N o n l i n e oR r e l o t i o n s h i p3s0 0 1 3 . 4 C o m p u t eAr p p l i c o t i o n s3 0 4 1 3 . 5 C o s eS i u d y E : n z y m eK i n e t i c s 3 0 7 ^
l l
rroDtems 5 | z
CHAPTER I4 Generol lineqr Leosf-Squores qnd Nonlineqr Regression 316 1 4 .I P o l y n o m i o Rle g r e s s i o n3 16 1 4 . 2 M u l t i p l eL i n e oR r e g r e s s i o n3 2 O 1 4 . 3 G e n e r oLl i n e oLr e o sSt o u o r e s 3 2 2 14.4 QR Foctorizotion ond the Bockslosh Ooerotor 325 1 4 . 5 N o n l i n e oR e g r e s s i o n 3 2 6 r 1 4 . 6 C o s eS l u d y F : i t t i n gS i n u s o i d s3 2 8 h
l l
rrontcms
. 1 . 1/
CONIENIS
'I 5.'l lntroduction to Interpotofion336 I5.2 NewronInterpoloring polynomiol 33g polynomiol j: .j tosron9etnrerpoloring 347 rJ.4 tnversetnterpolotion 350 I J.J txkopolotion o n d O s c i l l o t i o n s3 5 1 Problems 355
t6 cHAPTER Splines ond piecewise Inferpofofion I 6 . I l n k o d u c t i otno S p l i n e s 359 l o . z L i n e aSr p l i n e s3 6 1 1 6 . 3 Q u o d r o i i cS p l i n e s 365 1 6 . 4 C u b i cS p l i n e s 3 6 8
359
inMATLAB l9: liTewiseInrerpotorion
374 i 6.6 Multidimensionol Interpolotion 37g l6 Z CoseStudy:HeotTronsfer 3g2 rrobtems 386
PnnrFvr Infegrotion ond Differentiotion3g9 5.1 Overview 3g9 5.2 Port Orgonizotion 39O !H,\PTER| 7 Numericof fnfegrofion Formutos Sg2 'l Z.J lnhoduction ond Bocrground 393 I7.2 Newton-Cotes Formutos 396 17.3 TheTropezoidol Rule 39g 1 7 . 4 S i m p s o n ,Rsu l e s 4 0 5 17.5 Higher-Order Newfon_Cotes Formulos 4j j l7 6 lntegration with UnequolSegments 412 17.7 OpenMerhods 416 I 7 8 M u h , p l eI n t e g r o l s 4j 6 CompuringWork wirh Numericol lnregrorion 4j9 ;lJ;:r";udy: t I
!
t I
i{.
Numericof Integrotion of Functions 'l 8.I Introducfion 426 1 8 . 2 R o m b e r Iqn i e q r o t i o n 427
CONTENTS 1 8 . 3 G o u s sQ u o d r o t u r e4 3 2 I 8 . 4 A d o p t i v eQ u o d r o t u r e4 3 9 I8.5 CoseStudy:Root-Meon-Squore Current 440 Problems 444
CHAPTER I9 Numericql Differentiqtion
448
19.I 19.2 I9.3 ,l9.4 ,l9.5
Inhoduction ond Bockground 449 High-Accurocy Differentiotion Formulos 452 R i c h o r d s oEnx t r o p o l o t i o n4 5 5 Derivotives of UnequollySpocedDoto 457 Derivotivesond lntegrolsfor Dofo with Errors 458 19.6 PortiolDerivotives 459 I9.2 NumericolDifferentiotion with MATLAB 460 I 9 . 8 C o s eS t u d yV : i s u o l i z i nFgi e l d s 4 6 5 Problems 467
Pnnr5x
OrdinoryDifferentiol Equotions473 6.1 Overview 473 6.2 Porl Orgonizofion 477
CHAPTER 20 Initiol-Volue Problems
479
20.I Overview 481 20.2 EuleisMethod 481 20.3 lmprovemenls of Euler'sMethod 487 20.4 Runge-Kutfo Methods 493 20.5 Systems of Equotions 498 20.6 CoseStudy:Predotory-Prey Modelsond Choos 50A Problems 509
CHAPTER 2I Adopfive Merhods ond Stiff Systems
514
21 .'l AdoptiveRunge-Kutto Methods 514 2l .2 MultistepMethods 521 2l .3 Stiffness525 2l .4 MATLAB A p p l i c o t i o nB: u n g e Je u m p ew r i t hC o r d 5 3 1 2l .5 CoseStudy:Pliny'slntermittent Fountoin 532 r r o D l e m s3 J /
CONTENTS
xl
22 CHAPTER Boundory-Volue Problems
540
22.1 lntrodvction ond Bockground 541 22.2 lhe ShootingMethod 545 22.3 Finite-Difference Methods 552 P r o b l e m s5 5 9
APPENDIXA: EIGENVALUES 565 APPENDIXB: MATLABBUILT-INFUNCTIONS 576 APPENDIX€: MATIAB M-FltE FUNCTIONS 578 BIBLIOGRAPHY579 rNDEX 580
Modqling,CoTpute''i, qnd Erior Anolysis
t.t
MoTtvATtoN What are numericalmethodsand why shouldyou stridythem? Numericalmethodsare techniquesby which mathematicalproblemsare formulatedso that they can be solved with arithmeticand logical operations.Becausedigital computers excel at perform.ingsuchoperations.numericalmethodsare sometimesreferredto as computer mathematics. In the pre-computerera, the time and drudgeryof implementingsuchcalculationsse.-riously limited their practical use. However, with the advent of fast, inexpensivedigttul computers,the role of numerical methodsin engineeringand scientific problem solving has exploded.Becausethey figure so prominently in,:' much of our work, I believe that numerical methods should be a part of every engineer'sand scientist's basic education.Just as we a.ll must have solid foundationsin the other areasof mathematicsand science, we should also have a fundamentalunderstandingof numerical methods.In particular,we should have a solid appreciationof both their capabilitiesand their limitations. Beyond contributing to your overall education. .thog T9 several additibnat reasons why you shoutO .",,,,rr,,,r,,,r,,,,,, study numerical methods: 1. Numerical methods greatly expqld the types of , problems you can address.They are capableof handlinglarge systemsof equations.nonlineari, , d.l, and complicated geometriesthat are not uncommon in engineeringand scienceand that are often impossibleto solve analyticallywith standard calculus.As such"they greatlyenhanceyour problem-solving skills. 2. Numorical methods allow you to use "canned" so-ftwarewith insight. During your career,you will q*"
AND ERROR ANALYSIS PARTI MODELING,COMPUTERS,
3.
4.
5.
computerproinvariablyhave occasionto use commerciallyavailableprepackaged grarnsthat involve numericalmethods.The intelligentuseof theseprogramsis greatly enhancedby an understandingof the basic theory underlyingthe methods.In the absenceof suchunderstanding,you will be left to treat suchpackagesas "black boxes" with little critical insightinto their inner workingsor the validity of the resultsthey produce. Many problemscannotbe approachedusing cannedprograms.If you are conversant with numerical methods,and are adept at computer programming,you can design your own programsto solveproblemswithouthavingto buy or commissionexpensive software. Nr.rrnerical methodsare an efficient vehiclefbr learningto usecomputers.Becausenumericalmethodsale expresslydesignedfor computerimplementation,they areidealtbr illustratingthe conrputer'spowers and limitations.When you successfullyimplement numericalmethodson a computer,and then apply them to solve otherwiseintractable problenrs,you will be plovided with a dramaticdernonstrationof how computerscan serveyour professionaldevelopment.At the sarnelime, you rvilI alsolearnto acknowledge and control the errors of approximationthat are part and parcel of large-scale numericalcalculations. Numericalmethodsprovidea vehiclefbr you to reinforceyour understandingof mathernatics.Becauseone tunction of numericalmethodsis to reducehigher mathematics to basic arithmetic operations.they get at the "nuts and bolts" of some otherwise obscuretopics. Enhancedunderstandingand insight can result from this alternative perspective.
how numerical With thesereasonslls motivation.we can now set out to understand methodsand digital computerswork in tandemto generatereliablesolutionsto mathematical problems.The remainderof this book is devotedto this task.
1.2 PARTORGANIZATION This book is divided into six parts.The latter five partsfocus on the major areasof numerical methods.Although it might be temptingto jump right into this material,Part One consistsof four chaptersdealng with essentialbackgroundmaterial. Chapter 1 providesa concreteexampleof how a numericalmethodcan be employed to solve a real problem. To do this, we develop tt muthematicalmodel of a fiee-falling bungeejumper. The model, which is basedon Newton's secondlaw, resultsin an ordinary differentialequation.After first using calculusto developa closed-formsolution,we then show how a comparablesolution can be generatedwith a simple numerical method.We of numericalrnethodsthat we cover in end the chapterwith an overview of the major areers PartsTwo throughSir. Chapters2 and 3 provide an introduction to the MATLAB' software environment. Chapter2 dealswith the standardway of operatingMATLAB by enteringcommandsone at a time in the so-calledt'alculatornuttle.Thisinteractivemodeprovidesa straightforward meansto orient you kl the enviroumentand illustrateshow it is usedibr common opera-
I.2 PART ORGANIZATION Chapter-l shows how MATLAB's programming mode providesa vehicle for assembling individualcommandsinto algorithms.Thus,our intentis to illustratehow MATLAB servesas a convenientprogrammingenvironmentto developyour own software. ChapterI dealswith the irnportanttopic of error analysis,which must be understood for the effective use of numerical methods.The first part of the chapterfocuseson the roundoJferrors thar result becausedigital computerscannot representsome quantities truncationerrctrsthat arisefiom using an approximation exactly.The latter part addresses in place of an exactmathematicalprocedure.
{.1:i
ffilT**
i.t "
TFi +
i'i,r I r,if i
..,,lrfi;
U1
I
Mothemoticol Modeling, NumericolMethods, ond ProblemSolving
I
4 CHAPTER OBJECTIVES The prirnary objectiveof this chapteris to provide you with a concreteidea of what numericalmethodsare and how they relateto engineeringanclscientificproblem solving. Specificobjectivesand topicscoveredare o r o
r
Learning how mathematicalmodelscan be formulatedon the basisof scientific principlesto simulatethe behaviorof a simplephysicalsystem. Understandinghow numericalmethodsirlford a meansto generatesolutionsin a rnannerthat can be irnplementedon a digital computer. Understandingthe different typesof conservationlaws that lie beneaththe models usedin the variousengineeringdisciplinesand appreciatingthe diff'erence betweensteady-state irnd dynamic solutionsof thesemodels. Learning aboutthe difterent typesof numericalmethodswe will cover in this book.
YOU'VE GOT A PROBTEM upposethat a bungee-jumpingcompanyhires you. You're given the task of predicting the velocity of a jumper (Fig. l.l ) as a function of time during the free-fall part of thejump. This inlbrmationwill be usedaspart of a largeranalysisto determinethe length and requiredstrengthof the bungeecord for jumpers of differentmass You know from your studiesofphysics that the accelerationshouldbe equalto the ratio of the tbrceto the mass(Newton'ssecondlaw). Basedon this insightandyour knowledge
r
1
FIG For
f-^^ l I cc-
ium
I .I A SIMPTE MATHEMATICAL MODEL
Upwardforce d u et o a i r resistance
t
of fluid mechanics,you developthe following mathematicalmodel for the rate of change o l ' v e l o c i t yr . r ' i t rhe s p e ctto t i m e . ducd. dt''m
til
vtil
Downward forcedue to gravrty
where rr : vertical velocity (n/s). r : time (s), g : the accelerationdue to gravity (:9.81nls21, ca: a second-orderdrag coetficient (kg/m), and m: the jumper's mass(kg). Becausethis is a ditlerentialequation,you know that calculusmight be usedto obtain an analyticalor exactsolutionfor u as a function of /. However,in the following pages,we will illustrate an alternativesolution approach.This will involve developinga con.rputerorientednumericalor approximatesolution. Aside from showing you how the computercan be usedto solve this particularproblem, our more generalobjective will be to illustrate (a) what numericalmethodsare and (b) how they figure in engineeringand scientificproblen solving.In so doing, we will also show how mathematicaln.rodels figure prominentlyin the way engineersand scientistsuse numericalmethodsin their work.
flGUREl.l Forces ociingon o lreeJolling bungee iumpet.
I.l
A SIMPTEMATHEMATICAT MODET A motlrcnntical ntodelcan be broadly definedas a tbrmulation or equationthat expresses the essentialfeaturesof a physicalsystemor processin mathematicalterms.In a very general sense,it can be representedas a functionalrelationshipof the fonn D e o e n d e n ( - / i n d e' .n,e, n c l e n t forcine \ .v.a' .n.a. ,o_l e, : J [ v a n a D t e s. p u r a n ) e t el rusn.c t l o n s| \ ,f
(l.l)
where the de;tendentvariable is a characteristicthat usually reflectsthe behavioror state of the system:-the independetttvariablesare usually dimensions.such as time and space, along which the system'sbehavioris beingdetermined;theparametersareretlectiveof the system'sproperliesor composition;and thelbrring.functiottsareexternalintluencesacting upon it. The actual mathematicalexpressionof Eq. (1.1) can range from a sirnple algebraic relationshipto largecomplicatedsetsof diff-erentialequations.For example,on the basisof his observations,Newton formulatedhis secondlaw of motion, which statesthat the time rateof changeof momentumof a body is equalto the resultantforce actingon it. The ntathematicalexpression,or model,of the secondlaw is the well-known equation F:ma
(t.2)
whereF is the net force actingon the body (N, or kg nls"), m is the n.rass of the object(kg), a n dc i i s i t s a c c e l e r a t i o(nr n / s : ) .
6
MODELING,NUMERICAL METHODS, SOLVING MATHEMATICAL AND PROBLEM The secondlaw can be recastin the format of Eq. ( 1.l) by merely dividing both sides by m to give F
(1.-31
n1
where a is the dependentvariablereflecting the system'sbehavior,F is the forcing function, and nr is a parameter.Note that for this simple casethereis no independentvariable becausewe are not yet predictinghow accelerationvariesin time or space. Equation( 1.3)has a numberof characteristics that aretypical of mathematicalmodels of the physicalworld. . .
.
It describesa naturalprocessor systemin mathematicalterms. It represents an idealizationand sirnplificationof reality.That is. the model ignoresnegThus, ligible detailsof the naturalprocessand focuseson its essentialmanif'estations. minimal importance the secondlaw doesnot include the effects of relativity that are of when appliedto objectsand forcesthat interacton or aboutthe earth'ssurfaceat velocities and on scalesvisible to humans. can be usedfbr predictivepurFinally, it yields reproducibleresultsand,consequently, its poses.For example,if the force on an object and massare known, Eq. ( 1.3)can be usedto compLlteacceleration.
Becauseof its simple algebraicform, the solution of Eq. (1.2) was obtainedeasily. However,other mathernaticalmodelsof physicalphenomenamay be much more complex. and either cannotbe solvedexactly or requiremore sophisticatedmathematicaltechniques than simple algebrafor their solution.To illustrate a more complex model of this kind, Newton's secondlaw can be usedto determinethe terminal velocity of a free-fallingbody nearthe earth'ssurface.Our falling body will be a bungeejumper (Fig. 1.1).For this case, a model can be derived by expressingthe accelerationas the time rate of changeof the r,'elocity(tluldr) and substitutingit into Eq. (1.3)to yield duF dt
(1.4)
nr
where u is velocity (in metersper second).Thus, the rate of changeof the velocity is equal to the net force acting on the body normalizedto its mass.If the net force is positive,the object will accelerate.Ifit is negative.the objectwill decelerate.Ifthe net force is zero,the object'svelocity will remain at a constantlevel. Next, we will expressthe net force in terms of measurablevariablesand parameters. For a body talling witlrin the vicinity of the earth,the net force is composedof two opposing forces: the downward pull of gravity Fp and the upward force of air resistanceFy (Fig.1.1): F:
Fol
Fti
(1 . 5 )
If force in the downward direction is assigneda positive sign, the secondlaw can be u.sedto formulatethe force due to pravity as Fo:m8 whereg is the acceleration due to gravity(9.81m/s2).
(1.6)
I .I A SIMPLE MATHEMATICAL MODEI.
7
Air resistancecan be fbrmulatedin i.ivariety of ways. Knowledge from the scienceof fluid ntechanics suggests thata gtrodfirst approxirrration wouliJbe to assumethat it is proportional to the squareof the velocitl,, - l
f U:
-cdr)'
(1.7)
where r',1is a proporticlnalitvconstantcalled the drag coefticient(kg/m). Thus. the greater the fall velocity, the greaterthe uprvardfbrce due to air resistance.The parameterc./ accountslbr propertiesofthe ialling object,suchas shapeor surfaceroughness,that affect air resistance.For the presentc&s€,c,7might be a function of the type of clothing or the orientationusedby thejumper duringfree tall. The rlet fbrce is the differencebetweenthe downward and upwi.rrdforce. Therefbre, E q s .1 1 . 4 1 t h r o u g h( 1 . 7 )c a nb e c o m b i n e dt o y i e l d dt: ,
-,5
dt
ctt t
(1.8)
tn
Ecluation(1.8) is a ntodelthat relatesthe accelerirtion of a falling objectto the tbrces actingon it. It is a tli/Jerentialequtttiortbecauseit is written in terntsof the differentialrate of change(d uI dt 1 of the variablethat we are interestedin predicting.However,in contrast to the solutionof Newton'ssecondlaw in Eq. ( 1.3),the exactsolutionof Eq. ( 1.8)for the velocity of the jumper cannot be obtirinedusing simple algebraicmanipulation.Rather, more adt'ancedtechniquessuchas thoseof calculusnrustbe appliedto obtainan exactor analyticalsolution.For example,if thejumper is initially at rest(r.,: 0 at / : 0), calculus can be usedto solveEq. ( 1.8)for
:r,ff*"n( u(/) ,8,) yilt
(1.9)
/
where tanh is the hyperbolic tangent that can be either computed directlyr or via the more elementary exponential function as in
tanh-t:
e' - e-' e . ,* e \
(1.10)
Note that Eq. ( 1.9)is castin the generalfbrm of Eq. (1.1)wheret,(/) is the dependent variable.t is the independentvariable, ctt andm areparameters, andg is theforcing function.
EXAMPLE I .I
A n o l y t i c o lS o l u t i o nt o t h e B u n g e eJ u m p e rP r o b l e m ProblemStotement. A bungeejumper with a massof 68.1kg leapsliom a stationaryhot air balloon.Use Eq. ( 1.9)to computevelocityfbr the first l2 s of fiee fall. Also deterr-nine the terminal velocity that will be attainedfbr an infinitely long corcl(or alternatively,the jumprrasteris havinga particuiarlybad dayl).Use a dragcoefticientof 0.25kg/m.
I MATLABG'allows d i r e c rc a l c u l a t i o no f t h e h y p c r b o l i ct a n g e n vt i a t h c b u i l t - i nf u n c t i o nr d n h ( r ) .
MATHEMATICAL MODELING,NUMERICAL METHODSAND PROBTEM SOLVING Solution. Insertingthe parameters into Eq. (1.9)yields
which can be usedto compute
t, 3
0 2 4 6 B t0 12
u,mfs
o I B 72,92 33I I tB 42 4762 46 9575 4 S4 2 t4 50 6175 516938
According to the model, the jumper acceleratesrapidly (Fig. 1.2). A velocity of 49.4214m/s (about 110mi/h) is attainedafter 10 s. Note also that after a sufficiently lons
FIGUREI.2 Theonolyticol solution forthebungee problem oscompured in Exomple I 1 Velociry iumper increoseswith time ond osympfoiicolly opproocheso terminolvelociiy.
] .I A SIMPLE MATHEMATICAL MODEL
9
time, a constantvelocity.called the terminol velocitt',of 51.6983m/s (115.6mi/h) is reached.This velocity is constantbecause,eventually,the force of gravity will be in balThus. the net force is zero and accelerationhas ceased. ancewith the air resistance.
Equation( I .9) is called an anabtical or closed-formsolution becauseit exactly satisfies the oliginal diffbrentialequation.Unfortunately,tlrereare mirny matlrematicalnrodels that cannotbe solved exactly.In many of thesecases,the only alternativeis to develop a numericalsolutionthat approximatesthe exact solution. Nttntericalntethodsare thosein which the mathemirticalproblerr is refbrmulatedso it can be solvedby arithmeticoperations.This can be illustratedfor Eq. ( 1.8) by realizingthat by (Fig. 1.3): the time tateof changeof velocitycan be approximated
u ( / i + r -) u ( t i )
- Au - : - : Lt dt
dtt
(r.lr)
ti+t - ti
whereAu and At aredifferences in velocityand time computedoverflnite intervals,u(r1) is velocity at an initial time ri, and u(ria;) is velocityat somelater time f11. Note that du ldt = Lu I Lt is approximate becauseAr is flnite.Rememberfrom calculusthat dtt
dt
-
lim A1+l)
Au N
Equation( l. I I ) represents the reverseprocess.
FIGURE I.3 io / Theuseof o finitedifference thefirsfderlvotive of u wifhrespect to opproximote
AU
Approximateslope -u(r,) Ar_u(1,*,) Lt
li+t - ti
to
MATHEMATICAL MODELING,NUMERICAL MEIHODS,AND PROBLEM SOLVING Equation( 1.1I ) is calleclaJinite-diJJeren.ce opprcrirnationof the derivativeJt Iirnc /, . It can be substituted into Eq. (1.8)to give r'(1,,1)-u(t,l tr+t * t;
:g
.J _ _ t . t / 7 ) )tn.
This equationcan then be rearrangecl to yielcl
u(/i+r ) : u(/i)-1_ - 9lrt,,ff(/,+r - /i) [r
( r .r 2 )
Notice that the tenn in bracketsis the right-handside of the diff'erentialequationitself tEq. (1.8)1.That is, it providesa meansto computethe rateof changeor slopeof u. Thus, the equat.ioncan be rewritten as
u i + :r r + ' ! u
(t.13)
{1t
wherethe nomenclature u; clesignates velocityattinle /i and At : ti+t - ti. We cannow seethat thedifferentialequationhasbeentransformedinto an equationthat canbe usedto determinethe velocityalgebraically at ri+l usingthe slopeandpreviousvaluesof u andt. If you aregivenan initial valuefor velocityat sometime l;, you caneasilycomputevelocityat a latertime f 11. This new valueof velocityat l;11canin tum be employedto extendthe cornputationto velocityat l;12 and so on. Thus at any time along the way, New valne: old vahle* slopex stepsize This approachis tbrnrally calledEuler's metlnd. We'll discussit in more detail when we turn to diff'erentialequationslater in this book.
E X A M P L E1 . 2 '
N u m e r i c o lS o l u i i o nt o t h e B u n g e eJ u m p e rP r o b l e m P r o b l e mS t o i e m e n t . P e r f o r mt h es a m ec o r n p L l t a t i o an si n E x a m p l e1 . 1 b u tu s eE q . ( J . 1 3 ) to colnputevelocitywith Euler'smethod.Employ a stepsizeof 2 s fbr the calculation. Solution. At the startof the computation(/{):0), the velocity of the jumper is zero. Using this infbrmationand the parameter valuesfrom ExampleI . I , Eq. ( 1.13)can be used to corxputevelocityat 11- 2 s:
r,: o* fr.r' - H,o,rl x 2 : te.62rls 68.r L I For the next interval lfiom r : 2 b 4 sJ,the colnplrtationis repeated,with the result :
l-
0.2s
l ( ) . 6 2+ 1 9 . 8 1- , ( 1 9 . 6 2 t - l 68.t L
.1 r
" l:16.4117m/s
tt
I .1 A SIMPLE MATHEMATICAL MODEL
Terminalvelocity
F I G U R EI . 4 for thebungeeiumperproblem Compcrison of thenumericol ond onclyticol solutions The calculationis continuedin a similar fashionto obtain additionalvalues: ttS
0 2 4 6 B
r0 l2 oo
u, m/s 0 r9.6200 36.4137 46 2983 50 r802 5t 3123 5 r 6008 5r 6938
The resultsare plotted in Fig. 1.4 along with the exact solution.We can seethat the numerical methodcapturesthe essentialfeaturesof the exact solution.However,becausewe have employed straighfline segmentsto approximatea continuouslycurving function, there is some discrepancybetweenthe two results.One way to minimize such discrepanciesis to usea smallerstepsize.Forexample,applyingEq. (1.13)at 1-sintervalsresultsin a smaller error, as the straighrline segmentstrack closerto the true solution.Using hand calculations,the effort associatedwith using smaller and smaller step sizeswould make such numerical solutionsimpractical.However, with the aid of the computer,large numbers of calculationscan be performedeasily.Thus, you can accuratelymodel the velocity of thejumper without having to solve the differentialequationexactly.
r2
MATHEMATICAL MODELINGNUMERICAL METHODS, AND PROBLEM SOLVING As in Example 1.2,a cornputationalprice nrustbe paid for a nroreaccuratenumerical result.Each halving of the stepsize to attainmorc accuracyleadsto a doubling of the nurnber of computations.Thus, we seethat thereis a trade-offbetweenaccuracyand computational effort. Such trade-offsfigure prominently in numerical methodsand constitutean importantthemeof this book.
L/AWSlN ENGINEERING AND SCIENCE 1.2 CONSERVATION Aside from Newton's secondlaw. there are other major organizingprinciples in science and engineering.Among the most importantof theseare the conserv,ation lan:s.Although they form the basis for a variety of complicatedand powerful mathematicalmodels, the great conservationlaws of scienceand engineeringare conceptuallyeasy to understand. They all boil down to - decreases Change: increases
(1.r4)
This is preciselythe fbrmat that we empioyedwhen using Newton'slaw to developa force balancefor the bungeejumper tEq. ( 1.8)1. Although simple,Eq. (1.14)embodiesone of the most fundarnentalways in which conservation laws are used in engineering and science-that is. to predict changes with respect to time. We will give it a special name-the time-variable (or transient) computation. Aside from predictingchanges,anotherway in which conservationlaws are appliedis fbr caseswherechangeis nonexistent. If changeis zero,Eq. (I.14) becomes Change: 0 : increases decreases or Increases: decreases
(1.r5)
Thns, ifno changeoccurs,the increasesanddecreases nrustbe in balance.This case,which is also given a specialnarne-the stea(ly-state calculation-has many applicationsin engiincompressiblefluid flow in pipes,the neeringand science.For example,fbr steady-state flow into a junction musl be balancedby flow going out. as in Flow in : flow out For the junction in Fig. I .5, the balancecan be used to computethat the flow out of the fourth pipe must be 60. For the bungeejumper, the steady-state condition would correspondto the casewhere the net lbrce was zeroor [Eq. (1.8) with du ldt : 0l l1l$ :
f
6411-
(1.r6)
Thus. at steadystate,the downwardand upwardfbrcesarein balanceand Eq. ( 1.| 6) can be solvedfor the terminal velocity
AlthoughEqs.(1.14)and (1.15)might appeartriviallysimple,they embodythe two fundamentalways that conservationlaws areemployedin engineeringand science.As such,they will tbnn an importantpart of our et-fortsin subsequent chaptersto illustratethe connection betweennumericalmethodsand engineeringand science.
I.3 NUMERICAL METHODSCOVERED IN THISBOOK
r3
P i p e2 F l o wi n = 8 0
Pipe 1 F l o wi n : 1 0 0
Pipe 4 Flow out =
P i p e3 F l o wo u t : 1 2 0
F I G U R EI . 5 A f o w b o o n c ef o rs t e o d yi n c o m p r e s s ifbuei df l o t vo t t h ej u n c l i oonf p i p e s
Tablel. I summarizes somemodelsandassociated conservation lawsthatfigurepromrnently in engineering.Many chemical engineeringproblems involve mass balancesfor reactors.The massbalanceis derived from the conservationof mass.It specifiesthat the changeof mass of a chemical in the reactordependson the amount of mass flowing in minus the n.rassflowing out. Civil and mechanicalengineersoften focus on models developedfrom the conservation of momentum.Forcivil engineering,force balancesare utilized to analyzestructures suchas the simple trussin Table 1.1.The sameprinciplesare employedfor the mechanical engineeringcasestudiesto analyzethe transientup-and-downmotion or vibrations of an automobile. Finally.electricalengineeringstudiesen-rploy both currentandenergybalancesto model electriccircuits.The currentbalance,which resultsfrom the conservationof charge,is similar in spirit to the flow balancedepictedin Fig. 1.5.Justasflow mnstbalanceat thejunction of pipes,electriccurrentmust balanceat thejunction of electricwires.The energybalance specifiesthat the clrangesof voltagearoundany loop of the circuit must add up to zero. We shouldnotethattherearemanyotherbranches of engineeringbeyondchemical,civi,, electrical,andmechanical.Many of theseale relatedto the Big Four.For exalnple,chemical engineeringskills are usedextensivelyin areassuchas environmental,petroleum,and biornedicalengineering.Sirnilarly,aerospace engineeringhasmuch in cornmonwith mechanical engineering.We will endeavorto includeexamplesfrom theseareasin the coming pages.
I.3
NUMERICAT METHODSCOVERED IN THISBOOK We choseEuler's methodfor this introductorychapterbecauseit is typical of many other classesof numericalmethods.In essence, most consistof recastingmathematical operations into the simple kind of algebraicand logical operationscompatiblewith digital compllters.Figure 1.6 summarizesthe major areascoveredin this text.
l4 TABTEl.l
Field
MATHEMATICAT MODELING,NUMERICAT METHODS, AND PROBLEM SOLVING Devicesond typesof boloncesfhotore commonlyusedin ihe four moiororeosof engineering. For eochcose,lhe conservotion low on whichthe bolonceis bosedis specified. Device
Chemical engineering
OrganizingPrinciple
MathematicalExpression
Conservation of mass
Mass balance:
ffi
inort ff_--*
ourpur
Over a unit of time period Amass:inputs-outputs
I
Civil e n g in e e r ni g
Conservation of momentum
t structyz
ffi ?7fu.
+Fv
Forcebalance:
,,\
+ I
-Fn *
O+
\
+FH
I
V
,1m77,
-at/
At each node I horizontalforces (FH)= o I vertical forces (I'u) : 0
Conservation of momentum
Mechanical engineering
Forcebalance:
I Upward force I
lr=0 I
V Downward force m Li
Electrical engineering
ilH3
Conservation of charge
= downward force - upward force
C u r r e n tb a l a n c e :+ i , For each node I current (i) = 0
I'R'
Conservation of energy
Voltagebalance:
a{A&-l ,-, -R2,Jr * z- -Y
f
L--\A7\--J i:R: Around each loop I emf's - I voltage drops for resistors
>6->a:0
r5
I.3 NUMERICAL METHODSCOVERED IN THISBOOK
lal Part 2: Roots and optimization
f(xl
Roots: Solve for.r so thatfi-r) = 0 Optimization:Solvefor x so that/'(r) = 0
Optima
lbl Part 3: Linear algebraic equations Giventhe a's and the b's.solve for the.r's
f\x\
a ' , r xt a r 2 x . ab =, arrx, 1-a,x, = b2
lcl Part 4: Curve fitting
o
ldl Part 5: Integration and differentiation Integration:Find the area under the curve Differentiation:Find the slooe of the curve
lel Part 6: Differential equations Given dv Av : ,h N:flt'Yl solve for r as a function of r .,Ii+r= -]'i+ "f(ti,yJAr
FIGURE I.6 Summory of thenumericol methods covered inthisbook.
t6
MATHEMATICAL MODELING,NUMERICAT METHODS, AND PROBLEM SOLVING Part Two dealswith two relatedtopics:root finding and optimization.As depictedin Fig. 1.6a,root locotiorrinvolvessearchingfor the zerosof a function.In contrast,optimizarion involvesdetermininga valueor valuesof an independentvariablethat correspondto a "best" or optirnalvalue of a function.Thus, as in Fig. 1.6a, optimizationinvolvesidentifying maximir and minima. Although somewhatdifferent approachesare used,root location and optimizationboth typically arisein designcontexts. Part Three is devotedto solving systemsof simultaneouslinear algebraicequations (Fig. 1.6&).Such systernsare similar in spirit to l'ootsof equationsin the sensethat they are concemedwith values that satisfy equations.However, in contrastto satistying a single equatiou,a set of valuesis soughtthat simultaneously satisfiesa set of linear algebraic equations.Suchequationsarisein a varietyof problemcontextsand in all disciplinesof engineeriugand science.In particular,they originate in the mathenraticalmodeling of Jarge systemsof interconnected elementssuchas structures,electriccircuits.and fluid networks. However,they are also encounteredin other areasof numericalmethodssuchas curve titl i n g l r n dd i f f e r e n t i ael q u u t i o n s . As an engineeror scientist.you will often haveoccasionto fit curvesto datapoints.The techniquesdevelopedfor this pulposecanbe dividedinto two generalcategories:regression regressionis ernployedwherethere and interpolation.As describedin Part Four tFig. 1.6c'1, is a significantdegreeof error associirted with the data.Experimentalresultsareoften of this kind. For thesesituations.the strategyis to derivea singlecurve that representsthe general trend of the datawithout necessarilymatchingany individualpoints. In contrast,interpolutionis usedwherethe objectiveis to determineintermediatevalues betweenrelativelyerror-freedata points. Such is usually the casefor tabulatedinformation.The strategyin suchcasesis to flt a curve directly throughthe datapoints and r.rse the curve to predict the intermediatevalues. As depicted in Fig. 1.6d, Part Five is devoted to integlation and differentiation.A plrysical interpretationof ruurrcricctliltegratiott is tlre determinationof the area under a curve. Integrationhas many applicationsin engineeringand science,rangingfrom the determinationof the centroidsof oddly shapedobjectsto the calculationof total quantities In addition,nurnericalintegrationformulasplay an basedon setsof discretemeasurements. importtrntrole in the solution of diffbrentialequations.Part Five also covers methodsfor nume.ricaldifr'erentiation. As you know fiom your study of calculus,this involves the determinationof a function's slopeor its rate of change. equations(Fig. 1.6e). Finally.Part Si.xfocuseson the solutionof ordirro'vdi.fterential Such equationsare of greatsignificancein all areirsof engineeringand science.This is becausemany physicallaws arecouchedin termsof the rateof changeof a quantityratherthan rnodels the magnitudeof the quantity itself. Examplesrangefrom population-forecasting (rateofchangeof population)to tlreacceleration of a tallin-ebody (rateofchangeofvelocity). initial-valueandboundary-valueproblems. Two typesof problemsareaddressed:
t7
PROBLEMS
PROBTEMS l.l Usecalculusto velity that Eq. (1.9) is a solutionof E q (. 1 . 8 ) . 1.2The following infbrmation is availablc for a bank acc0unt: Dote
Deposits
Withdrowols
2 2 41 3
327.26
2t6B0
3 7 86 1
, r ( nr <
r 0 68 0
t 2 7. 3 1
3 5 06 r
5ir
Bolonce I 5 I 2 .3 3
6/l
7/l
Decay rate : -tc whereft is a constantwith units of day I. Thercfore,according to Eq. (1.14),a rrass balancefbr the reactorcan be wntten as dc
Bit 9ll
Usetheconservation of cashto computethe balanceon 6/ l, 111.811, and9/1. Showeachstcpin thc computation. ls this a steady-state or a transientcomputation? 1,3Repeat Example1.2.Computethc velocityto t: l2 s, witha stepsize of (a) I and (b) 0.5 s. Can you make any statement regardingthe crrorsof thc calculationbasedon the results? 1.4Ratherthanthe nonlinearrclationshipof Eq. ( 1.7),you mightchooseto model the upward fbrce on the bungee j u m p car sa l i n e a r e l a t i o n r h i p : rl f tt :
1.7 Thc amountof a uniformly distributedradioactivecontaminant containedin a closed reactor is measuredby its concentrationc (becquerel/literor Bq/L). The contaminant decrcasesa1 a decay rate proportionalto its concentration; that is
C U
wherer'': a first-orderdrag coefTicient(kg/s). (a) Usingcalculus,obtain the closed-fbnn solution fbr thc casewherethejurnperis initiallyat rcst (u : 0 at 1: 0). (b) Repeatthe numericalcalculationin Example 1.2 with the sameinitial condition and oarametervalues.Use a valueof 12.5kg/s fbr c'. 1.5For the free-talling bungee jumper with linear drag (Prob.I .4).assumea first jumper is 70 kg and hasa drag coefficient of l2 kg/s. If a secondjurnperhasa drag coefficient of 15kg/sand a massof 75 kg, how long will it take her to reach thesamevelocityjumpcr I rcachedin l0 s'l 1.6 For the fiee-falling bungce jumper with linear drag (Prob. 1.4),compulethe velocityof a free-tallingparachutist usrng Er-rler's methodfbr thc cascwhcre rr : 80 kg and c' : l0 kg/s.Perfbrm thc calculation from / : 0 to 20 s with a stepsizeof I s. Use an initial conditionthat the parachutist hasanupwardvclocity of 20 m/s at /: 0. At r: l0 s, assume thatthe chuteis instantaneously deployedso that thc dragcocllicientjumps to 50 kg/s.
dt / changc\ t .i"ntm: ta,rls \ /
kt / decrease \ U l d e c a S \ /
(a) Use Euler's mcthod to solve this equationfrom t : 0 to I d w i t h k : 0 . 2 d r . E m p l o y a s t e p s i z eo f A r : 0 . I d . The concentration at /:0 is l0 Bq/L. (b) Plot the solution on a semilog graph (i.e., ln c versus/) and detenninethe slopc.Intcrpretyour results. l ll A storagetank (Fig. Pl.8) containsa liquid at depth) where,r': 0 when the tank is half full. Liquid is withdrawn at a constantflow rate Q to meetdemands.The contentsare resuppliedat a sinusoidalrate3Q sin2(t).Equation(1.14) can be written fbr this systcrnas
-d(+Av'\ AT
/ c h a n e ei n \ | ,' l: \ votume /
F I G U RP EI . 8
: 3 0 s i n ' ( r )-
O
tinflowt - (outflow)
r8
MATHEMATICAL MODELING,NUMERICAL METHODS, AND PROBLEM SOTVING
or, sincethe surfaceareaA is constant r/t'O.O : :3: s i n - f t )_ : drAA
- rr) o : - Ir': c,dr : nrc,,(rz
Use Euler's method to solve for the depth _vfron-rr : 0 to I0 d with a step size of 0.5 d. The parametervaluesareA : 1200m2 and p : 500 m3/d.Assumethat the initial condition i s . y: 0 . 1.9 For the same storagetank describedin Prob. 1.8, suppose that the outflow is not constantbut rather dependson the depth. For this case,the differential equationfbr depth can be written as dl' dt
nr the heat capacity,and thc changein temperatureby the followingrelationship:
a(ltr')r O . = . 1 ; s i n - (trA
Use Euler's method to solve for the depth )' fiom t : 0 to 10 d with a step siz-eof 0.5 d. The parametervaluesareA : 1200m2,O:500 mr/d, and cv: -300.Assurnethat the initial condition is _r: 0. 1.10 The volume flow rate through a pipe is given by Q : rA, whele u is the average velocity and A is the crosssectionalarea.Use volume-continuityto solvefor the required areain pipe3 of Fig. P I . 10. l.ll A group of 30 studentsattenda classin a room which lneasuresl0 m by 8 m by 3 m. Each studenttakesup about 0.075 mr and gives out about 80 W of heat (l W = I J/s). Calculatethe air temperaturerise during the first l5 minutes of the class if the room is completely sealedand insulated. Assume the heat capacityC,. tbr air is 0.7 18 kJ/(kg K). Assume air is an ideal gas at 20 "C and 101.325kPa. Note that the heatabsorbedby the air O is relatedto the massof the air
The massof air can be obtainedliom the ideal sas larv:
Plt :
nt
-P7 Mwt
where P is the gaspressure,V is lhe volume of the gas,Mwt is the molecularweight of the gas (1br air 28.97 kg/krnol), and rRis the ideal gas constant[8.31,1kPa m]/(kmol K)]. 1.12 FigureP1.12depictsthe variousways in which an averilgeman gainsandloseswaterin oneday.One liter is ingested as food, and the body metabolicallyproduces0.3 liters. In breathingair, the exchangeis 0.05 liters while inhaling, and 0.4 literswhile exhalingover a one-dayperiod.The body will also lose 0.2, 1.4.0.2.and 0.35 litersthroughsweat,urine, feces,and throughthe skin, respectively.To maintainsteady state,how much water must be drunk per day? l.13 In our example of the fiee-falling parachutist,we assumed that the accelerationdue to gravity was a constant value of 9.8 m/s2.Although this is a decentapproxinration when we are examining falling objectsnear the surtaceof the earth, the gravitational lbrce decreasesas we lrove above sea level. A rnore general representationbased on Newton's inversesquarelaw of gravitationalattractioncan be written as R: s ( x ) : g ( 0 ) ( l-( + . r ) '
Skin
Qr.in=
Qz'rt: 20 m3/s Food Drink u.,ou, = 6 m/s
A:=? F I G U R EP I . I O
Metabolism
F I G U RPEI . I 2
r9
PROBLEMS whereg(r) : gravitationalaccelerationat altitude.r (in m) measured upward fronr thc earth'ssurfacetm/s2).gtO) : gravitational accelerationat the earth'ssurface(! 9.8 rn/sr), a n dR : t h ec a r t h ' sr a d i u s( = 6 . 3 7 x 1 0 6m t . (a) In a fashionsimilar to the derivationof Eq. (1.8), use a forcebalanceto derive a ditlerential equationfor velocity as a function of time that utilizes this more complete representation of gravitation.However. lbr this derivation, assumethat upward velocity is positive. (b.1For the casewhere drag is negligible,use the chain rule to expressthe differentialequationas a function of altituderatherthan time. Recall that the chain rule is du
du d-r
dt
d.r dt
(c) Use calculusto obtain the closed form solution where u = u,,at.r : 0. (d) UseEuler's rnethodto obtain a numericalsolutionfrom r : 0 to 100,000m using a stepof 10,000m where the initial velocity is 1400m/s upward.Compareyor,rrresult u ith the analyticalsolr"rtion. l. l{ Supposethat a sphericaldroplet of liquid evaporatesat a ratethat is proportionalto its surfacearea. dV :: dt
-kA
rvhereV: volume 1mm3),t : time (hr), k : the evapol'ation rate(mm/hr), and A : surface area 1mmr). Use Euler's methodto conrputethe volume of the droplet from I : 0 to l0 min using a step size of 0.2,5min. Assume that ft : 0.1mm/min andthat thedropletinitially hasa radiusof 3 mm. Assess the validity oi your resultsby determiningthe radius
of your final computedvolume and verifying that it rs consistentwith the evaporationrate. l.l-5 Newton's law oicooling saysthat the temperatureof a body changesat a rateproportionalto the differencebetween its temperatureand that of the surroundingmedium (the ambient temperature).
d -; T : - k ( T - T " ) at
where Z: the temperatureof the body ("C), r : time (rnin), k : the proportionalityconstant(per minute), and 7, : ths arnbienttemperature("C). Supposethat a cup of coft-eeoriginally has a temperatureof 68 'C. Use Euler's method to computethe temperaturefrom I : 0 to l0 min using a step sizeof I min if I.. : 2l "C and ft : 0.017/min. |.16 Afluid is pumpedinto the networkshownin Fig. P1.16. lf 0, : 0.6.O.,: 0.4.Qt:O.2. andQo: 0..1mr/s.determine the other flows.
ilr*?-i=-i--I iiri:;$
bt'ili o,l "i "i "'i i-.er-:--s...;-*-*L*-i FIGURE PI.I6
\
iiEi;4+'" ,l
i.
MATLABFundomentols
2.1 CHAPTER OBJECTIVES and overviewof The primaryobjectiveo1'thischapteris to providean introductiorr how MATLAB's calculatormodeis usedto implcmentinteractivecornputations. Specificobjectivesand topicscoveredare ' . . . .
Lcarninghow realanclcornplexnurrrbcrs areassignedto variables Learninghow vectorsand matricesareassignedvaluesusingsinrpleussignrnent. .l l'ttnctitlns. the colon ope'rator', and thc irs;p.r,:cand 1oq1,;piic:,--thc priority rulcsfirr constructingmathernatical cxpre-ssions. Llnderstanding ol'built-in lr-rnctions and how you can lcarntnore Gaininga gencralundcrstanding a b o u t h e mw i t h M A ' I ' L A B ' sH e l p f a c i l i t i e s . Learninghow to usc vectorsto crcirtca sinrplclinc plot baseclon an equation.
YOU'VE GOT A PROBLEM n Chap. l. we usctlir firrcebalanceto detcrnrinethe tcrminalvelocityof a fiec-falling I o h j e c ll i k c i r h L r r r g cj uer n p c r . I T
tsi V t,t
(m/s').m : mass(kg), wher-er.,,: ternrinalvelocity(nr/s),,q: gravilationalaccelerertion and t',,: a dragcoefl'icient(kg/m).Asidefrom prcdictingthe terminalvelocity,this equato computethe dragcoefficicnt tion can alsobe rearranged m8
( 2 .I ;
ut
20
ffi=?N ,f zr'+
4_ %. uE-1 5t %I zl x+ =
2r MAT4gllyllerynENl 2 r THE TABTE2.I m,kg u,,ml s
Dotoforthemossondossociotedtermino|velociliesofonumberofiumpers BO9 653 gll 929 721 6A2 5l l 836 4 7 7 5 4 557 la \ 509 (1,,1
of knclwnmass,this velocityof a numberol .iumpers terminal the nleasure we Thus, if The clatain Table2' I werecolestimatethe drag coeificient' tt.l *t-' o provides equation l e c t u ' ldo r l h i s p u r p o s e ' ^ m , ^ n ^ ^ ^ , . ^ , , . n r rl n analyze r n : r l v . ztr ' s r rdclht u . B e y o n d how MATLAB canbe uscdto leartr will we chapter, In this showinghowMATLABcanbeenlployedtocolnputeqtrantitieslikeclragcoetTicients'we willalsoillustratchowttsgraphicalcapabilitiesp,*i.t.adtlititlnalinsightintosuchanalyses.
2.r M A T L A B i s a c o m p u t e r p r o g r a l x t h a t p r o v i < l c s . t hiteprtlvides u s e r w i ta h avery c o nnice v e l rtool i e t r tto e ninrple. virontlrentfclr ln particular, typcs.ol.ctilculaticrns. many perfornirrg
* *' 1P, rute " "!i:H:i:iilll1l;|;"vtoope :',:,'^ ::::::::::):i:I;:;: ii' ffi:lr ngprots n ati crc u'T*.,ln ;::'# ":1 "'" ilJffill'.":ili: ff.:ll,'Ji:'.:il{; "'to createMAft-AB progralns'
commandscan be usecl Chap.3, we show how sr'rch o n e f u r t h e r . n o t c ' . [ h i s c h a p t e r h a s b e e n w r i tThe t e n a s a l relficient a n c l s - t way l r r c xtoc lbeconle .cise.Thatis,yott o1'youlcornpLrtcr. m.st fiont in ,,tring while it shoultlrcad oroficientist'oacttrallyinl;llctlentthcctltntrratrclsonMATLABasyouprocccdthrotrghthe following material' witrclows: fUeif-A,B usesthreeprimary anddata' winclow'Uscdto entercommancls . Conrtnancl graphs' and plots display . C."pt,i.* windtlw' Usedkr and edit M-filcs' creute tt'r gaii Usecl . winclow' wind.ws' ln chap' 3 we usc of thc c.mmand anclgraphics In this chapter,we wilr r.rake crcatcM-iiles' prompt *iii ut. the edit window to openwith thc commancl tht tu'ntunti window will Afier startingMd;;;' bcingdisPlaYcd y o L lt y p c i n c o n t lrI a s c q u c n t i i tl ll t s h i t r na s oper''rtcs o1'MATLAB nlode The calculator you can think of it as operyou get'r-result cotnurancl' each For line' by Lh::s' manclsline tn For exanlplc'if you typc ^ii"g f if,. a uery l'ancycalcttlator' -_-,
51
Lo
rcsult' MATL.AB will disPlaYthc a,ttt-,=,,
1c'i':'r ' M A | l , A B s k r p s a l i n c b c t w c c l l t | - r | a h e l ( l , r r l . ) l t n d t h c n u n l b c r . ( - ] ) . H( e) rr: rr r- '.1$i(1..i (l L, m l r firtis: 1lAl cuhd b! .l rl :n' rk' 'l j n c s t o r I l i n e ' sa r e i n c l u d c t l w i t h t h c a u n , , a r t. u h . i t h e rr r t a n t co.ciscness.varuaun comlrtnds.
ffi qA ,.-ILJ'
8 b
zt\ 4x
4
AL ^r-
?t 5t !8
22
MATLABFUNDAMENTALS Notice that MATLAB has automaticallyassignedthe answerto a variable,ans. Thus, you could now use ans in a subseouentcalculation: '-
ars
+
II
with the result 50 MATLAB assignsthe resultto ans wheneveryou do not explicitlyassignthe calculation to a variableof your own choosing.
2.2 ASSIGNMENT Assignmentrefersto assigningvaluesto variablenames.This resultsin the storageof the valuesin the memorylocationcorresponding to the variablename.
2.2.1 Scolors The assignment of valuesto scalarvariables is sirnilarto otherconlputerlanguages. Try typing 1 _ A
Note how the assignment echoprintsto confirmwhat you havedone: 4 Echoprintingis a charactelistic of MATLAB. lt carrbe suppressed by terminatingthecommand line with the semicolon(; ) character. Try typing >> A
=
5;
You can type severalcommandson the sameline by separating them with comnlasor semicolons.If you separatethem with commas,they will be displayed,and if you usethe semicolon,they will not.For example, >> a -
4,A = 6;x
= . 1;
twe
4 MATLAB treatsnames in a case-sensitivernanner-that is. the nan)e a is not the same as the name a. To illustrate this. enter >>
short Iong short e long e long eh^rt
and then enter
long
>>A
bank
Seehow their valuesare distinct.Thev are distinct names.
frT-Tbl
"L x jj€-
zfTtr ='1.t- st 4 zl-+ Z] X+? .-f
J
g
a
_A
-z th -r
15/ I
end
eng
23
2.2 ASSIGNMENT
We can assigncomplex valuesto variables.since MATLAB handlescomplex arithmetic automatically.The unit irnaginary number ir preassignedto the variable i. "/J Consequently, a complexvaluecan be assignedsimply as in =
>> x
2+i*4
2.0000 + 4.0000i It shouldbe notedthat MATLAB allows the symbol j to be usedto representthe unit imaginary numberfor input.However,it alwaysusesan i for display.For example, = 2+1*4
>> x
2.0000 + 4.0000i Thereare severalpredefinedvariables.fbr example.pi. pi
>-
3 . r4r6 Noticehow MATLAB displaysfbur decinralplaces.If you desireadditionalprecision, enterthe fbllowing: >>
format
long
Now whenpi is enteledthe resultis displayedto l5 signilicantfigures: >> pi
1 . 1 4 1 5 9 2 b 5 - 15 8 9 7 9 To return to the four decimal version, type >> format
short
The f
twe
Result
Exomple
- c h o ]t
S c o e d f i x e dp o i n t f o r m c tw i t h 5 d i g i t s S c c r i e df r x e d - p o i nfto r m o iw i t h 1 5 d i g l h f o r d o u b l e o n d Z d i g i t sf o r s i n g l e F l o o i i n g ' p o i nfto r m c tw i i h 5 d i q i t s F l o o i i n gp o i n t l o r m o tw i f h l 5 d i g i i s f o r d o u b l e o n d / d l g i t sl o r s i n g l e B e s lo f f i x e d o r f l o o f i n g - p o i nl ol r m o tw i t h 5 d i c t l t s B e s to [ [ i x e d o r f l o o t i n gp o i n l f o r m o fw i t h 1 5 d l g i t sf o r d o u b l e andTaglsforsingie E n g i n e e r l n gf o r m o fw i t h o l l e c s l5 d i g i t so n d o p o w e r i h o t i s c r r n u t l p l eo f 3 Engineen r g f o r m o tw i i h e x c c l l y 1 6 s i g n i f i c o ndt i g i t so n d o p o w e r t h o t s o m ut i p l eo f 3 F i x e dd o l l o r so n d c e n t s
3. r416 i.1415q1b5J58979 l.1415e+0tr0 3 . 1 4l 5 9 2 6 5 3 5 8 9 7 9 3 e + 0 0 0 3.r416 3 . 1 4 1 5 9 2 . 6 5 ] , 5 8 997
Lcrng srrort
e
lonlJ e shorl: g lonq g short
eng
long eng bank
ffiE\ z\ -_-
el +
a --:-----;:a, + .__..__i . I
'"1F{'J
Z} X+
1.1416er000 3 . 1 4 1 5 9 26 5 3 5 8 9 79 e + 0 0 0 1
1A
-a.[h
--T a-*!1,€
4
el 5t I
24
MATLABFUNDAMENTALS
2.2.2 Arroys, Vectors qnd Mqtrices An arrat' is a collection of values that are representedby a single variable name. Onedincnsirnalarral't arc callcd tu'/ort and two-dinensional anavs are c'allednrutnce:. The scalarsusedin Section2.2.1 areactuallya matrix with one low andone column. Bracketsare usedto enterarraysin the comnranilmode.For exan.rple, a row vectorcan be assisnedas fbiiows: 123451
Note that this assignment overridesthe previousassignment of a : 4. practice, In row vectorsare rarely usedto solve rnathematical problerns.When we speakof vectors.we usuallyret-erto column vectors,which are more contmonlyused.A column vectorcan be enteredin severalways.Try them. or _-..>b-
|2;
1; 6; 8; I0
l
or. by transposing a row vectorwith the ' operator, >> Lr =
|
2, 4
6
I
10
I'
Tlre resultin all threecaseswill be b= 1. 4 b S 1t)
A rnatrixof valuescan be assignedas lblkrws:
r'23 456 789 In addition.the Enterkey (carriagereturn)can be usedto separirte the rows. For example. in the followingcase,the Enterkey woulclbe struckafterthe 3. the 6 andthe I to assignthe matrix: .->A..
lr23 456 7 8 9l
ffi qA1, zf :5yst LIZ
zt -+
4
"L x jl':-
a_J---*=
_r -{
vI
Z] X+ , -
A
<, lh -r
?l 5t I
2.2 ASSIGNMENT
25
At any point in a session,a list of all currentvariablescan be obtainedby enteringthe v;hocommand: > > rvho yariabl
Your
al
es
.rrc:
ans
b
x
or. with moredetail.cnterthc whos command: >;.whos f Jd r { i e
Sr ze
A
lx3
12
clouble
array
I x5
40
cloubLa
array
dns
LXL
l)
5xi '1 xl
x(comfiLex) Clrancl t-otal
is
Byte::
Class
tlouLr Lc: ;.Lrray
8
21
4Ct
il,trrb l e
i)i I'ay
I6
doubLe
arr.r.y
elenrrrnl.s; LrsinqJ 175
bytes
Note that subscriptnotationcan be usedto accessan individualelcmentof an array. For exarnple.the firurthelemento['thc columnvccfort) can be displayedas :>
b(4) .
,tlt si
8 F o r a n a r r a y , A ( n, rn ) s e l e c t s l h e c l e m c n t i n m t h r o w a n d t h c n t h c o l u r n n . F o r e x a m p l e , A(:,1) 6
Thsre are sevcralbuilt-in functionsthat can be used1ocreatcmatrices.For cxarnplc, the ones llnd zero:i l'unctionscreatcvcctol'sor matricesfilled with ones and zeros. respectively. Both havetwo argumcnts, thc first tbr the nulnberof rows and the secondtbr thc number-ofcolumns.For example.to creatca 2 x 3 matrixtl1'zeros: E :
7 , c I o s ; ( 2 ., - ) J
00 UU
Sintilarly, the or-resfLtnctioncan be used to create a row vector of oncs: u
= o n c : : ;( j ,
- l)
2.2.3 The Colon Operotor The colon opefatoris a powerfll tool fbr crertingand manipulatingarrays.If a colon is used to separatetw'o nunrbers,MATLAB generaleslhe n u m b e r sb c l w e e nl h e m u s i n ua n
-
ffial z$4 zl' -+
zf Ll 2
'|-A
I la-
;n-Zt*
5t
s
-i
Z] X+
L
vI
<, - [h Y
t('€ - ?'' 5l I
26
MATLABFUNDAMENTALS
l.J
incrementof one:
whi For
-:
l-
-
1.(
I
If colonsare usedto separatethreenumbers.MAILAB generatesthe numbersbetweenthe first and third numbersusing an incrementequal to the secondnumber: >> t = L:0.5:3
2.3 rnr
f-
1.0000
2.0000
1.5000
2.5000
ope
3.0000
com
Note that negativeincrementscan also be used >-" f
=
Ifn
i0:-1:5
t0
9
B
1
6
5
Aside from creatingseriesof numbers,the colon can also be usedas a wildcard to select the individualrows and columnsof a matrix.When a colon is usedin placeof a specific subscript,the colon reprcsentsthe entire row or column.For example,the secondrow of the matrix A can be selectedas in >> A(2,:)
We can also use the colon notation to selectivelyextract a seriesof elementsfrom within an array.For example,basedon the previousdefinition of the vector t :
Alsc
> > ' 1 -( 2 : 4 ) 981 Thus, the second through the fourth elements are returned.
Res 2.2.4 The rlnspace ond losspace Functions
plv
The iinspacre and logspace functionsprovideotherhandytoolsto generatevectorsof spacedpoints.The 1 inspace functiongenerates a row vectorofequally spacedpoints.It has the form
thes ing
I i n s p a c e (x l . x 2 , n ) which generatesn points betweenxl and x2. For example '> linspace(0,1,6)
Th AS
0
0.2000
0.4000
0.6000
o.BOOO
1.0000
lf the n is omitted,the function alromatically generates100points. The logspace l-unctiongenerates a row vectorthatis logarithmicallyequallyspaced. It has the form logspace(x1,
x2,
ffi= ?l -+
Ll 2. * J
). -t
n)
Zl eNst
"L x
ZI X+?
vI
-tlh
2-
3l
=$8
2.3 MATHEMATICAL OPERATIONS
27
which generatesn logarithmicallyequally spacedpoints betweendecadesl0'r and 10": For examp)e, >>
e (-I
logspac
0.1000
,2 , 4)
1.0000
10.0000
100.0000
If n is omitted,it automaticallygenerates50 points.
2.3
MATHEMATICAL OPERATIONS Operationswith scalarquantitiesare handledin a straightforwardmanner,similar to other computerlanguages.The common operators.in order of priority, are Exponenliolion Negotion M u l t i p l i c o t i o no n d d i v i s i o n L e f td i v i s i o n ' l A d d i t i o no n d s u b l r o c t i o n
These operators will work in calculator fashion. Try ,r
2*pt
6 .2,832 Also, scalar real variables can be included: >> Y - Pt/4; >> y ^ 2.45
0.5533 Results of calculations can be assigned to a variable, as in the next-to-last example, or simply displayed, as in the Iast example. As with other computer calculation, the priority order can be overridden with parentheses. For example, because exponentiation has higher priority then negation, the following result would be obtained: >> Y =
4 ^ 2
y-
_r6 Thus,4 is first squaredand then negated.Parentheses can be usedto overridethe priorities as in >> Y = \-4) L6 : Lefi division appliesto matrix algebra.It will be discussedin detail later in this book.
-----
zl'+
ffi= r\ a +
,"1lJ t
-
-=-:-T
*-6 -l
__..-..-..i, - "L x .l
5[ s\
-{
Z] X+
4
<, [h
6re %&
2a
MATTABFUNDAMENTAIS Calculations can alsoinvolvecomplexquantities. Herearesomeexamplesthatusethe valuesof x (2 + 4i) andv ( I 6) definedpreviously: '>,'3
* x
ans 5.0004 >>'1
+I2.0000i
/;<
0 . 1 0 1 10
0.2000r
^ ,2
>> x
ans -12.0000
+1r,.0000-r
>> x + y an!; lB.O0tJ0 + 4.00001 The real power of MATLAB is illustrated in its ability to carry out vector-rnatrix c a l c u l a t i o n sA . l t h o u g h w e w i l l d e s c r i b es u c h c a l c u l a t i o n si n d e t a i l i n C h a p . 8 . i t i s w o r t h introducing sorne of those manipulations here. The irtner product of two vectors (dot product) can be calculated using the * operator, :.:,d*b
r10 and likewise, the outer pnrluct >:, b * -
allS
2 4 6 8 10
4 8 12 16 2t)
b 12 18 24 l0
8 16 24 3'2 4L)
I0 20 30 4t) 50
To further illustrate vector-matrix mLrltiplication,llrst redefine a and l:,: .
,
ti
2
l;
and >> b =
f4 5 tjl';
Now. try >>
a
*
l0
A
t6
4?.
Z
ffir qAb zl' -&
LJt
ffi
Z uN x
A
i-
x+
<, - [h
z_! 1J'€- sJ 5t 3$&
29
2.3 MATHEMATICAL OPERATIONS
>>A*b 32 71 r22 Matrices cannot be multiplied if the inner dimensions are unequal. Here is what happens when the dimensions are not those required by the operations. Try >> A * a MATLAB
automatically displays the error message:
??? ETrar using =-> mtimes Inner matrix dimensi.ons must aQfree. Matrix-rnatrix multiplication is carried out in likewise fashion: >> A * A 30 66 102
35 B1 126
42 96 150
Mixcd operations with scalarsare also possible: >> A/pi ans = 0.3183 'I .2132 ?,.2,?,82
0.5366 I.5915 2,.\46\
0.9549 1.9099 2,.8648
We must always remember that MATLAB will apply the simple arithmetic operators in vector-matrix fashion if possible.At times, you will want to carry out calculations item by item in a matrix or vector. MATLAB provides for that too. For example,
30 66 r02
35 81 t2,6
4?, 9tr 1 50
resultsin matrix multiplicationof a with itself. What if you want to squareeachelementof e? That can be donewith >> anS
A
.^
2
=
r49 16 49
2a 36 64 81 ^ The . precedingthe operatorsignifiesthat the operationis to be carriedout elementby element.The MATLAB manualcalls thesearray operatlons.They are also often referred Loas element-hy-cIemen t t tparutit tn.s.
ffi z\4 ,"I LJ' -2.(-h
Zl- gt st -l
z_|lh
"L 4
<, lh x _-T - 9f 5l ,t\€ -l-
I
30
MATLABFUNDAMENTALS MATLAB containsa helpful shortcutfor performingcalculationsthat you've already done.Pressthe up-arrowkey. You shouldget back the last line you typed in. >> A
.^
2
PressingEnter will perform the calculation again. But you can also edit this line. For example,changeit to the line below and then pressEnter. >> A
.". 3
1821 64 I25 343 5r2
216 129
Using the up-arrow key, you can go back to any command that you entered. Press the uparrow until you get back the line F * -
Alternatively,you can type b and pressthe up-arrow once and it will automaticallybring up the last commandbeginningwith the letter b. The up-arrow shortcutis a quick way to fix errorswithouthavingto retypethe entireline.
2.4
USEOF BUIIT.INFUNCTIONS MATLAB and its Toolboxeshavea rich collectionof built-infunctions.You can useonline help to find out nroreaboutthem. For example.if you want to learnaboutthe 1og function, type rn >> helP LOG
log' lrlatural
LOG(X)
is
Complex See
also
logarithm. the
nat-ural
results LOG2,
lograrithm produced
are
LOG10,
EXP,
if
of X
the is
elements
not
of
X.
posrtive.
LOGM.
For a list of all the elementaryfunctions,type '>
help
elfun
One of their importantpropertiesof MATLAB's built-in functionsis that they will operatedirectly on vectorand r.natrixquantities.For example,try >> log (A)
0 1.3863 1.9459
0.6931 1.6094 2.0194
1.0986 r.79'LB 2.1912
and you will see that the natural logarithm function is applied in array style, element by elenrent,to the matrix A. Most functions. such as sqr t, abs, sin, acos, f anh, and exp, operate in anay fashion. Certain functions, such as exponential and square root, have matrix definitions also. MATLAB will evaluate the matrix version when the letter m is appended to
'\--
zf+
ffi= qA+,
,"1{z
a eys\ -1
-l
Z] X+
"L 4
<'- [h Y
ll'€
- eJ 5t &
FUNCTIONS 2,4 USEOF BUILT-IN
3l
the function name.Try >> sqr:tm(A)
A.4498 + 0.1623i 1.0185 + O.AB42i I.5813 - 0.594A1
0.5526 + 0.2068i 1.2515 + 0.A22Bi 1.9503 - 0.1611i
0.5555 I.4844 2.3134
0.3481l 0.0385i + 0.21I7i -
There are several functions for rounding. For example, suppose that we enter a vector: >> E =
-1.5
l-1.6
-I.4
I.4
1.5
I.6l;
The round function rounds the elements of E to the nearest integers: >> round(E) -2-2-1r22 (short fbr ceiling) function rounds to the nearestintegerstoward infinity:
The ceil
>> ceil
(E)
1
-1
2
2
2
function rounds down to the nearestintegers toward minus infinity:
The f loor >>
1
f loor
( F t)
22
2rl
There are also functionsthat perform specialactionson the elementsof matricesand arrays.For example,the sumfunctionreturnsthe sum of the elements: >> F = l3 5 4 6 Il; >> sum(F)
19 In a similar way, it should be pretty obvious what's happeningwith the following commands: >> min(F),max(F),mean(F),prod(F),sort
(F)
1
6
3.8000
360
13456
ffi
Z} x --T<, lh ,f\€ - 9f >l 4 -l-
&
32
MATTABFUNDAMENTALS A common useof functionsis to evaluatea fornrula for a seriesof arguments.Recall that the velocityof a free-fallingbungeejumper canbe computedwith [Eq. (1.9)]: lom
/
loe,
M, gft
\
, : , / a t a n h,l/ r - r I V c.r \V trt / due to gravity(9.81 m/s:),n is mass(kg), where u is velocity(m/s),g is the acceleration c,,is the drag coefficient(kg/m), and r is time (s). Createa columnvectort that containsvaluesfrom 0 to 20 in stepsof 2: ;'> l- =
2.5 G
Th bo
[0:2:20]'
t.0 2 4 6 B l0 t2 l4 15 1B 20 Check the number of items in the t array with the lerrgLh function:
Yor
>> I cnqtlr (L ) =
allS
1I
Assignvaluesto the parameters: =
>:.9
9.81;
in -
68.1;
crl
=
0.25;
M A T L A B a l l o w sy o u t o e v a l u a t ea f o r m u l as u c ha s r : . 1 ( r ) , w h e r et h e f b r r n u l ai s computedfbr each value of tl.rer array,and the result is assignedto a correspondingposition in the u array.For our case, -/ :
s q L t - ( g * n / c ' d ) * 1 - a n h ( s q r t ( g * c c l/ n ) * t
) i
( (
t) 1B.'t292 33.r118 42 .01 62 46 .951 a 49 .42r4 50.6175 51.1871 51.4560 51.5823 br.6416
I
-
ffir zl Llz
zl'+
.r1
-\
7f 5[ st -{
L
-l
Z] X+?
vI
x
<, - [h
j_f€ - sJ 5t
33
2.5 GRAPHICS
2.5 GRAPHICS createa MMLAB allows graphsto be createdquickly and conveniently.For example,to enter above, data the graphof the t and v arraysfrom (t,
:,., nlol
./)
lne gfapn appears ln tne grapnlcs wrnoow ano can De pnnteo or translerreo vra tne clrpboard to other programs.
60 50 40 30 20 10 0
8
1214
10
16
18
You can customizethe grapha bit with commandssuchas the following: >>
til,re('Plot of ('Valucs
:':.' xlabel
, -> y l . L b c l
('Valucs
v
ver':jus
of
L')
ol
v')
t')
.:'> qrid
Plot of u versus f
60 50
a
: .' l
t
'!ry
l
l
i
40
I l
l
I
o
: 30
6
20
i
:
j
i
I
!
I,
i
.-..--- t. "- -
:
10
"-t--* l
I i
l
0
?,r'
ffi z\4+ ,"lqfjeTl
I
8 10 12 Valuesof t
14
16
18
a-* z,J ^:, x+
<, lh --4 l,f.€- ef >l
-l-
zl
3$&
34
MATLABFUNDAMENTALS TAB E 2.:2 Specifiers for colors,symbols,ond linetypes. Colors Bl,re Green Red Cyon A^agenio Yello'"v Blccl.
Symbols
Line Types
Poinl
Solid Dolied Dcshdot Doshed
Circle 'MO TK
Pu l s Slcr Squore Diomond T r i o n ge ( d o v r n ) Triongle{up) Triongie(leftJ T ri o n g l e { r i g h t ) Penlogronr nexogfcrn
m v k
The 1rlot- conlmandclisplays a solidline by default.If you wantto plot eachpoint with a symbol' you can includea specifierenclosedin singlequotesin the pior functio.. Table2'2 Iiststhe availablespecifiers. For example,if you want to useopencirclesenter r-:.l:jlot
(t,
v,
,o,)
MATLAB allowsy.ruto displaymorethanonedataset on the sameprot.For example, if you want [o connecteachclatarnarkerwirh a straight rineyou courdrype :'i.I,1ol,
(L,
yt
t,
\r,
,o')
It shouldbe Inentionedthat,by clefault,previousplots areerasedeverytirnethe plor commandis implernented. The hol d c,nconrmancl hoidsthecurrcntplot anclall axis propertiesso that additionalgraphingcommancls can be addedto the existingplot. The h.tit of f cornmandreturnsto the def'aultmocre.Fclr exampre,if we had typed the tbilowing cornmands, the flnal plot woukl only displaysyrnbols; :''
pIoL
>> plot
(1,
.r)
(t,
v,
,o,)
ln contrast,the f
-
ploL
(t, lLoLd on (t, Plot-
-,.) v,
'o')
::-, ho1c1 off
In additionto ho1.l, anotherhanrlyfunctionis subplor, which alJowsyou lo split the graphwindow into subwindows()rp(ute,r.It has the svntax subploL (n, n, pJ This cornrnandbreaksthe graphwindow into an m-by-nmatrixof small axes,and selects the p-th axesfor the currentplot.
-
ffi aAI
=,:st --7i 5[ 7l Lt r J-l J'-l
zt'-+
Z-} X+
z-t\ -x v!
a. - [h
rf € - 1 5t
2.5 GRAPHICS
35
We can demonstratesubplot by examiningMATLAB's capabilityto generatethreedimensionalplots. The simplest manifestationof this capability is the p1or3 command which has the syntax p1ot3 \x, y,
z)
wherex, y, and z arethreevectorsof the samelength.The resultis a line in three-dimensional spacethroughthe pointswhosecoordinatesarethe elementsof x, y, anrJz. Plotting a helix providesa nice exampleto illustrateits utility. First, let's grapha circle with the two-dimensional plot function using the parametricrepresentation: -r: sin(t) cos(r). and.y We employ the subplot commandso we can subsequently add the three-dimensionalplot. >> l- = 0:pi/50:10*pi; > > s u b p l o r -( 1 , 2 , I ) ; p 1 o t ( s j n ( L ) , c o s ( r ) ) >> axis squaLe >> r.irle (' (a) ') As in Fig. 2.la,the result is a circle. Note that the circle would have been distorted if we had not used the axis sou.er e command.
F I G U R E2 . I A twoponeplotof (o)o two-dimensionol circleond (b)o three-dimensionoJ helix (b)
ffi=b
z['+
,"I LJ'
5t(st 1t -l
Z] X+
"L Y vI
<, lh
_-T j} € - eJ 5t 3$8
36
MATLABFUNDAMENTALS Now, let's add the helix to the graph'sright pane.To do this, we againemploy a parametl'icrepresentation: x: sin(/),.)': cos(1),andu : / ' > s u b p L o t (- I , 2 , 2 ) ; p 1 o t 3 ( s i n ( L ) , c o s ( 1 . ) , L ) ; ,) >> rirle('(b) The resultis shownin Fig. 2.l b. Can you visualizewhat'sgoingon?As time evolves. the .r and .y coordinatessketchout the circumferenceof the circle in the r-r' plane in the plot. However.simultaneously. the curverisesvertisamefashionas the two-dimensional cally as the : coordinateincreaseslinearlywith time. The net resultis the characteristic springor spiralstaircase shapeof the helix. of graphicsthat are useful-fbr example.plotting obiects Thele ale other f'eatr.rres insteadof lines,familiesol curvesplots,plottingon the complexplane,log-logor semilog plots,three-dimensional meshplots,and contourplots.As describednext,a varietyof resolll.ces areavailableto learnabouttheseas well as otherMATL,AB capabilities.
2.6
OTHERRESOURCES ol'MATLAB that we will be using The fbregoingwas designedto lbcus on thosef'eatures overviewof all in the remainderof this book.As such.it is obviouslynot a comprehensive in learningrnore,you shouldconsultone If you are interested o1'MATLAB's capabilities. of the excellentbooksdevotedto MAILAB (e.g..Palrn.2005;Hanselmanand Littlefield. 2 0 0 5 :a n dM o o r e . 2 0 0 7 ) . by an extensiveHelp lacility thatcan be accessed Further,the packageitsell'includes clicking on the Help nrenuin the comnrandwindow.Thi.swill provideyou with a nutnber of ditl'erentoptionsfbr exploringand searchingthloughMATLAB's Help material.In addition, it prclvidesaccessto a numberof instructivedetnos. As describedin this chiLpter. help is also availablein interactivernodeby typing the lrclp conrmandfbllowedby the narneo1'acommandor function. If you do not know the narne,you can use the lookfor commandto searchthe of text.Fol example,supposethatyou wantto find all MATLAB Help files lbr occurrences you couldenter and functionsthat relateto logarithrns, the cornrnands l-,kt
,r
..,'t
irhn
anclMATLAB will displayall ret'erences that includethe word loq.rri t-hnr. Finally.you can obtainhelp tionr The MathWorks.lnc., websiteat www.mathworks books,and technical .corn.Thereyou will l'indlinks to productinfbrmation,newsgroups, uselitl well ofother resources. supportas as a variety
ffiar --=-al'
z[ Ll 2
zl'+
L1
-l_t
r
5y 5t
-r
zl ,x+
"L x 4
a, - [h
1.|€- 1 5t =$8
37
2.7 CASESTUDY
EXPLORATORY DATAANALYSIS Your textbooks are filled with formulas developed in the past by Bockground. renowned scientistsand engineers.Although theseare of great utility, engineersand scientists often must supplementtheserelationshipsby collectingand analyzingtheir own data. Sometimesthis leads to a new forrnula. However, prior to arriving at a final predictive equation, we usually "play" with the data by per{ormingcalculationsand developingplots. ln most cases,our intent is to gain insight into the pattemsand mechanismshiddenin the data. In this casestudy,we will illustratehow MAILAB facilitatessuch exploratorydata analysis.We will do this by estimatingthe drag coefficientof a free-falling human based on Eq. (2.1) and the data from Table 2. 1. However, beyond merely computing the drag coefficient, we will use MATLAB's graphicalcapabilitiesto discernpatternsin the data. canbe enteredas The datafrom Table2.1 alongwith gravitationalacceleration
Solution.
, 9)..9 (tir..l 8t:1.91; tr-jrr1.o f,0.ll 7.1.l 91.1 'rl..ll; :;'rrt::l!-1 !,5.i !,1 4l.l .4 4|J.1 !0.9 ,--. r; 9.fl1;
The drag coefficientscan then be computedwith Eq. (2. l). Becausewe are performing operationson vectors,we must include periodsprior to the operators: element-by-element
(r. '.1, r
1)..tt,il')
l).')t) lr)
We can now use someof MATLAB's built-in functionsto ceneratesome statisticsfor the results: - . . - 1 , J , 1 r 7 q = 1 1 r ( ' . (l (a i l l ';1,1 ,. i Li,-.1
l i , i i r = t i ri r r (
.l r
, n , . ,r ,
, , 1 , 'q1 ( r ' r l )
l) . .l llr;4 r l c l r r ri r r
-
i)
r ' c l n t . r: . 0. J12!
Thus, the averagevalue is 0.2854 with a rangettom 0.25l l to 0.3125kg/m. Now, let's start to play with this data by using Eq. (2 l) to makea predictionof the teuninal velocity basedon the averagedrag: ',..
v l ) ] , . ' a l . , : l a l r L ( g * r l t r t r , l , r ' " ' ). 1
Vpl t r:Ci 'rl-t,(lL'r '.1),.i
1i,..iii,)/
4'1. /!.1l'-r'r.ll1r',tr:,
] ,) f "
Notice that we do not have to use periodsprior to the operatorsin this formula? Do you understandwhy? We can plot thesevaluesversusthe actualmeasuredterminal velocities.We will also superimposea line indicating exact predictions(the l;1 line) to help assessthe results.
ffi= qA+, z(r Ll 2
,e[r +
tr4
-\
e[ st 2-{
Z] X+?
x
-z [h
/_l€
- eJ 5t ?$
I
38
MATLABFUNDAMENTALS
Plot of predictedversusmeasuredterminal velocities
3ss €0)
o-' 50 45L
50
49
48
56
54
53
52 51 Measured
Plot of drag coefficientversusmass
0.35 o)F o<.
o
q)
t!
U.J
o
o
o.Y
c,)
O
oo r!o
0 . 2| 60
65
70
85
80 75 Mass (kg)
95
PROBTE
FIGURE2.2 withMATIAB Twoololscreoted
2.1 A simple electricc pacitor, and an inductor on the capacitor4(l) as as
Becausewe are going to eventually generatea secondplot, we employ the subplot command: ' ' : : u l . , y r l o L( 2 , ) , 1 ) ; p l o t x I a b , - l ( ' l r , : , , t : ; L r r i : c l)'
( " / t - , v p r ( . 1 ( ,l ' . ) ' , ' , ' 1 . , v l - )
q(t):
.,
('Irr i.:rlic lL-'tl') 'l'l ol ol I i t lc] ( I)t ('(l i ( t..crd
q t ) g - R t / ( 2s1
I i.rlrt'I
Lrurar(l l,
wherer:time,qo:tl L: inductance,and ( generatea plot of this fi
As in the top plot of Fig.2.2,becausethe predictionsgenerallyfollow the l: I line, you might initially concludethat the averagedrag coefficientyields decentresults.However, notice how the model tends to underpredictthe low velocitiesand overpredictthe high. This suggeststhat ratherthanbeing constant,theremight be a trendin the dragcoefficients. This can be seenby plotting the estimateddrag coefficientsversusmass: x l ; r i r e I ( ' n r , r : r f , r( k q r ' t ' , ' l , i ' l : rl, ( ' , - ' : : 1 i t n a l ' t , r l L l r , t g r : t , [ ' l ' I i c t l l r l ( ] i q / 1 l l) L i 1 . L e 1 ' ; ; L i r l - O I c 1 L . i . - l] r o l r i I i . i c t l L \ " r r : . i 1 1 l rf li l , l
4o:12,R:50'L-j 2.2 The standard norrr bell-shapedcurve that c
,)
Use MATLAB to ger z: -4 to 4. Labelth scissaasz.
The resulting plot, which is the bottom graph in Fig. 2.2' suggeststhat rather than being constant, the drag coefficient seems to be increasing as the mass of the jumper
ffi= llAb zl'+
: :
"L x
Z] X+?
vI
<, - [h
rtE-:l
5 I
39
PROBLEMS
continued increases.Basedon this result,you might concludethat your model needsto be improved. At the least,it rnight motivateyou to conductfurther experimentswith a larger numberof jumpers to confirm your preliminaryfinding. In addition, the result might also stimulateyou to go to the fluid mechanicsliteratureand leam more aboutthe scienceof drag.If you did this. you would discoverthat the parameter c.,is actually a lumped drag coefficient that along with the true drag includes other factors such as thejumper's frontal areaand air density: CoPA
(2.2)
where C, : a dimensionlessdrag coefficient,p : tir density (kg/mr), and A : frontal area (m2), which is the areaprojectedon a plane normal to the direction of the velocity. Assuming that the densitieswere relatively constantduring data collection (a pretty good assumptionif thejumpersall took off from the sameheighton the sameday),F,q.(2.2) suggeststhat heavierjumpersmight have largerareas.This hypothesiscould be substantiatedby rneasuringthe frontal areasof individualsof varying masses.
PROBTEMS 2.1A simplcelcctlic circuit consistingol' a rcsistur,a capacitor, andan induc(oris dcpictcdin Fig. P2.l. T'hcchargc onthecapacitor ri(t) as a lirnctionol'limc can bc cor.nputcd as
2 . f U s c t h c I i r : r p , r c e l ' u n c t i o nt o c r c a t c v c c t o r s i d c n t i c a l t o t h c l i r l l o w i n g c r c a t c c lw i t h c o k l n n o l a t i o n : (a) t : !:{r:-JlJ (b) r:
t,:,.1
2.4 Usc colon notution lo crcatc vectors idcntical to the l i r l l o w i n g c r c u t c dw i t h t h c I i n : , [ ) , ] .r : l i r n c t i o n : ( a ) r ' : I i r s l ) . r c ' { Ir l , I , t t ( b ) | : 1 r n s l r . r c e ( r ,0 , I I )
(#)' ,]
whcre I = time,r7o: thc initial chargc.1l : the rcsistancc, I = inductance, and C : capacitancc.Usc MATLAB to generate r plot o1'thislirnctionliorn / - 0 Io 0.7, givcn that q o - 1 2R . : 5 0 ,L : 5 , a n d C : l 0 r . 2.2Thestandardnorrnalprobabilitydensitylirnction is a bell-shaped curvethat can be rcprescntcdas
2 . 5 I l ' a l i r r c c I - ( N ) i s a p p l i c d t o c o n r p r c s sa s p r i n g , i t s d i s placcmcnt,r (rn) can ol'tcn bc rnodclcd by Hookc's law: Ij:lr
Inductor It-\-
-
/
-,
l-
:-'-
\ ! / / ; - .
\/ llt
Resistor UseMATLAB to gencrate a plot ol' this fbnction liom ;=-,1 to.1. Labcl thc ordinateas lrcquencyand the abscissa as:.
F I G U R EP 2 . I
ffir qA+, zt'+
,"1{t
6 e[st -l
-l
Z] X+
x
<, - [h
1-l€- eJ 5t 8,
40
P
MATLABFUNDAMENTALS
where k : the spring constant(N/m). The potentialenergy storedin the spring U (J) can then be computedas l-
U:
-(.r2
Five springsare testcdand the following data compiled:
F,N
ll
r,rr
0013
)2 0020
l5
9
0C09
0010
2.13 You conractthe jur Table 2.1 and measure values, which are order conespondingvaluesin T
2.8 lt is general practice in enginceringand sciencc that equationsbe plotted as lines and discretedata as syrnbols. Here is somcdatafbr conccntration({) \'elsustimc (l) for the photodegradation of aqueousbromine:
t, min c, PPm
t0 414
2A 26
3C t6
40 t3
50 i0
A,m2 o45A O4OlO
60 c5
t2 0012
(a) If the air density is p compute values of tl cD. (b) Determine the averag resultingvalues. (c) Develop a side-by-si C, versusrr (right sic and titles on the plots 2.14 The following para that contractsexponentia
This datacan bc dcscribedby the lbllowing function: LlseMATLAB to storeF and,t as vectorsand thcn conrputc vcctorsof the spring constantsand the potentialenergics. Use thc max l-unctionto dcterminethe maxirnumpotential cncruy. 2.6 The densityoi treshwater canhe computedasa lunction ol'tcmperaturc with the firllowinecubiccquation: p:
- 5 . - 5 2 8x9 l 0 - 8 f ( l - 8 . - 5 0 1 x6 t o ' - 6 7 < ? s i 6.5622x l0 11 + 0.99987
(._ 4.g4.,.00r.1/ Usc MATLAB to crci\tc a plot displayirrgboth the data (r-rsingsquarcsymbols)and the function (using a dotted l i n e ) .P l o tt h c l u n c t i o nl b r r : 0 t o 7 5 r n i n . 2.9 The senii 1o9i, lunctiont)pcratcs in an idcnticalfashion to thc frlot, lirnction cxccpt that a logarithrnic(basc-10) scale is used lirr thc l axis. Usc this l'unctionto plot the data ancl l'unctionas dcscribcdirr Prob. 2.t1.Explain thc result s. 2.1() Hcrc is sorncwirrd tunncl data lirr lirrcc (I-) vcrsus v e l o c i t y( r ' ) :
- srnI nt , cosl .v=e .r:c
('C). Usc whcre p : dcnsity (g/cnr31and 16' : tcmperat.urc MATLAB to gencratea vcctor of tcrnpcraturesrangingliorn 32 "F to 82.-1'F usingincrcrncnts of 3.6 oF.Convcltthrsvcctor tclclcgrecs Celsiusanclthe'ncomputca vcclorof dcrrsitics -i"1.. u , m l s 1 0 ? . 0 3 0 4A 50 60 7A B0 hascdon the cubic lbrmula. Crcatea plot ol'p vcrsus 't5 7A 3BO N t 2 2 A 8 3 0 1 450 I t , 5 5 0 6 r 0 : 519(7'F 32). Rccallthat T( 2.7 Manning'scquationcanbc uscdto compulcthc vclocity 01'wirtcrin a rectanglllar opcn channcl: This clatacan bc describcdhy thc lirllowingl'ur.rction: BH \t' /s/ . I" : 0.2711ttt')x+t rt \B+2H/
oo0to 0 0008 0 0c103
t5 3 '25
l9 '24 t5
Storethesevaluesin a matrix whereeachrow reprcscntsone oi'thc channelsandcachcolumn represents one of the paranrctcrs.Writc a single-lineMATLAB statcmentto conputc a column vectorcontainingthe vclocitiesbasedon the virlues in the parameter matrix.
.r5 .r'l . Sln,I=-t-+ .r! 51
,r7 71
(b) rt
q = 4:2:L0; - 17 B 4; 3 >> sum(q) * r(2, >> r
Use MATLAB to creatca plot o1'thcsine(solid linc) along r.vitha plot o1'thcscriescxpansion(dashedline) up ttr andincluding the terni -r7171.Usc the Lruilt-intirnctionf .rct1)r ial in conrputingthe scriesexpansion.Make the rangeof the abscissali'onr,r : O to 3tr 12.
a. th
4-t
-----j
zl 4+
7l Llz *2f
z['+*
-r
"
5[st ::-
;:,
.l
Z] X+?
jlg-
.
>> x ^ 3; >>Y=B-x
.r'') +-9l
+fi6-qrlal
l,
Use subplor to genera (.r, .y) in the top pane and (x,.v,l) in the bottompane 2.15 Exactly what will t MATLAB commandsare I (a)>>x=z;
whercU: r,clocity(nr/s),S: channelslopc,n: roughncss LJsr'MATLAB to crcatc a plot displayingboth thc data (nr),and /1 : dcpth(rn).Thc lirllo',v- (usingdiantondsymbols)and thc llnction (usirrga dottcd cocfl'icient, Ii: yu;.1,1.r ing datais availablclbr I'ivechannels: l i n e ) .P l o tt h c l u n c t i o nl i r r u : 0 t o 9 ( )n r / s . 2.ll Thc loc_{locr lirnclionopcratcsin an idcnticalI'ashion to the lirnction cxccpt(hatlogarithnricscalcsarc uscd fil()'. n5BH lirr both thc,r ancl.r'axcs.Usc this lunctionto plot thc dau 0 035 0 0001 t0 2 and lunctionas dcsclihcdin Proh.2. I0. Explaintlrc rcsults. B I 0.020 0 0002 2.12 Thc Maclaurinscricscxpansiontirr thc sinc is
00r5 0 030 a a22
-ll
1 5t I
4l
PROBTEMS 2.13You contactthe jumpers used to generatethe data in Table2.1 and measure their frontal areas.The resulting values,which are ordered in thc same sequenceas the conesponding valuc-sin Table 2. l, are
2.16 The trajecbry of an object can be modeledas R. ):(tando).\--r'*)t, IU;COs.d0
where _v: height (m), d,, : initial angle (radians),r : horizontal distance (m), g = gravitationalacceleration ( : 9 . 8 1 m / s r ) ,u , , : i n i t i a lv e l o c i t y( m / s ) ,a n d y , , : i n i t i a l find the displacementfbr r',, : Q (a) Il the air densityis p : 1.225kg/mr, use MATLAB to height. Use MATLAB to r,,:30 nr/slbr initial angle-s rangingliom l5 to 75o in and drag coefficicrrt computevaluesof the dimensionless increrncnts of 1,5".Employ a rangeof horizontaldistances cD. from -t : 0 to 100m in incremcnts of 5 m. Thc resultsshould (b) Deternrine the average,minimunrand rnaxirlumol'thc in an array where thc first dimension(rows) be assernblcd resulting values. distances.and the seconddimension (c) Dcvelopa side-by-side plot ofA versusrr (lcft sidc)and correspondsto the (columns)correspondsto the difl'erentinitial angles.Use C.,versusrr (right side).lncludedescriptivcaxis liibels this matrix to gcncratea singlc plot of thc hcightsversus andtitlcson the plots. horizontaldistanceslbr eachof the initial angles.E,rnploya 2.11The lbllowing paramctricequllirrrrsgenerrlc-l hclix among the dill-crentcases.and scale legenclto di.stinguish r'xponentially as it cvolvcs thatcontriicts thc plot so that thc minirnumheightis zero usingthe azis .r:e ol'sinI commancl. \'=c CosI dcpendence ol'chcmicalreactionscan 2.17 The lcmperature wilh theArrhaniu.tt'tlutrtittn; bc ctrrnprrtetl
A,nt2 0.454 0 401 0 453 0 485 0 532 a 474 A 4B6
Usesubpl,rr to gcneratca two-dimcnsionallinc plot ot' (-r,i') in the top pane and a three-dirncnsional line plot ol' (,r,r, :) in thc bottom panc. 2.15Exactlywhal will be displaycd alier thc lirllowing MATLABcourmandsarc typcd'l -
2;
-
lt
q
.
. 1: 2 : 1 0 ;
>> r
.
f]
(a) -'> x :':' X' >-../ (b)t-
l; x
I
> > : i L l l r ( c 1 )*
zt +
4; r(2,
)
6
2.);
3)
,jF-\
k - Aa t'itRI,,1 (or tiewherck: rcactionratc(s r).A : thc prccxponential quency)lactor,E: activationenergy(J/mol),R - gasconstant [8.314J/(rnolc . K)], and l, : absolutetemperaturc ( K ) . A c o r n p o u n d h a s El :x l O s J / m oaln d A : 7 x 1 0 1 6 . Usc MATLAB to gcncratevalucsol reactionratcs1brtemperalurr-s rangin-u lrom 273 to 333 K. Usc subplot to gc'ngraphol'(a) /
ffi= z\4+ zl
: F+
"L x z,_!
<, lh
u=4 3$8
43
3,I M-FITES
We also learnedthat a numericalsolutionof this eouationcould be obtainedwith Euler's method: Ui+l: ui +
dut dt
Lt
This equation can be implementedrepeatedlyto compute velocity as a function of time. However, to obtain good accuracy,many small stepsmust be taken.This would be extremelylaboriousand time consumingto implementby hand.However,with the aid of MATLAB, suchcalculationscan be performedeasily. So our problem now is to figure out how to do this. This chapterwill introduceyou to how MATLAB M-files can be usedto obtainsuchsolutions.
3. I
M-FIEs The most common way to operateMATLAB is by enteringcommandsone at a time in the command window. M-files provide an alternative way of performing operationsthat greatly expand MATLAB's problem-solvingcapabilities.An M-file containsa seriesof statementsthat can be run all at once.Note that the nomenclature"M-flle" comesliom the fact that suchfiles are storedwith a .m extension.M-files comein lwo flavors:scrintfiles and functionfiles.
3.l. t ScriptFiles A scriptfile is merely a seriesof MMLAB comnrandsthat are savedon a file. They are useful for retaininga seriesof commandsthat you want to executeon more than one occasion.The scriptcan be executedby typing the file name in the commandwindow or by invokingthe nrenuselections in the edit window: Debug,Run.
EXAMPTE 3.I
S c r i p tF i l e Problem Stotement. Developa script lile to computethe velocity of the lree-falling bungeejumper. Solution. Openthe editorwith the menuselection:File,New, M-file.Type in the fbllowing statenrents to computethe velocityof the fiee-tallingbungeejunrperat a specifictime ( 1 . 9 )]: [recalE l q. 9 = 9.81; m = 68.1; L - 12; cd = 0.25; v = sqrt(g * m / cd) * tanh(sqrt-(g * cd / m) * L) Savethe file as scriptdemo . rn.Returnto the commandwindow and typc >>scriptdemo
The result will be displayedas 50.6175 just as if you had typedeachof its linesin the comntandwinclow. Thus,the scriptexecutes
ffi ---_
--At zf Lt. t -{ -l
zl'+
5[ st ^l
zf x+
L
zJ
x
<, - [h
lj€-
eJ 5 8,
PROGRAMMING WITHMATLAB
44
As a final step,determinethe value of s by typing >>g
9. BlOO So you can seethat even though q was definedwithin the script.it retainsits value back in the commandworkspace. As we will seein the following section,this is an importantdist i n c t i o nb e t w e e ns c r i p t sa u d f u n c t i o n s .
3.1.2 FunctionFiles FunctionJilesareM-files thatstartwith thewclrdf unction. In contrastto scriptfiles,they can acceptinput argumentsand returnoutputs.Hencethey are analogousto user-deflned functionsin programminglanguages suchas Fortran,VisualBasicor C. generallyas The syntaxfor the functionfile can be represe'nted funcLion
outr'.rr-
=
funclanlc
( . : r - g l Ji ^ . - - t)
Z he)pconme.nLs s L.l t emell L.s auLva r- - v'al ue;
where outvar'= the name of the output variable,furcr.rme = the function's name, valuesthat are passedinto argt)st = the function'sargumentlist (i.e.,comma-delirnited the function.l,?reTpcornnerts= tcxt thatprovidestlreuserwith infbrmationregardingthe function(lhesecan be invokedby typing t1elp f uncnane in the cornmandwindow),and thitt is assignedto ourvar. thatcomputethe r,.r-/uer srare.mL-lLs= MATLAB statements Beyondits role in describingthe function,the first line ol'the he/pcorrurrcnts,called by the lookf or command(recallSec.2.6).Thus, the Hl line, is the line that is searched you shoulclincludekey descriptivewclrdsrelatedto the file on this line. The functioncan thenbe run by typing The M-flle shouldbe savedas furrcnarne.m. f uncnamein thecommandwindow asillustratedin the fbllowingexarnple.Note thateven your computer'soperatingsystem may not be. though MATLAB is case-sensitivc. W h e r e a sM A T L A B w o u l dt r e a tf u n c t i o nn a r n e sl i k e f r : e el a l l v e i a n d F ' r e c t r a l l v e l a s two diff'erentvariables,your operatingsystemmight not.
EXAMPLE 3.2
r.l
r u n c l t o nr i l e ProblemStotement. As in Example3. l, compufethe velocityof the free-fallingbungee jumper,but now we will usea functiorrfile for the task. Solution. Type the following statements in the file eclitor: f rrnction
v
=
% f reef al1vel:
f reef aLLvel- (L, m, bungee ',-elocitiz
cd) rviLh
% v=freefallrrel(L,m,cd)
conputes
Z
c:f an
sjecond-order
the
objecL
frc-e-faL1 wiLh
drag velocity
second*order
drag
input:
l-ir_,
=
ffi --l
zl 4+ zt\
?JLJ' zl'+
!t'i 5t
-jy -l
^l
af xh
"L Y
a. [h -r
VT jjg-:l
5t
3$8
45
3.I M-FILES % r = rime (s) (ks) m=mass % 2 cd = second-order drag coef ficrent % output: (m/s ) % v = dovrnward velocity 9 = 9.81; v = sert(g
of ? acceleration * m / cd)*tanh(sqrt(g
Save the file as f reef al lvel and type in >> freefallvel
gravity * cd,/
(kSi m)
m) * t);
. m. To invoke the function, return to the command window
\12, 68. 1, 0.25)
The result will be displayed as
50.6i75 One advantage of a firnction M-file is that it can be invoked repeatedly for diff'erent argument values. Suppose that you wanted to compute the velocity of a 100-kg jumper after 8 s: (8, 100, 0.2t)
>> freefallvel
53.187B To invoke the help comments type >;. hel.p freefallveL which results in the comments being displayed freef;r1lve1: bungee velocity with second-order drag .,'=freefallvel(t,m,ccl) compuLes the free fall veloclty of an object with second-order i nput- : t = time (s) m = nass (kS) (kglm) cd = second-order drag coefficient ourpu[: (m/s) v = downward velocrty
dr:ag
If at a later date, you tbrgot the narne of thjs functiorr, but remembered that it involved bungeejumping, you could enter >> lookfor-bungee and the following information would be displayed f r e e f a l .1 . m :
% f reef al1:
bunqee velocity
with
second-order
clr:agt
Note that, at the end of the previous example, if we had typed >>g
ffi= z$4+
zr+
ztf-ai3'" -\
= "L vI
3l F+
<,- - lh Y
jtE-
?l 5t I
WITHMATLAB PROGRAMMING
46
the following messagewould have beendisplayed or variable
??? Undefined function
'S' .
So even thoughs had a value of 9.81 within the M-file, it would not have a value in the command workspace.As noted previously at the end of Example 3.1, this is an important distinction betweenfunctions and scripts.The variableswithin a function are said to be local and are erasedafter the function is executed.In contrast,the variablesin a script retain their existenceafter the script is executed. Function M-files can retum more than one l'esult.In suchcases.the variablescontaining the resultsare comma-delimitedand enclosedin brackets.For example,the following function,staf s.m, computesthe meanand the standarddeviirtionof a vector: Imean, stdev] = stats (x)
function
n = length(x); mean = sum(x) /n; stdev .. sqr t (sum( (x-mean) .^2/ (n-1) ) ) ; Here is an example of how it can be applied: '>
!' -
lB 5 1D 12' 6 1.5
3.2
4l;
m= 7.5000 S=
2 .873'1 BecausescriptM-files have limited utility. functionM-flles will be our prirnaryprogramming tool fbr the remainderof this book. Hence,we will ofien refer to function M-files as simply M-flles.
3.1.3 Subfunctions M-files, Functionscan call otherfirnctions.Althoughsuchfunctionscan exist as separate they rnay also be containedin a singleM-file. For example.the M-file in Example3.2 (without conrments)could have been split into two functionsand saved as a single M-file': rr = f reef al Lsubf unc (L, f unction = veL (t, m, cd);
n r,
ccl)
v
end function n
a
r . r t l
9
v
=
(t,
= vel
v
m,
cd)
al.
sqrt
*
(q
m ,z cd) *tanh(sqrt
(g
*
cd
/
m)
*
t);
end
I Note that althoughend statementsare t.rptionalin single-functionM-lilcs, we like to includethem whcn s u b f u n c t i o nasr e i n v o l v c dt o h i s h l i g h tt h e b o u n d a r i ebs e t w c e nt h c m a i n f u n c t i o na n d t h e s u b f u n c t i o n s .
ffi qA+, zt Ll 2
zl'+
LA
jt -l
ZN
z,J Lt€ - ?l 5t 3$8 Z] X+?
el' st
-l
<, - th
"Y
3.2 INPUT.OUTPUT
47
This M-file would be savedas f reefallsubfunc.m. In suchcases,the first functionis called the main or primary function. It is the only function that is accessibleto the command window and otherfunctionsand scripts.Al1 the otherfunctions(in this case,ve1) are referred to as subfunctions. A subtunctionis only accessibleto the main firnction and other subfunctionswithin the M-file in which it resides.If we run f reef al lsubf unc from the commandwindow the result is identicalto Example3.2: ;'> freefa]lsubfunc (12, 68.!, 0.25) 50.6175 However,if we atternptto run the subfunctionvel, an errormessage occurs: >>
vel(I2,68.1,.25)
?? ? Undefined
3.2
command/ funct
ion
've1 '
INPUT.OUTPUT As in Section3.1, informationis passedinto the functionvia the argumentlist and is output via the function'sname.Two otherfunctionsprovideways to enterand displayinformationdirectlyusingthe commandwindow. T h e r r , l ' r r l F u n c t i o n . T h i s f u n c t i o na l l o w sy o u t o p r o m p tt h e u s e rf o r v a l u e sd i r e c t l y from the commandwindow.lts syntaxis ,r = input \'promptstring') The lunctiondisplaysIhepronptstring, value fiom the keyboard.For example,
waits for keyboardinput,and then returnsthe
m = input ('Mass (kS)r ') When this line is executed,the useris promptedwith the message Mass
(kg):
If the userentersa value,it would thenbe assigned to the variablen. The inpur funclion can also returnuserinput as a string.To do this, an ,s, is appendedto the function's argunlentlist. For example, name = input ( 'Enter your natne: ' , ,s , ) The ,ti. 1,Function. This functionprovidesa handyway to displaya value.Its syntaxis disp(rza-Zue) where yalue = the valueyort would like to display.If can be a numericconstantor variable,or a string messageenclosedin hyphens.Its applicationis illustratedin the following example.
zt+
'"1-r
ffiE\ z\4+
zf\ -x
:
zl x+
4
<, [h
--T ,t€ - ef >t
-l-
I
48
PROGRAMMI NG WITHMATLAB
EXAMPLE 3.3
A n I n t e r o c t i v eM - F i l eF u n c t i o n ProblemStqtement. As in Example3.2,computethe velocityof the free-fallingbungee jumper, but now usethe inprut and disp functionsfor input/output. Solution. Type the fbllowing statementsin the file editor: runct ion free[-, L: % f reef a1li: int.eracti're bungee velocity ? Z eo
freefalli
inter.active computatjon of the free fal1 rrelocity of an object \ r itt h s e c o n d - o r : d e r d r a g , % acceleration of grraviLy 9 - 9. B1; ( l < g ): ' ) ; m = input('Mass (kqlm) : ') ; cd = input ('Drag coefficient ' ) ; (s): t =:-nput('Time disp('') (m/s) :' ) drsp('Velocity * m / cd)*tanh(sqrt(g drsp(sqrt(g
* cd / m) * t))
Save the file as f reef a11 i . m.To irrvoke the function. return to the command window and type >> f reef .rl I i Mcrss (kg): 6g.l Drag coeffrcierrt Time (s): 12.
(kg/m) :
0 .2 5
Velocit,lr (nr,/s): 5 0 . 6 1 ' 15
T h e t 1 , r r r r l 1 F u n c t i o n . T h i s f u n c t i o n p r o v i d e s a d d i t i o n a lc o n t r o l o v e r t h e d i s p l a y o f infbrmation. A simple representationof its syntax ts fprinLf
('formaL',
> < , ,. . . )
where form.: t is a string specifying how you want the value of the variable x to be displayed. The operation of this function is best illustrated by examples. A sirrrpleexample would be to display a value along with a message.For instance,suppose that the variable velocity has a value of -50.6175.To display the value using eight digits with four digits to the right of the decimal point along with a message,the statement along with the resulting output would be >> fprinLf
('T'he velocity
The velocit.7
is
%8.4f n/s\n',
velocity)
i s 50.6175 mr's
This example should make it clear how the format string works. MATLAB starts at the left end of the string and di:;plays the labels until it detectsone of the symbols: %or \. In our example, it first encounters a % and recognizes that the following text is a format code. As in Table 3.1 , the format codes allow you to specify whether numeric values are
ffi zl' +
zl Llz .-r J
ffii: :.t -l
"L x -,\'IN Y+
vI
a,- . [h
j-l € - sJ 5t I
3.2 INPUT.OUTPUT TABTE3.I
49
Commonlyusedformotond controlcodesemployed w i t ht h e f p r i n t f f u n c t i o n .
Formoi Code
Descripiion
zd
zf Zg
l n l e g e rl o r m c i S c i e n t i f l cf o r m o fw l t h l o w e r c o s ee S c i e n t i cf f o r m o tw i t h u p p e r c o s eE D e c l m o lf o r m o t T h e m o r e c o m p o c to [ % e o r % f
Conirol Code
Description S l o r fn e w l i n e Tob
\Il
\t
displayedin integer,decimal,or scientificformat.After displayingthe valueof veLociry, MATLAB continuesdisplayingthe characterinformation(in our casethe units: m/ s) until it detectsthe symbol \. This tells MATLAB that the following text is a control code. As in Table 3.I,the control codesprovidea meansto perfbrm actionssuchas skippingto the next line. If we had omittedthe code \n in the previousexample,the comrnandprompt would appearatthe end of the labelm/s ratherthanon the next line as would typicallybe desired. The fprintf functioncan alsobe usedto displayseveralvaluesper line with diff'erent formats.For example, ('%5d %10.3f %8.5e\n',100,2*pi,pi)
>> fprintf 100
6.283
;
3.14159e+000
It can also be usedto display vectorsand matrices.Here is an M-file that enterstwo setsof valuesas vectors.Thesevectorsarethencombinedinto a matrix.which is thendisDlavedas a tablewith headines: fprrnt fdemo function '2 x = lI I 4 5l; 2A.4 12.6 r7.B BB.7 L2a.4l; x;yl j -
fprintf fprintf
(' y\n' ) x ('%5d %10.3f\n', z) ;
The resultof runningthis M-llle is >>
f pr:int X 1 2 3 4 5
--
z[+
Y 20.400 12 .6AA 17.800 88.700 120.400
ffi qAb :I
,/--5 ---"'-
z(\ Ll 2 -zl '-A
f de rno
--:> - ^, a I
e[st
x
<, - lh
rf €- sJ 5t
_.1
Z} Fh
I
50
PROGRAMMING WITHMATLAB
3.2. I Creoting ond AccessingFiles MATLAB has the capability to both read and write data files. The simplestapproachinvolves a special type of binary file, called a MAT-fi\e, which is expresslydesignedtbr implementationwithin MATLAB. Such files are createdand accessedwith the save and load commands. The save commandcan be usedto generatea MAT-file holding eitherthe entire workspaceor a few selectedvariables.A simple representation of its syntaxis save fifename varT var2 ...
varn
This commandcreatesa MAT-file named fi-tename.mar that holds the variablesyarl through varn. If the variablesare omitted,all the workspacevariablesaresaved.The l oad commandcan subsequentlybe usedto retrievethe file: load
fr lename
varl
var2
...
varrl
which retrievesthe variablesrzar-1throughvarn from f ilenante.mat. As was the case with save, if the variablesare omitted. all the variablesare retrieved. For example,supposethat you use Eq. ( 1.9) to generatevelocitiesfor a set of drag coefficients: > > 9 = 9 . B 1 ; m = B Ot ;= 5 ; >> cd=[.25 .261 .245 .28 .213]'; >> v=sqrt (g"m ./cd) .*tanh(sqrt(g*cdlm) *t) ; You can then createa tile holding the valuesof the dragcoefficientsand the velocitieswith >>
save
veldrag
v
cd
To illustratehow the valuescan be retrievedat a later time, removeall variablesfrom the workspacewith the clear command, >>
p F
clear
u al
At this point, if you tried to display the velocitiesyou would ger rhe result: >>
3.3
v
??? Undefined function
or variable
'v'
However,you can recoverthem by entering >>
load
!
T m CI
veldrag
n
Now, the velocitiesare availableas can be verified by typing
a
>> who Your
variables
3 TI
are:
cdv
Although MAT-files are q,rite useful when working exclusivelywithin the MATLAB environment,a somewhatdifferent approachis requiredwhen interfacingMATLAB with otherprograms.In suchcases,a simpleapproachis to createtext files written in the widely accessibleASCII format.
ffi= qA+
,
zl -h
--a zl Llt ._r _l
5[ sl
-i
Z] X+
"L x z,J
<, [h -r
j.l'€- ?l 5t &
PROGRAMMING 3.3 STRUCTURED
5l
ASCII files can be generatedin MATLAB by appending ascii to the save command. In contrastto MAl-files where you might want to savethe entire workspace,you would typically savea singlerectangularmatrix of values.For example, >> A=15 I >>
save
9 2;3 6 3 9l;
simpmatrix.txt
ascii
In this case,the save commandstoresthe valuesin A in 8-digit ASCII form. If you want to storethe numbersin doubleprecision,justappend-ascii -doubLe. In eithercase,the file can be irccessedby other proglams such as spreadsheets or word processors.For example,if you openthis file with a text editor,you will see 5 . 0 0 0O O 0 O e + O 0 O7 . 0 00 0 0 0 0 e + 0 0 9 . 0 0 0 0O0 O e + 0 0 0 2 . 0 0 0 00 0 O e + 0 0 0 3.0000000e+000 6.0000000e+000 3.0000000e+000 9.0000000e+000 Alternatively,you can read the valuesback into MATLAB with the load command, >>
load
sirnpmatrix.txt
BecausesimpmaLrix. txt is not a MAT-file,MATLAB createsa doubleprecisionarray namedafter the fi l ename..' '
,;irpt-rt. ' i>:
simpmaLr ix 5192 zF,
= l9
Alternatively, you could use the load command as a function and assign its values to a variable as in >> A = loeid(simpmatrix.txt) The foregoing material covers but a small portion of MATLAB's file managementcapabilities. For example. a handy import wizard can be invoked with the menLrselections: File, lmport Doto. As an exercise,you can denronstratethe import wizards convenienceby u s i n g i t t o o p e n s i m p m a t r i x . t x t . I n a d d i t i o n ,y o u c a n a l w a y s c o n s u l th e l p t o l e a r nm o r e about this and other l'eatures.
3.3
STRUCTUREDPROGRAMMING The simplestof all M-files perform instructionssequentially.That is, the program statementsare executedline by line startingat the top of the function and moving down to the end. Becausea strict sequenceis highly limiting. all computerlanguagesinclude statepaths.Thesecan be classifiedas mentsallowingprogramsto takenonsequential . .
The branchingof t-lowbasedon a decision. Decisions (or Selection). Loolts (or Repetition).The looping of flow to allow statementsto be repeated.
3.3.1 Decisions The i r Siructure. This structureallows you to executea set of statementsif a logical conditionis true.Its generalsyntaxis if
condition statemenf-s
ffi z\4 ,"I LJ' zl'+
9t" 5\
-J
3l x+
"L 4
<, lh x --4 - 9f 5l ,f\€ -l-
&
52
PROGRAMMING WITHMATLAB wherecondr tron is a logicalexpression that is eithertrueor false.For example,hereis a simpleM-tile to evaluatewhethera gradeis passing: gldder(g oder (grade ) : determines whether
Iuncrion % grader %
% input: r" grade
= numerical-
qrade value
is of
passing grade
(0-100 )
% output: % if
drsplayed
message
grade
>= 50 di sp ( 'pass rr-rg grrade ' ) end
The following illustratesthe result >> glader(95.6) passingr grade For cases where only one statementis executed, it is ofien convenient to implenrent the i f structure as a single line, if
grade
> 60,
djsp('passing
gr.rdc'),
end
This structure is called a single-lirrc rf For cases where rnore than one statel.nentis irnplemented, the multiline if structure is usually preferable because it is easier to read. E r r o r F u n c t i o n . A n i c e e x a m p l eo r l ' t h e u t i l i t y o f a s i n g l e - l i n ei f i s t o e m p l o y i t f o r r u d i rnentary error trapping. This involves using the cr:ror: function which has the syntax, er ror (msgr) When this firnction is encountered,it displays the text message,.rsEr and causesthe M-tile to terminate and return to the conrmand window. An example of its r"rsewould be where we might want to terminate an M-file to avoid a division by zero. The fbllowing M-file illr"rstrates how this could be done: function f = errortest(x) 11 1 == 0 , er ror (' 'zcrc:value f = I/x.;
encountered'
) , cnd
If a nonzero argument is used, the division would be implemented successfully as in >> errotLest(l0)
0.1000 Howevel fbr a zero argument! the function would ternrinate prior to the division and the error messagewould be displayed in red typef'ace: >> error test (L ; ??? ErroL usit'tg ==> err-or:test zero value encountered
ffi= zr+
'"1
"L x
:
zl x+
4
<, - lh Lrt9 - 9n 5
-tl -l-
I
53
PROGRAMMING 3.3 STRUCTURED TABTE3.2
Summoryof relotionoloperoiorsin MATLAB Operotor
Exomple unit -= a< 0
Relotionship Equol Not equcl Lesslhon G r e o l e rl h o n L e s st h o n o r e q u o l l o G r e o i e rl h o n o r e q u o l t o
'r
s>t
3.9 .:= 3/3 r: >= 0
Logicol Conditions. The simplestform of the condr Lron is a singlerelationalexpression that comparesfwo valuesas in vaf ue.
relaL
i,on vaf ue,
variables,orexpressions wherethe yaf ue-scan be constants, and the relaLion is one of Table the relationaloperatorslisted in 3.2. MATLAB alsoallowstestingof morethanone logicalconditionby cmployinglogical We will ernphasize the fbllowing: operators. .
- lNot).Usedto perfbrmlogicalnegationon an expression. - exp res-s r o,t
If the expre.sr;ion is true,the resultis false.Converscly,if the expressjon is false, the resultis true. * (And). Usedto perlbnn a logicalconjunctionon two expressions.
.
+;1p7 essr.,r
d
pxp/
| .';S,rOn
If both e)<pre.ssrorsevaluateto true, the resultis true. If eitheror both expre-ssrons evaluatesto false,the resultis false. (Or).Used to performa logicaldisjunctionon two expressions.
.
FXp/
ps.-;'on
I
PxLrt r-. :'io"1
If either or both express,-i on!r evaluateto true, the result is true. lhble 3.3 summarizesall possibleoutcomeslor eachof theseoperators.Just as fbr arithmeticoperations,there is a priority order lbr evalualinglogical operations.These TABTE3.3
for logicoloperotors the possibleoutcomes A truthtoblesummorizing Theorderof priorityof ihe operotorsis shownol employedin MATLAB. the loo of thetoble. Highesr
Lowesl xly
x&y
ryx
T T T F
I
I F F
"L x 4
1-1,€
5t
8,
PROGRAMMING WITHMATLAB
54
are from highest to lowest: -, a and I . In choosingbetweenoperatorsof equal priority, MATLAB evaluatesthem from left to right. Finally, as with arithmetic operators,parenthesescan be used to override the priority order. Let's investigatehow the computeremploysthe prioritiesto evaluatea logical expres=t,andy='b',evaluatewhetherthefollowingistrueorfalse: s i o n I. f a = - 1 , b - 2 , x a * b > 0 & b == 2 & x > 7 | .'(y > ,d ) To make it easierto evaluate,substitutethe valuesfor the variables: I * 2 > 0 & 2 =_ 2 & 1 > 7 | _(,b, > ,d') The first thing thatMMLAB doesis to evaluateany mathematical expressions. ln this example,thereis only one: -7 x 2, -2
>
A & 2
==
2
t''I
> 1
-('b'
>
'd')
Next, evaluateall the relationalexpressions -2 > A & 2 -= 2 & I > I T F & & tr
1 -('b' l-
;' 'd') F
At this point, the logical operatorsare evaluatedin priority order.Sincethe - hashighest priority,the lastexpression(-r) is evaluatedfirst to give F&T&F
T
The a operatoris evaluatednext. Since there are two, the left-to-right rule is appliedand the first expression(p r r) is evaluated: F&FIT
The r againhashighestpriority FIT Finally,the I is evaluatedas true.The entireprocessis depictedin Fig. 3.1. The j r . r ' t : : , , S t r u c t u r e . T h i s s t l u c t u r ea l l o w sy o u t o e x e c u t ea s e to f s t a t e m e n itfs a logical condition is true and to executea secondset if the condition is firlse.Its general syntaxis if
r:ondjtion stat€ment:,-,
else Stdae'nlenLS.., end
T h e i r . . { , t i ; r l t S t r u c t u r e . I t o f i e n h a p p e n s t h a t t h e t a l s e o p t i zo rnnoi f . . . e l s e structureis anotherdecision.This type of structureoften occurswhen we have more than two options lbr a par-ticularproblem setting.For such cases,a special form of decision structure,the if. . . efseif hasbeendeveloped.It hasthe Seneralsyntax if
condition, statements, elseif conditron.. statements,
ffi= qA+, z(| Ll 2
zl'+
LA
_\
7[e1.s\ -l
^l
zl /+?
"L x 4
<, - [h
ll'€- eJ 5t 3$8
EXAMPLE 3.4
PROGRAMMING 3.3 STRUCTURED
a*b
1^, --
>u&
ll
-r*2
Z
55
| -( Y '
x>'7
d..
II
I
>0
&
2&
Substitute constants
t
i
I '('b'
7>'7
I
t Evaluatemathematical expressions
t
-2
>0
&2==2
&1
>7
&
I I
tl tl
ii
F&T
>'d')
II
-F \---YJ
T
I
Fl
&
Evaluaterelational expressions
t
Fl
\.-.........-.Y#
F
| _('b'
Evaluatecompound expressions
+
T
T FIGURE 3.I A slep'by'srep evollotion o{ o conrplexdecision.
elseif
condition-,
statementsr
else statements..l5.. end
EXAMPLE 3,4
i t Structures Problem Stotement. For a scalar,the built-in MATLAB sign function returnsthe sign of its argument(- I, 0, I ). Here's a MATLAB sessionthat illustrateshow it works; >> sign (25.6) 1 >> sign (-4.776) -1 >> sign(0) 0 Develop an M-file to perform the same function.
zt+
,"1{
ffi qA+,
zt\ aY
.
Z]
x+
4
a[h
I
_-T ,f\t - ef
-l-
3$ 8,
56
WITHMATLAB PROGRAMMING Solution. First, an i f structurecan be usedto return 1 if the argumentis positive: % mysign(x) >
x
if
returns
1
if
x
gTreater
is
zero.
than
0 =
sgn
= nYSiQIn(x)
sqn
function
1;
end
This function can be run as >> mysign(25.6) 1
Although the function handles positive numbers correctly, if it is run with a negative or zero argument, nothing is displayed. To partially remedy this shortcoming, an if. . .e1se structure can be used to display -1 ifthe condition is false: sgtn = mysign (x) f unction 1 if x is greater than zero. returns % mysign(x) I if x i-s less than or equal Z if x > 0 sgn = 1; ^t
to
zero.
^-. _ -
-^-
- 9 r !
_t.
end
This function can be run as >> mysign(-0.176) -1 Although the positive and negative cases are now handled properly, -1 is erroneously structurecan be used to incorporate returned if azero argument is used.An rf . . . elseif case: this final funct-ion sgn = mysign(x) 1 if returns % mysign(x)
z z
x
is x
O if
x
is is
gtreater less eq-al
than than to
zero.
zero.
7.aro.
lf x > 0 sgn = 1; x < O elseif sgn = -1; else sgn = O; end
The function now handlesall possiblecases.For example, >> mysign(0) 0
ffi!3r qA+,
il{#ff: er'
"L x 4
zl xh
%8,
57
PROGRAMMING 3.3 STRUCTURED
T h e s r v i t - c hS t r u c t u r e . T h e s w i t c h s t r u c t u r e i s s i m i l a ri n s p i r i tt o t h e i f . . . e 1 s e i f structure.However,ratherthan testingindividual conditions,the branchingis basedon the value of a single test expression.Dependingon its value,different blocks of code are implemented.In addition.an optionalblock is implementedif the expressiontakeson noneof the prescribedvalues.It has the generalsyntax I ..sf r.xprcss
sr.ricch case
I on
r z a Ju e ,
"-ctaaemencs. y a - 1u e . , case staIe.menrs.
oInerwrse s I a a emel]
f .sol lr..r rr:i,
end
As an example,hereis functionthatdisplaysa message dependingon the valueof the stringvariable,gra de. gracle
=
svrifch case drsp case
'R'; qracle: 'A' ( 'Excellent' 'B'
)
clisp('Good') 'C'
case
disp L r15ri
( 'Mc,-diocre' ) D
disp('Whoops') 'F' case drsp
( 'Wou1d
like
fries
wj th
your
order?'
)
otherw]se disp(,lluht
)
end
"Good" would be displayed. When this codewas executed,the message Vorioble Argument List. MATLAB allowsa variablenumberof argumentsto be passed to a function.Tlris featurecan come in handy for incorporatingdefaultvaluesinto your functions.A d{oult vulueis a numberthat is automaticallyassignedin the eventthat the userdoesnot passit to a function. As an example,recall that earlierin this chapter,we developeda function freef a 1 1ve i, which had threearguments: =
v
freefallvel
(t,m,cd)
Although a userwould obviously needto specifythe time and mass,they might not have a good idea of an appropriatedr;..9coefficient.Therefbre,it would be nice to have the program supply a value if they omitted it from the argumentlist. MATLAB hasa functioncallednarsin that providesthe numberof input arguments suppliedto a function by a user.It can be usedin conjunctionwith decisionstructureslike
ffi ,-TLJ' -b
b
Z-x N
z.J
3!h ,f.€ - 9f 5l la^,
l -
PROGRAMMING WITHMATLAB
58
the if or switch constructsto incorporatedefault valuesas well as error messagesinto your functions.The following code illustrateshow this can be done for fr^eefallvel: function ?
=
v
%
(t,
freefalLvelt
freefallvef
m,
cd)
with second-order bungee vefocity (t,m, cd) computes free-fa11 the
:
v=freefallvel
of
with
an object
% input: Z L = time (s) %m=mass(ks) (default = 0.21 cd = drag coefficient % % outpul-: (mrs) v = downward velocity % swi t ch nargrin case O enter trme and mass') error('Must ^--^
drag.
second-order
kg/m)
a
error case cd
en1-er nass ' )
( 'Must
2 = 0.21 ;
end A = v
drag velocity
% acceleration
9.81; = .sqrt(g
*
m,/
of
cd)*Lanh(sqrt(g
gravity * cd
/
m)
*
t);
Notice how we have used a switch structureto either display error messagesor set the default,dependingon the numberof argumentspassedby the user.Here is a command window sessionshowingthe results: >'> f reef allvc. l (12,68.I,
A .25)
50.61.75 >>
( 12,6B.l)
freefallvel
48 .81 41 >>
??'. Err(,r Must >>
(12 )
freefallvel
enter
mass
f reef al.lvel
??? Must
Error enter
f reefallv-l
us.ng
() ltrofallvol
*sing time
and
EXAMPLE 3.5
mass
Note that narsin behavesa little diff'erentlywhen it is invoked in the command window. In the commandwindow, it must include a string argumentspecifyingthe function and it returnsthe numberof argumentsin the function. For example, >> nargin
('freefallvel'
ffir
,fl1fu= -\
zt'--h"
)
"L x zJ 7) F+
<. - [h.
il'€
-
- :l 5 I
3.3 STRUCTURED PROGRAMMING
59
3.3.2 Loops As the name implies, loops perform operationsrepetitively.There are two types of loops, dependingon how the repetitionsare terminated.A for loop endsafter a specifiednumber of repetitions. A whi I e loop endson the basisof a logical condition. The ror . . . encl Structure. A f or loop repeatsstatementsa specificnumberof times.Its g e n e r asl y n t a xi s f o r i n d e x - s t a r t : s t e D : .f i n i s h ".1-uteme',rs The f or loop operatesas follows. The rndex is a variable that is set at an initial value, start. The programthencomparesthe rndex with a desiredfinal value,fin jsh. If the index is lessthanor equalto the frnish, the programexecutesthe sLatemenrs. When it reachesthe end line that marks the end of the loop, the index variableis increasedby the srep and the programloopsbackup to the for statement. The processcontinuesuntil the index becomesgreaterthan the f ini sh value.At this point, the loop terminatesas the programskipsdown to the line immediatelyfollowing the end statement. Note that if an incrementof 1 is desired(asis ofien the case),the s t ep can be dropped. For example, for
i
= 1:5
djsp(i) end
When this executes,MATLAB would displayin succession, 1 , 2 , l, 4, 5. In other words,the def-aultstep is 1. The size of the srep can be changedfrom the defaultof 1 to any other numeric value. It doesnot haveto be an integer,nor doesit haveto be positive.For example,stepsizesof A . 2 , - I , o r - 5 , a r ea l l a c c e p t a b l e . If a negativestep is used,the loop will "countdown"in reverse.For suchcases,the loop's logic is reversed. Thus,the f .inj sh is lessthanthe start and the loop terminates when the index is lessthan the f inish. For example, for i = l0: disp rj r
1:1
end
When this executes,MMLAB would displaythe classic"countdown"sequence: 10 , 9 , B, 1, 6, 5, 4,3,2, 1. E X A M P L3E. 5
U s i n g o 1 . r L o o p t o C o m p u i et h e F o c t o r i o l ,
ProblemStotement. Developan M-file to computethe factorial.2
0!: I ll-
I I v
) -')
2 Note that MATLAB has a builrin function f acLoriat
zl'
,"1{t -&
that oerformsthis comnutation
ffi= z$4 b
st 71et I
^
ZI\ x _-T<,- lh 9l ,f\€ 4
>l
-t-
z I x+ /
&
WITHMATLAB PROGRAMMING
60
3l:lx2x3:6 4l:lx2x3x4:24 5!:1x2x3x4x5:120
A simple function to implement this calculation can be developed as
Solution.
fout = factor (n) function ( n ) : % factor Computes the producc of % -
--
for x
all
Lhe integiers
from
I
to
n'
I
t.
I
i = l:n = x * i;
end = x;
fout end
which can be run as >>
f actor: (5 )
12a This loop will execute5 tirnes(fiom I to 5). At the end of the process,x will hold a value o f 5 l ( r n e a n i n g 5 f a c t o r i a l oI l x 2 x 3 x 4 x 5 : 1 2 0 ) . Notice what happensif n : 0. For this case,the f or loop would not execute,and we would get the desiredresult,0! : I
Vectorizotion. The for loop is easy to implement and understand.However, for MATLAB, it is not necessarilythe most efficient meansto repeatstatementsa specific numberof times.Becauseof MATLAB's ability to operatedirectly on arrays,vectorizotion providesa much rnoreefficient option. For example,the filllowing f or structure: i
n. t- = 0:0.42:5C
for r
=
y(i)
i
+ l; = cos(r);
end
can be representedin vectorizedform as t
= 0:0.02:50;
y
= cos(t);
It shouldbe noted that for morc complex code,it may not be obvious how to vectorizethe code.That said,whereverpossible,vectorization is recommended.
T Ir
Preollocotion of Memory. MATLAB automaticallyincreasesthe size of arays every tirne you add a new element.This can becometime consunringwhen you perform actions suchas addingnew valuesone at a time within a loop. For exarnple.hereis somecodethat
fiT=fiar qA+ zf Ll L
zl'+
.r'.
.{
A e[st -.1
Z] X+
"L x
sl SI
<, Ih
VT jl'€-
-r
sJ 5r I
3.3 STRUCTURED PROGRAMMING
6t
setsvalue of elementsofy dependingon whether or not valuesof t are greaterthan one: L
-
U:.UI:);
F^LOr
l I
l:lerg.hrL/
t ( i ) >l = I/t-(r); Y(i) else r f
y(j) end end
For this case,MATLAB rnustresizey every time a new value is determined.The follow_ ing code preallocatesthe proper amount oi -".o.y by using a vectorizeclstatement to assignonesto y prior to enteringthe loop. 1, - 0:.01:5; y - one-q(si:e(t)); for i = l:lengLh(t) rf r (i )>l y(i) - Iir(i); encj end Thus, the anay is onry sized once. In addition, preailocation herps reduce memory frag_ mentation, which also enhancesefficiency. The r.;hLt L:,Structure. general syntax is
A while
lclop repeats as long as a logical condition is true. Its
vrh,l. e^ndirian st;tte,rnert,s end The statemencs betweenthe whiLe anclthe end are repeatedas long as the condr jon t is t r u e .A s i m p l ee x a n r p l ei s vO
while disp
x
>
0
(x)
encl
When this cocleis run. the resultis E 5 2 -1 T h e r r ' i l r : ' ' ' 1 r r r - : ' r sk t r u c t u r e ' A l t h o u g ht h ew h i l e s t r u c t u r ies e x t r e m e r u y s e f u rt,h e f'act that it always exits at the beginning lf the structureon a false result is somewhat constraining'For this reason'languagessuch as Fortran90 and visual Basic have special structuresfhat allow loop terminationon a true condition anywherein the loop. Although suchstruduresarecurrenrlynor availablein MATLAB, tt er. iun.tion;ii;."" be mimicked
-WT-Tbl zl
zt -+
-tA-\
aA+, a s[sl Ll 2 -{
L
4
af x+?
x
<. lh
lj€
- eJ 5t %8
62
PROGRAMMING WITHMATTAB by a specialversionof the while loop. The syntaxof this version,called a while... break structure,can be written as r,vhile
(1)
stafements if
conditron,
break,
end
.'rXtutements wherebreak- terminatesexecutionof the loop.Thus. a singleline i f is usedto exit the loop if the conditionteststrue.Note that as shown,the break can be placedin the middle of the loop (i.e., with statementsbeforeand atier it). Sucha structureis calledarnicltestloop. If the problem requiredit. we could placethe break at the very beginningto createa prel(.stloop.An exampleis (1)
while If
<
x
-;
0, -
w
break,
end
l.
end
Notice how 5 is subtractedfrom x on eachiteration.This representsa rnechanismso that the loop eventuallyterminates. Every decisionloop must havesucha mechanism.Otherwise it would becornea so-calledinfinite loop that would neverstop. at the very end and creAlternatively.we couldalsoplacethe i f . . . break statement ate a posttest lo)P,
3
(1)
whrle
q.
if
x
<
(.), break,
end
end
EXAMPLE 3
It shouldbe clear that, in fact, all threestructuresare really the same.That is, depending on wherewe put the exit (beginning,middle,or end)dictateswhetherwe havea pre-, nrid- or posttest.It is fhis simplicity that led the computerscientistswho developed Fortran 90 and Visual Basic to favor this structureover ofher forms of the decision looo suchas the conventionalv;hile structure. The 1,,,,,',,-. Commond. Thereare often timeswhen you might want a programto temporarily halt.The commandpause causesa procedureto stopand wait until any key is hit. A nice exampleinvolvescreatinga sequence of plots that a userrnight wanf to leisurely perusebefbremoving on to the next.The following codeemploysa f or loop to createa sequenceof interestingplots that can be viewed in this manner: for n = 3:10 mesh (magic (n) ) pause end
The pause canalsobe formulatedaspause (n ), in which casethe procedurewill halt fbr n seconds.This featurecan be demonstratedby irnplementingit in conjunction with severalother usefulMATLAB functions.The beep commandcausesthe computerto emit a beep sound.Two other functions.t ic and toc, work togetherto measureelapsedtime.
ffiat
-
=L
,r, 4 1+%-q'tr -zt+ zl ,N17
x
<, - [h
j]\€ - ?l 5t I
63
3.4 NESTING ANDINDENTATION
The tic command saves the cuffent time that toc later employs to display the elapsed time. The following code then confirms that pause (n)works as advertisedcomplete with sound effects: tic beep pause (5) beep toc When this code is run, the computer will beep. Five seconds later it will beep again and display the following message: ELapsed time
is
5.006306 seconds.
By the way, if you ever have the urge to use the command pause ( inf ), MATLAB will go into an infinite loop. In such cases, you can return to the command prompt by typing
Ctrl*C or Ctrl+Break. Although the foregoing examplesmight seema tad frivolous, the commandscan be quite useful.For instance,tic and toc can be employedto identify the partsof an algorithm that consumethe most executiontime. Further,the Ctrl*C or Ctrl+Break key combinations comein real handyin the eventthatyou inadvertently createan infiniteloop i n o n eo f y o u r M - f i l e s .
3.4
NESTINGAND INDENTATION We needto understandthat structurescan be "nested"within eachother.Nc.sllrrcref-ersto placingstructureswithin other structures. The following exampleillustrtrtes the concept.
E X A M P L3E. 6
N e s t i n gS t r u c t u r e s ProblemStqtement. The rootsof a quadraticeqLration /(x):uxl+lrx+r' can be determinedwith the quadraticformula
-b +.,t8-- 4aa LA
Developa functionto implementthis fbrmulagivenvaluesof the coeff'cients.
v
Solution. ktp-down r/e.rlgnprovidesa nice approachfor designingan algorithm to compute the roots.This involvesdevelopingthe generalstructurewithout detailsand then refining the algorithm.To start,we first recognizethat dependingon whetherthe parameter a is zero,we will eitherhave"special"cases(e.g.,singlerootsor trivial values)or conventionalcasesusingthe quadraticfbrmula.This "big-picture"versioncan be programmedas
a
f - u n c t i . o nq u a d r o o t s ( a , b , c ) % quadroots:
llt
%
Lth nit ]e.
z
quadroots
roots of (a, b, c) :
quadratic real
and
o'quod'ot
equation complex j-
aqua'
rool-s jon
% input: 9o
o
ffi
z$4
?JLI' zl+
lo.ond
J J
"t
otdet
coefficient
:I 5\
Z] Xh
"L x 4
<, [h
--4 1.1':- sl 5t I
64
PROGRAMMING WITHMATLAB Z b = fir-st-order Z c = zera order % output: Z rI = real part Z i1 = imagrinary Z 12 = real part % i2 = imaqinary 1t
==
d
coefflcient coefficient of first root part of flrsL root of second root part of second root
u
%special
cases
els<: "ir'r
'^rl-L1a
e
'dr,tl
end
Next, we developrefined codeto handlethe "special" cases: %special cases if b -'= 0 e;srngle root 11
|
V,
else?tr i vial
sol ut ior-r
error('Trivial
solutron.
Try
again'
)
end
And we can developrefinedcodeto handlethe quadraticformulacases: %qu.rdrat ic - b ^ 2 if cl >- O
f or:mul a 4 * a *
d
c;
rooLs %real r1 = ( b + sqrt(d) ) ( b - sqrt.(d)) t'2 -
/ /
12 * (), *
a) a)
el se %complex roots = rl b / \2 * a) i1 = serL(.rbs(d)) r)
/
(2
*
a)
r-l
end
We can thenmerelysubstitutetheseblocksback into the simple"big-picture"framework to give the final result: quadrooLs
functron ?
quadroots: quadroots
%
(a,
roots of (a ,b, c) :
%
b, c) quadratic real of
input: % a = Z b -
equafion
and compl-ex roots quadratic equation
Z
%
c
second-order first
-
zero-aYder
% ouLput: % rI Z
i1
order
=
real
part-
imaginary
coefficient coefficient coefficient of first part- of
roo[ first
ffi= zl+
,"1{z
5fi st 7t l-l
3l x+
root
"L x 4
<, [h Lrtg -
:.I
9r\ 5 -J
I
3.4 NESTING AND INDENTATION % Z ai Fr
a
65
12 = r.eal- part of second root 12 = imag,inary part of second root ==
u
Sspecial cases ifb-=0 ?single roor 1L--c/b else ?trivial solution ( ' ?rivial solut ion. g^l * l u: r . o .
?ry
agaln , )
e-Lse Sguadratic formula d=b * c; 4 *a Ediscriminant rf d >= 0 ?rea1 roois ri- = (-b + sqrt(d)) / (2 * a) 12= (-b-sqrr(d)) / (2*a) else Scomplex roots rl-=-b/(2*al i1 = gqy; (abs (d) ) / (2 * a) r)
-
v1
l . ! .
-
- f
a
end end
o.r highrightedby the shading,notice how indentationhelps to make the , . underlying logical structurecrear.Also nodc;how "modular,,the structuresare. Here rs a command window sessionillustratinghow the function f"rfo.rs >> quadroots(1,1,1) 17 = -0.5000 =
i1
0.8660 12_0.5000 -0.8660 >> quadroots (l, 5, 1 ) -0.2087 4.7913 >> quaCroots (0,5,1) -0.2000
ml zI zl Llz zl' +
,
t-r
_\
+
4-{
5[st
Z] X+
"L x
a,- . [h
VT Lf €- 1 5t '$8
66
PROGRAMMING WITHMATLAB j
>> quadroots (0, 0, 0 ) ??? Error using ==> quadroots TriviaL solution. Try again
PASSINGFUNCTIONS TO M.FIIES
3.5
Much of the remainderof the book involves developingfunctionsto numericallyevaluate other functions.Although a customizedfunction could be developedfor every new equation we analyzed,a betteralternativeis to designa genericfunction and passthe particular equationwe wish to analyzeas an argument.In the parlanceof MATLAB, thesefunctions are given a special name:function.functictns.Before describinghow they work, we will first introduceanonymousfunctions,which provide a handy meansto define simple userdefinedfunctionswithout developinga full-blown M-file.
3.5.1 Anonymous Functions Anonyntous.functionsallow you to createa simple function without creating an M-file. They can be definedwithin the comrnandwindow with the following syntax: ihandLe = @(argJist) expressro,r where f.harrdle - the functionhandleyou can use to invoke the function,argTist = a comma separatedlist of input argumentsto be passedto the function.and express l on = any singlevalid MATLAB expression. For example, >>
fl-@(x'Y)
x^2
+ Y^2;
Once thesefunctionsare defined in the commandwindow, they can be usedjust as other functions: >>
f1(3,4)
ans.=.
\ Asidefiom the variablesin its argumentlist, an anonymousfunctioncan includevariables that exist in the workspacewhere it is created.For example, we could createan a n o n y l n o ufsu n c t i o n / t l ) : - L l l u s >>
a
-
>>
4; = 2; f2=@ (x)
>>
f2 (3)
>> b
ans
a*x^b,'
= 36
Note that if subsequentlywe enter new valuesfor a and b, the anonymousfunction doesnot chanse: f 2 (3 ) ans
ffiar qAb zf Ll t zl -&
''l
-\
T -l
--c,lh -
5[st
3l x+
Y
j-l'€- eJ 5t 3$8
t t
FUNCTIONS 3.5 PASSING TOM.FILES
67
Thus, the function handleholds a snapshotof the function at the time it was created.If we want the variablesto take on new values,we must recreatethe function.For example,having changeda to 3, >>
f2=@x
a*x^b;
with the result >>
f2 (3)
ans
27
It shouldbe notedthat prior to MATLAB 7, inline functionspertbrmedthe same role as anonymousfirnctions.Forexample.the anonymousfunctiondevelopedabove,f 1, could be written as >>
+ y^2',
fl=rnline:('x^2
'x','y');
Although they arebeing phasedout in favor of anonymousfunction,somereadersmight be usingearlierversions.and so we thoughtit would be helpfulto mentionthem.MATLAB helpcan be consultedto learnmoreabouttheiruseand limitations.
3.5.2 FunctionFunctions Function functions are functionsthat operateon other functionswhich are passedto it as input arguments. The functionthat is passedto the functionfunctionis ref'erredto as the passed.function. A sirnpleexampleis the built-in function fp1oL, which plots the graphs of functions.A simplerepresentation of its syntaxis fplot\iun,lins) where fun is the tirnctionbeing plotted betweenthe -r-axislimits specifiedby 7 rms = [xmin xmax].For this case,f un is the passedfunction.This functionis "smart" in that it automaticallyanalyzesthe functionand decideshow many valuesto use so that the plot will exhibitall the function'sf'eatures. Here is an exampleof how fplot can be usedto plot the velocityof the free-falling bungeejumper.The f unctioncan be createdwith an anonymousfunction: .
vrl
'Jrrj
s q r t ( 9 . B 1 * 6 i l . I / 0 . 2 5 ) * L a n h ( s c l r t -( 9 . B 1 * A . 2 5 , / 6 8 1 . ) *t) ; We can then generatea plot from / : 0 to l2 as >>
fplot
(vel,
f2 I )
[0
The resultis displayedin Fig. 3.2. Notethatin theremainderofthisbook,wc,will havemanyoccasions to useMATLAB's builrin functionfunctions.As in the following example,we will also be developingour own.
ffi= ----
,"I LJ' zl+
--1t _-l
5[ s\
-
<, [h x --T - 91 5l ,r\€
"L z,J
-l-
-l
Z] X+
I
68
PROGRAMMING WITHMATLAB
FIGURE3.2 A ploio{velocily generoted versus trrne wilhthefplor funclion
E X A M P L3E. 2
n gF u n c t i oFnu n c t i o n B u i l d i no gn d l m p l e m e n t io ProblemStqtement.DevelopanM-filefunctionfunctionto determine value theaverage jumpervelocityovertherange of a functionovera range.lllustrate its usefbr thebungee f t o mr : 0 t o l 2 s :
u(r): ffo"n(rH,) whereg:9.81,m:68 Solution. The averagevalue of the function can be comnutedwith standardMATLAB commandsas >> t=linspace (A, I2) ; >> v=sqrf (9. 81"68. 1,/0 >> mean(v)
*tanh (sqrt (9. 81"0.25 / 68. 1 )
36.0870
lnspectionof a plot of the function (Fig. 3.2) showsthat this resultis a reasonableestimate of the curve's averageheieht.
ffir qAb
<, lh -r
-
z[ Ll 2 --zl ?,1+
t"l
"L x j.l'€ -
5[ st -r,ar
-l
Z] X+
4
eJ 5t I
3.5 PASSING FUNCTIONS TOM.FILES
69
We can write an M-file to perform the samecomputation: funcr o
ion
e"
I uncavg {o, b, n )
lavg ^.
f ',-^r-
f-vg
u
v
L
!
s Y
f rra.l:.
ro
n
funcavg(o,b,n):
l-,ohL
compuLes average of function over
% % input: 2 a = lower bound of ranqe % b = upper bound of range % n - number of intervals % output: % favg = average value of functron x = linspace(a,b,n); y = func(x); favg - mean(y) ; end
value d range
function f = func(t) f-sqrt (9. B1*68.I / 0. 25 ) *tanh (sqrt (9. B1*4.25 / 68. 1 ) *t ) ; end The main function first uses linspace to generate equally spaced -r values across the range. These values are then passed to a subfunction func in order to generate the corresponding y values. Finally, the average value is computed. The function can be run from the command window as >> funcavS
(0, 12,60)
36.4121 Now let's rewrite the M-file so that rather than being specific to f unc, it evaluates a nonspecific function name f that is passed in as an argument: [*nct.on 9
r
iavg
r nr"a.
%
{r,o,o,n)
fun-dvg
r'7^,
,aa
f rrrnl
favg-funcavg(f
:nr
,a,b,n)
:
%
hoiqhL
comput-es of
Z
input: f =
to
be
evaluated
%
a
=
lower
bound
of
range
%
b
-
upper
bound
of
range
%
n
= number
?
z
of
over
val.ue a
range
intervals
outpuc: l'aV9 = ,vQ) oQe
% x
function
average
function
=
Y = favg
volue
of
f une t ion
llnspace(a,b,n); f (x); = mean(Y) ;
Becausewe haveremovedthe subfunctionf unc, this versionis truly generic.It can be run fiom the commandwindow as >> vel-@ (t ) sqrt ( 9. B1* 68.I/ 0.25 ) *tanh >> funcavg (vel , A, 12 ,60)
ffi z$4 zt+
l{t
( sqrt
( 9. B1*4.25 / 68.1 ) *t ) ;
"L x
+,
3l x+
4
-a, lh
--T
,f\€ - 9f 5l
-l-
3$8
70
PROGRAMMING WITHMATLAB
36 . 0 r 2 ' 7 To demonstrateits genericnature,funcavg can easily be appliedto anothercaseby merely passingit a different function. For example,it could be usedto determinethe average value of the built-in sin function between0 and 2z as >> funcavg(Gsin, 0, 2*pi, 180) 6.3001e 017 Doesthis resultmakesense? We can seethat f uncavg is now designedto evaluateany valid MATLAB expression, We will do this on numerousoccasionsthroughoutthe remainderof this text in a numberof contextsrangingtiom nonlinearequationsolvingto the solutionof differentialequations.
3.5.3
Possing Porqmelers
Recall from Chap. I that the terms in mathematicalmodelscan be divided into dependent and fbrcing functions.For the bungeejumper and independentvariables,pararneters, (u) variable,the is the dependentvariable,time (t) is the independent model,the velocity (c,,) (rr) gravitational and the constant(g) is the rnass and dragcoefTicient areparameters, to investigatethe behaviorof sr.rchmodelsby perfbrcing function.lt is cornmonplace variablechanges ana\-sis.This involvesobservinghow thedependent fbrmingtr sensitivitlt as the parameters and forcingtunctionsarevaried. In Example3.7,we developeda functionfunction,f uncavg, and usedit to determine the averagevalue of the bungeejumper velocity fbr the casewherethe parameterswereset L\ttn :68. I and c,,- 0.25.Supposethatwe wantedto analyzethe samefunction,but with Of course,we could retypethc functiclnwith new valuesfbr each dif'ferentparameters. case,but it would be preferabletojust changethe parameters. into anonymous As we learnedin Sec.3.5.1,it is possibleto incorporateparameters functions.For example,ratherthan "wiring" the nurnericvalues,we could havedonethe fbllowing: >> m-b8.I;r:d-().2,5; >> -ze1=@(t,) sqrt (9.81*m/cd) >>
f uncavg
*Lanh(sqrt
(9.81*cdz'm)
*t)
;
(ve1, 0 ,1,2,6A)
36.A121
However,if we want the parameters to take on new values,we must recreatethe anonymous function. MATLAB offers a better alternativeby adding the term varargin as the function function'slast input argument.In addition,every tirne the passedfunction is invoked within the function function,the term vararsin{: } shouldbe addedto the end of its argumentlist (notethe curly brackets).Here is how both modificationscan be implemented for conciseness): fbr f uncavg' (clmittingcomnrent.s
ffi= ,t';rrA {
=1_h"
x = "L zJ
Zl F+z
I
3.6 CASESTUDY
7l
function favg = funcavg (f, a, b, n, varargin) x - linspace(a,b,n); y = f(x,varargin{:}); f avg = mean(Y) ; When the passed function is defined, the actual parameters should be added at the end of the argument list. If we used an anonymous function, this can be done as in .
vol=t?(r,m,edt
SqrL(q.Bl*m
edr*Lanh(sqrr_r9.Bl"cd
ml*Lr;
When all these changes have been made, analyzing different parameters becomes easy. To implement the case where m : 68.1 and c1 - 0.25, we could enter >> funcavg(ve1, 0,12, 64, 68. 1, 0.25) ans = 36 . Ar21 An altemative case, say m : c h a n g i n gt h e u r g u m e n t s :
100 and cd :
0.28, could be rapidly generated by merely
>> funcavg(ve1, 4,f2, 60, 100, 0.28)
38.9345
BUNGEEJUMPER VELOCITY ) t I
Bockground. In this section,we will use MATLAB to solve the free-falling bungee jumper problem we posedat the beginningof this chapter.This involves obtaininga solution of
1
du cd .t -:8--UaIm
) e
Recall that, given an initial conditionfor time and velocity,the problem involved iteratively solving the formula, U i + l: u 1 *
v)n its ed
du' -At AI
Now also remember that to attain good accuracy,we would employ small steps.Therefore, we would probably want to apply the formula repeatedlyto step out from our initial time to attain the value at the final time. Consequently,an algorithm to solve the problem would be basedon a loop. Solution. Suppose that we started the computation at / = 0 and wanted to predict velocity at t : 12 s using a time stepof A/ : 0.5 s. We would thereforeneedto apply the iterative equation 24 times-that is,
t2
n---24 0.5
ffi q
,-ILJ' ?,tr+
b
{e[ --{
st -r
Z] X+
"L x 4
3$ 8,
72
PROGRAMMING WITHMATLAB
continued where n : the number of iterations of the loop. Becausethis result is exact (i.e., the ratio is an integer),we can usea for loop as the basisfor the algorithm.Here'san M-file to do this including a subfunction defining the differential equation: f uncLion vend = velocityl % % Z Z % % % Z % t % L 1,/
(dt,
t,i, rf , vi )
vel.ocityl: Iiuler solutj.on for bungee velocity vend = veloc-Ltyl (dt, ti, Lf , v1) [.uler uret]rod soluLion of bungerj u m p e ] r z eL o c i t y rnpuL: dL ,= t.rme step (s) ti = initi.al Lime (s) Lt = f,in.rl time (s) (rn/s) vi = initial val.ue of dependent vari.rble outDut: (n/s) v e n d = v e l . o c . iL v a t t f = Li; =
Vl,'
n =
(tf
Far
i
-
ti)
/
dL;
1.n
dvdt = deriv(v) ; v : v + dvdt * dt; t = t + dt; end Varnd = V,' end f unct ion d.; - deri-z (v) dv = 9.BL - (A.25 / 68.1) end
x v''2;
This function can be invoked from the command window with the result: >> vel,oci Lyl (0.5 ,0 , 1?,,l)) dns = 50.9259 Note that the true value obtained from the analytical solution is 50.6175 (Example 3.1). We can then try a much smaller value of dt to obtain a more accurate numerical result: > > v e . o c r i . r - y (10 . 0 0 1 , 0 , 1 2 , O )
50.6i.Bi
Althoughthis functionis certainlysimpleto program,it is not foolproof.In particular,it will not work if '.hecomputation intervalis not evenlydivisibleby thetime step. To cover such cases,a while . break loop can be substitutedin place of the shadedarea(notethatwe haveomittedthecomments for conciseness):
ffi= zt+
,f -\
= 4F+
\'r?+,
73
3.6 CASESTUDY
continued vend = velociLy2
function t = ti; ] - \
-
(dt,
t1,
t;
end
tf,
vi )
^ f .
while (1) if t + dt > tf, dvdt = deriv(v) v=v+dvdt*h; -
1-
t
r
if t >end vend = v; end
h = tf ;
-
h.
tf,
break,
end
f unct ion clv = deriv (v ) - 9.81. - (0.25 / 68.L) end
dv
*
v""2;
As soon as we enterthe while loop, we usea singleline if structureto test whether adding t + dt will take us beyondthe end of the interval.If not (which would usually be the caseat first), we do nothing. If so, we would shortenup the interval-that is, we set the variablesteph to the intervalremaining:tf * t. By doing this, we guaranteethat the last stepfalls exactly on t f . After we implementthis final step,the loop will terminatebecause the condition t >= tf will test true. Notice that before entering the loop, we assignthe value of the time step dt to another variable h. We create this dummy variable so that our routine does not change the given value of dt if and when we shortenthe time step.We do this in anticipationthat we might need to use the original value of dt somewhereelse in the event that this code were integrated within a larger program. If we run this new version,the result will be the sameas for the versionbasedon the for structure: ; ' > v e l o c i L y 2 ( 0 . 5 , 0 , l : 1, 0 )
5A.9 2\9 Further, we can use a dt that is not evenly divisible into tf :'-- velociLy2
-
ci:
(0. 35, A, 12, A)
50 - 834r1 We should note that the algorithm is still not foolproof. For example, the user could havemistakenlyenteredast3psizegreaterthanttrecalculationinterval(e.g., tf - ti = 5 and dt = 2 0). Thus, you might want to include error traps in your code to catch such errors
e
and then allow the user to correct the mistake.
ffi ,"I LJ' -b
ffii --I
\
x
<. [h
^r-
sJ 5t
^l
Z] X+
3$&
74
PROGRAMMI NG WITHMATLAB
PROB
continued As a final note, we should recognize that the foregoing code is not generic. That is, we have designed it to solve the specific problem of the velocity of the bungee jumper. A more generic version can be developed as f L r n c - ' t i o ny e n d = , o d e s i r n p ( d y d r , C t , y = yi.; h = dt; L - ti; (1) whiIe j 1' t + c]t- ; Lf, h =, t-f L; t.rrd .t -./ + dyclL(1) * I'ii f
ts
-
t- '=
i{
t i,
t,f , y j )
break,
cos"r: I -
Create an M-file to ir putes and displays thr series is added. In o sequencethe valuesfc 'l COS.r: x2 cos-r: I -
h.
tf,
3.1 The cosine funct infinite series:
r:nd
t
c-nrl = Y'
Yenrl
-2 COS,f :
Notice how we have strippedout the parts of the algorithm that were specific to the bungeeexample(including the subfunctiondefining the differentialequation)while keeping the essentialfeaturesof the solutiontechnique.We can thenusethis routineto solvethe bungeejumper example,by specifyingthe differentialequationwith an anonymousfunction and passingits function handleto odesimp to generatethe solution - - . r l . r < . l t i. a ( v )
q . I l . - \ ( . ) .2 \ / 6 f . r . L ) * u " " - ' . o r , l e : r 1m p ( c 1 v c 1 l ,(,r . l , a r , l : l . , l l )
,.: ,lI
We could then analyzea different function without having to go in and modify the M-file. For example,if .y: lQ al t :0, the differentialequationdy/dt : -0. l.y hasthe analytical solutionjy : l0e-0 r'. Therefore,the solutionat t : 5 would be y(5) : l0e*0 r('5) 6.0653.We can use oclesimp to obtainthe sameresultnumericallyas in ' ' : . o d . : i i n r r ,( 0 ( y )
-0.
11.
up to the order term o ceding, compute and d true - se
-
'r(l-92b'r
l * y , l l . i J { . ) ' :(,1 , ! , . 1{ r )
;*
:
"/oeffOr
-
ll
I -
As a test case,employ up to and including e x t o1 1 4 r . . 3.2 An amountof mon interest is compounde worth F yielded at an determined from the fo F:P(l*i)"
Write an M-file that wi vestment for each year function should includt rate i (as a decimal),an future worth is to be ca a table with headingsa gramforP:$100,00( 3.3 Economic formulr payments for loans. Su money P and agree to interest rate of i. The ment A is
b.0i,4'l
i/ | a;\r
A-
ffi
,*ufleqqte zr+
--
zl F+
x
-a,th -r
jl€ - :l 5t 3$ &
P
"'
"t
(1+i),-t
PROBLEMS
75
PROBtEMS we Dre
3.1 The cosinefunction can be evaluatedby the following rnfinite series: x2 .t4 c o s . r : I - -) t + | | Createan M-file to irnplementthis folrrrula so that it computesand displaysthe values of cos x as each term in the seriesis added. In other words, compute and display in sequence the valuestbr C O s - r :I cos-r: I cos.r: I -
the
y the ) anal5t
+-
where Z,u""u: the averageannual temperaturc.In.,,s: thc peaktcmpelaturc,o : thc frequencyof thc annualvariation (:2n1365). and tn"^r.: day ol' the pcak tempcrature (=205 d). Paramcters fbr someU.S. towns are listcd here:
City
4l
upto thc orderterm ol'youl choosing.For eachol-thc preceding,computeand display the perccntrelativeerror as
T.error:
7 : 4."^n * (Q"ar.-- I."nn) cos((,(/ - lp"ol))
x4
,
:
-"Y
) the [nc-
2l
Write an M-file to compLrte A. Test it with P : $55,000and an interestrateof 6.6% (l : 0.066).Conlpu(eresultsftrr l : 1 , 2 , 3 , 4 , a n d5 a n dd i s p l a yt h er e s u l t sa s a t a b l ew i t h h e a d ings and columnsfbr n and A. 3.4 The averagedaily temperaturefor an area can be approxirnaledby the followingfunction:
T-.on ("C)
T"""r ("C)
221 231 52 t0 6 la7
283 336 221 )7.6 229
AAiomi, FI Yumo,AZ B i s m o r cN k ,D Seoiile,WA Boston,MA
lrue - senesapproxrmatlon
^ 1007,
true
Dcvelop an M-file that cornputesthe averagctcmperature two claysol' thc year {irr a particularcity. Tcst it hetrvcen As a testcase.employ the prograrnto corllputecos(L5) fbr (a) in Miarni. FL 1l : 0 to -59)and lirr January-Fcbruary is, tcrm up to and including eight tcnrs-that up to thc (b) JLrlyAugusttcnrpcraturc in Boston,MA (t: 180to 242). r'* f141.. 3.2 An amountol rnoneyP is investcdin an accountwherc 3.5 Figurc P3.-5shows a cylindrical tank with a conical inlerest is conrpoundcd at thc cnd ol'thc pcriod.Tlrc luturc' basc.lf'the liquid levcl is quitc low, in the conicalpart.thc yielded worthF at an intercsl ratc I aficr n periodsrnay be v o l u m ei s s i m p l yt h e c o n i c a vl o l u m eo l ' l i q u i d .l f t h e l i q u i d level is rnidrangcin the cylindricalpart,thc totrl volLrncol' liom thc lbllowing fbrmula: deternrined liqLridincludcsthc fillcclconicalpart and thc partiallytillcd F=P(l+i)" c y l i n d r i c apl a r t . Usc dccisionalstructures to writc an M-f ile to compule WriteanM-filc that will calculatethe futurcworth of an inthe tank'svolumc as a flnction ol'given valucsof'R and 11. vestment lbr each ycar lrorn I through rr. The input t0 the Desisnthc tunctionso thatit rcturnsthe volurncfirr all cascs function shouldincludethe initial investmentP, thc interest rateI (asa decinral),and the numbcrof yearsn firr which the worthis to be calculatcd.The output shouldconsistof future atablervithhcadingsand columnsfirr n and F. Run the progtamfor P : $100,000,t : 0.06,and n : 7 years. 3.3Economictbrmulas are available to compute-annual payments for loans.Supposethat you borrow an amountof money P and agreeto repay it in n annrralpaymentsat an inlerest rateol l. The tblmula to computc the annual payment A is
NT
t-p
W
iil + l)" (l+i),-l
FIGUREP3.5
ffiE\ zJLJt -&
+,
ZN 4Y
-1 st
4
fi:-
sp 5t
l -
3$8
76
PROGRAMMING WITHMATLAB
where the depth is less than 3R. Return an error message The difliculty arisesfor the other cases.The following table ("Overtop") iI you overtopthe tank-that is, d > 3R. Test it summarizesthe possibilities: with the lbllowine data: xy0 R d
1.5 I
t5 2
t5 45
t<
<0 <0 <0 :0 :0 :0
>0 <0 :0 >0 <0 :0
ton l"-,zxl * r loo lr,/.rJ- 1T T
rl2 -n /2 0
Note that the tank's radiusis R. 3.6 Two distancesare required to specify the location of a point relative to an origin in two-dimensionalspace (Fig. P3.6): Write a well-structuredM-file to calculater and 0 as a func. The horizontal and vertical distances(-r, y) in Cartesian tion of -r and_r'.Expressthe final resultsfor d in dcgrees.Test your programby evaluatingthc fbllowingcases: coordinates. . Thc radiusand angle(r, d) in polarcoordinates. xJr| It is relatively straightfbrwardto computeCartesiancoordinates(-r,.y) on the basis of polar coordinatcs(r,6). The reverseprocessis not.sosirnple.The radiuscan be cornputcd by the fbllowing lbrmula:
where U: velocity(m cocfficient,B: width ing data is availablefo
0.035 0 020 o 0t5 0 030 o 422
,:l*?-+v, If the coorilinateslic within the first and lburth coordinates(i.e.,x > 0), then a sirnplelbrmula can bc uscd to compute9: rl : tan I (I) \_r,/
0 0
0.0
OO 00t 0.0 00(
I
3.7 Dcvelop an M-filcr to detcrmine polar coordinatesas descrihedin Prob. 3.6. However, rathcr lhan designingthe function to cvaluatea single casg,passvectorsof ir and). Havethe f unctiondisplaythe resultsas a tablewith columns forx,.y, r, and().Testthe prograrnfbr the casesoutlinedin Prob.3.6. 3.11Dcvelop an M-lllc function that is passeda numenc grade liom 0 to 100 and returnsa lettcr grade accordingto the schcmc:
Write an M-file that cor channels.Enter thesev umn represents a param nel. Have the M-file di computedvelocityin ta colurnn.Includeheadin 3.10 A simply suppo F i g . P 3 . 1 0 .U s i n g s i n g along the heamcan be e
-0 Criterio
A B C D
+ 7 5 ( x- 7
90 < BO 1 /O . 60 <
n u m e r i co r o d e < l 0 C n u m e r i cg r o d e < 9 0 n u m e r l cq r o d e . B O n u m e r i cc r o d e < / 0 n u m e r i cg r o d e < 6 0
F
By dcfinition,the sing follows: (x-tt)":lt*-o l0
to computethe velocity 3.9 Manning'scquationcan be r.rsed of water in a rectangularopen channel:
FIGUREP3.6
Jsr
DevelopanM-filethat< distance alongthebeam the beam.
BH \2'
il \.8+2H/
ffi= n = ,f 4+-4-n-vI -\ ?,t'+ +Y
3l x+
<, - [h
jl€ - sJ 5t I
77
PROBLEMS
20 kips/ft
:-c
F I G U RP E3 . I O
where cylinder U: vclocity(rn/s),S: channclslope,n - roughness 3.1l ThcvolumcV of liquidin a hollowhorizontal r andlcngthL is related to thedeptho1'theliquid B: width (m), and H : dcpth(rn).The tbllow- ol'radius coefficient. hbv lbr llve channels: ingdatais availahle
l-, ,/r V-lt-cor'[L \r
n
0035 0020 00r5 0030 0422
r0
0 000t 0 0002 0 0010 0 000/ 0 0003
B 20 24 t5
hr | /
rr
--l htr2 / rlt h,IlI
Dcvclopan M-fllc to cl'ciltea plot ol'volume versusdepth. Tcst thc program l-orr - 2 rn and L: -5m. 3.12 Dcvclop a vectorizedvclsion ol'thc lbllowing codc: 1,:ildrL=U;
tc,ncl-ilfl; rri
5;
L ( L) -ts1,.t1L;
Write anM-filcthatconrputcsthc vclocitylirl cachol'thcse Enterthesevaluesinto a matrix whcrc cach colchannels. umn reprcscnts a prramctcrand cach row rcprcscntsa channel. Havethc M-l'ilcdisplaythc input data along with thc c l r t ( l vclocityin tabularfirm whcrc velocity is the l'ilth computed 3.ll Thc "dividc and average"method,an old-timcmcthod lncludchcadingson thc tablcto lahcl thc colurnns. column. lirr approximating thc squareroot ol'any positivcnumbcra, 3,10A simply supportcdbcant is loadcd as shown in can bc firrrnulatcdas Fig.P3.10.Using singularityf'unctions.thc displacerrncnt .r + ulx along thebcamcan bc cxprcsscdhy thc ccpration: 2 -s t5 ' Writc a wcll-structurcd M-l'ilc lunction bascd on a :'*l . . (.r x) lrr.r)--l'r-0)'-(r oo w h i l - - . . . b | e a k l o o p s t r u c t u r ct o i r n p l c m c n t h i s a l g o 57 r rithur.Use ploper indentationso that thc structnrcis clcar.At '\ 23tt.25 t * ? 5 ( r 7 ) r+ clrch\tcp c\tinrirtcthc crrrrrin y,rurirpprtrrillluli()n irs 6 t._,,,1 Bydetinition, thc singularitylunctioncan be cxprcsscdas " - t -I t r n r u ' l follows: Rcpeatthe loop until a is lessthan or cqual to a spccified (. rtl" when,r > a value.Designyour programso that it returnsboth thc rcsult ,, _,,,,, _ J I tt and the errur. Makc surc that it can cvaluatethc squareroot when -r < a I I o1'nunrbers thatarc equalto and lcssthanzero.For the latter versus casc,displaythe resultas an imaginarynumber.For examDevelop anM-filc thatcrcatcsa plot ol-displacemenl 21.Test your program distance alongthe beam,,r. Notc that,t : 0 at the leli end ol' plc, thc squareroot of -4 would retr.rrn b y e v a l u a t i nug- 0 , 2 , 4 , a n d- 9 f b r c : I x l 0 r . thebeam. .r/lr
^
ffi q A b
,"I LJ' zl'+
-._
tt
.\t'ltl
st
3l x+
Z} z,J I
PROGRAMMING WITHMATLAB
78
3,14 Piecewisefunctions are sometimesuseful when the relationshipbetweena dependentand an independentvariable cannot be adequatelyrepresentedby a single equation.For example,the velocity of a rocket might be describedby l1t2-5t ll00-5/ 50t+2(t-20)2 l52oe 0.2(t10) 0
0 30
w h e r em o : t h e m o n t h ( l - 1 2 ) , d a : t h e d a y ( l - 3 1 ) , a n d leap : (0 for non-leap year and I for leap year).Test it for Januaryl, 1999,February29,2000,March 1, 2001,June21, 2002, and December31, 2004. Hint: A nice way to do this combinesthe for and the switch structures. 3.17 Develop an M-file function to determinc the elapsed days in a year.The first line of the function should be set up as function nd = days(mo, da, year)
othcrwise
w h e r ei r o : t h e m o n t h( l - 1 2 ) , d a : t h e d a y ( l - 3 l ) , a n d year : the year.Test it for January l,1999, February29, 2000,March l, 2001,June21,2002,andDecember31,2004. 3.18 Developa lunctionfunctionM-flle thatreturnsthe differencebetweenthc passcdiunction's maximum and minimum value givcn a range of the independentvariable. In addition,havethe lunctiongenerate a plot of the functionfbr the range.Test it for the following cases: function xr = rounder(x, n) ( a ) . l ( 0 : l 0 e - 0 2 s ' s i n- ( l4 ) f r o mt : o t o 6 T . Testthe prograrnby roundingcachof thc fbllowing to 2 dec- (b) /(x) : es'sin(l/x) liom -r : 0.01 to 0.2. -0. 125, inraldigits:r : 467.9587, 467.958'7,0.125, 0. 13,5, (c) The built-inhumpslunclionliom x : 0 to 3. a n d- 0 . 1 3 5 . 3.19 Modily the functionfunction odesimp developedat -1.16 Develop an M-filc function to dctcrmine the elapsed the end ol'Sec. 3.6 so that it can bc passcdthc argumcntsof days in a year. The f'irst linc ol' thc function should be set the passedlunction.Tcstit lbr thc lbllowing casc: up as Develop an M-flle function to compute u as a function of /. Then, use this lunction to gcneratea plot of u versus/ fbr t5to50. 3.15 Developan M-file lunctioncallcdrounder to rounda number-r to a specifiednumberof dccimal digits,n. The first line of the f'unctionshouldbe setup as
(c<1/m)*v^2; d v c l t - @ ( v , m , c : c 1) 9 . B L :':' oclesirnp (dvtlt , 0. 5, 0, L2 ,0 , tt9.I, A .25) >>
f uncrt i.rn
nd
=
days
(mo,
_q
zt'+
,"1{t
cla,
le.rp)
ffir
+,
71--syst -.t
^l
Z] X+
3$8
.
;.1ffi
""j
'rt6;"',,1li,ii
: i irl:r 1,,1
."lii-: i
i 1 ) ,a n d :st it for I u n e2 1 , do this
Roundoff Errors ond Truncotion
elapsed J be set
ll), and
nry 29, t,2001. ; the difrd miniiable. In ction for
CHAPTER OBJECTIVES The primaryobjectiveof this chapteris to acquaintyou with the major sourcesof errorsinvolvedin nurnericalrnethods. Speciticobjectivesand topicscoveredare
:loped at mentsof
' ' ' '
r)
' ' ' t '
Understanding the distinctionbetweenaccuracyand precision. Learninghow to quantityerror. Learninghow errorcstimatescan be usedto decidewhen to terminatean iterrti','e calculation. Understanding how roundofferrorsoccllrbccausedigitalcornputers havea limitedahility to represent nunrbers. Understanding why lloating-pointnumbershave[irnitson their rangeand precision. Recognizingthat truncationerrorsoccurwhenexactmathematical fbrmulations ilrc represcnlcd by approxirnations. Knowing how to uscthe Taylorseliesto estimatetruncationerrors. Understanding how to write lbrwarcl,backward.and ccnteredfinite-difTerence approximations of first and seconcl derivatives. R e c o g n i z i ntgh a te l ' f i r r tlso r n i n i m i z el " r u n c r r t i e o rnr o r sc a ns o m e l i m eisn c r c a s e roundofTerrors.
YOU'VE GOT A PROBLEM n Chap. I, you developeda numericalmodel for the velocity of a bungeejumper.To solve the problemwith a computer,you had to approximatethe derivativeof velocity with a llnite difl-erence. !]=!!dr Lt
u(ti+r)-u(ri) t,+t - ti
79
80
R O U N D O FA F N D T R U N C A T I OENR R O R S
Thus, the resultingsolution is not exact-that is, it has error. In addition,the computeryou use to obtain the solutionis also an imperfecttool. Bethe magnitudes causeit is a digitaldevice,the computeris limitedin its ability to represent precision yields Consequently, the itself and of numbers. machine resultsthatcontainenor. your your both mathenlatical digital So approxir.nation and computercauseyour lrprediction problem How rnodel to Your is: sulting be uncertain. do you deal with suchunyou introduces to sonle and certainty'?This chapter approaches conceptsthat engineersand with this scientistsuseto deal dilemma.
4.I
ERRORS constantlyfind themselves havingto arccomplish objectivesbased i Engineersand scientists goal, perf'ection rarely if ever atis a laudable it is on uncertaininformation.Although For tact that the developed liom Newton's second law tained. rnodel example,despitethe jumper's never practice predict fall. is an excellentapproxirnation, it would in cxactly the would resultin deA varietyof factorssuchas windsand slightvariationsin air rcsistance high or low, thenwe viationsftom the prediction.If thesedeviationsare systematically new rzrndomly might needto devclopa modcl.However,il-theyare distributedandtightly groupedaroundthe prediction,thenthe deviationsrnightbe considered negligibleandthe Numerical introduce similar discrepancies model deemedadequate. approxirlationsalso i n t o t h ea n a l y s i s . quantification, This chaptercovcrsbasictopicsrclatedto tlreidentification, and rninimizaticlnof theseerrors.Ccneralinformationconcernedwith the quantificationol'erroris reviewedin this section.l-his is fbllowed by Sections1.2 and4.3, dealingwith the two (dueto computerapproximations) andtrunmajorfirrmsof numericalerror:rclundof'l'crror (due We alsodescribehow strategies to recationerlor to mathenraticalapproximations). roundolT. we increase Finally, briefly discusscrrorsnot duce truncationerror sometinres Thess includeblundcrs,rnodel directlyconnectedwith the nurnericalnrethodsthemselves. errors.and datauncertaintv.
I
I i I
4.1.1 Accurocy ond Precision with The errorsassociated with both calculations can be characterized arrdmeasuremenls rcl'ersto how closelya corlputedor mearegardto theiraccuracyand precision.Att'unrc'.v ref'ersto how closelyindividuitlcomputed suredvalueagreeswith thetruevalue.Prct'1.\'iorr or measuredvaluesagreewith sachother. Theseconceptscan be illustratedgraphicallyusing an analogyfiom targetpractice. The bulletholeson eachtargetin Fig.4. l can be thoughtof as the predictionsof a numerical technique,whereasthe bull's-eyerepresents the truth.Inaccurucy(alsocalledbirr.i)is defirreclas systematicdeviationfrom the truth.Thus, althoughthe shotsin Fig. 4.lc are more tightly groupedthan in Fig.4.la, thc two casesarc equallybiasedbecausetheyate both cerrteredon the upperleft quadrantof thc target.Imltrecision(alsocalledurrc:ertaintl), on the other hand,ref'ersto the magnitudeo1'thescatter.Therefore,althoughFig. 4. lD and d are eqLrallyaccurate(i.e..centeredon thc bull's-eye),the latteris more precisebecause the shotsaretightly grouped.
I e a
to
w L(
rh
8r
4.I ERRORS
Increasingaccuracy
ol.Beritudes n errol'. our rerch unersand o 0)
a
s based )ver atrnd law :r'sfall. It in delhen we I tightly and the :pancies r c lm i n i 'error is the two s to reors not . rlodel
o)
FIGURE 4.I i l l u s t r o t i nt hg e c o n c e p l so l o c c u r o c yo n d p r e c i so n . frommorksmonship Anexomple ( o ) i n o c c u r o t cen d r m p r e c i s e( ,b l o c c u r o t eo n c ii n r p r e c i s e( c, Ji n c c c u r o t o end precise, n - d { r J )o ( ' L r o ' ee r ', J p e c : s e .
r\umerrcalmethoclssnoulcloe sufiicrenrlyaccura("e or unorused("omeet irc requtreparticular problem.They also shouldbe preciseenoughfbr adequatedesign. nrentsof a In tlris book, we will use thc collectiveterm errur to representboth the inaccuracyand implccision t l l 'o u r p r e d i ci to n s .
4.1.2 Error Definitions ed with or rneamputed rractice. numerbia.s)is 4.I c' are they are rtainty), l.l b and because
Numericalerrorsariscfrom the useof approximations to rL'present exactmathematical operationsand quantities.For sucherrors,the relationshipbetweenthe exact,or true,result and thc approximation carrbe lbrmulatedas True value: approximation f error
( 4 .l )
By rearrangingEq.(4. l), we find that the numericalerror is equal to the discrepancy b e t w e e nt h el r u t hi t n dt h eu p p r o x i m a l i o n a .s i n 6 r : t r u ev a l u e- u p p r o x i m a t i o n
(4.21
whereCr is usedto designate theexactvalueof theerror.The subscriptt is includedto designatethat this is the "true" error.This is in contrastto other cases,as describedshortly. wherean "approximate"estimateof the enor must be employed.Note that the true error is commonly expressedas an absolutevalue and referredto as the absoLuteerror. A shortcomingof this definitionis that it takesno accrount of the order of magnitudeof thevalueunderexarnination. For example,an errorof a centimeter is muchrnoresignificant
82
ERRORS ROUNDOFF AND TRUNCATION
4.
if we are measuringa rivet than a bridge. One way to accountfor the magnitudesof the quantitiesbeing evaluatedis to normalizethe error to the true value,as ir-
wl be pe to
True fractionalrelativeerror :
true value - approxlmatlon true value
The relativeerrorcan alsobe multipliedbv 100%,toexpressit as
Th to aln
true value - approximation cl
-
true value
1004/t
where e, designatesthe true percentrelativeerror. For example,supposethat you havethe taskof measuringthe lengthsof a bridgeand If the true valuesare 10.000and rivet and come up with 9999 and 9 cm, respectively. a 10 cm, respectively, the enor in both casesis I cm. However.their percentrelativeerrors Thus,althoughbothnteacanbe computedusingEq. (4.3) as0.0I 7oand l0olo,respectively. for rivet is much greater.We the 1 cm, the relative error error of surcmentshavean absolute job measuringthe bridge. of probably that we have done an adequate wor.rlcl conclude to be desired. for rive( leaves sornetl.ring whereasour estimate the with a 1 to signifythatthe Noticethat for Eqs.(4.2)and (4.3),t and e aresubscripted rivet For exanrple of the and the bridge,we wercprothe value. the error is basedon true such infbrrnation is rarcly available. in situations with this value. Howcvcr, actual vided wlren we deal with functionsthat value will only be known nrethods, the true For nurnerical when we investigate the theothe will typically bc case can be solvedanalytically.Such in real-world apHowever, particular fbr simple systerns. of a techniqLre reticalbehavi<,rr plications,we will obviouslynot know the true answcra priori. For thesesiturtions.an alternativeis to norrnalizetheerrorusirrgthebestavailableestimateof thetrue value-that i s , t o t h ea p p r o x i m a t i oint s e l f ,a s i n error' approxrmate l00a/( approximation wherethe subscripta signifiesthat the error is norrnalizedto an approximatevalue.Note Eq. (4.2)cannotbe usedto !:alculatetheerrttrtermin alsothatfbr real-worldapplications, the nunteratorof Eq. (4.4).One of the challengesof numericalrnethodsis to determine crror estimatesin the absenceof knowledgeregardiugthe true value.For example,certain nunrericalmethodsuseiteratiotrto computcanswers.In suchcases,a presentapproximaThis processis performedrepeattion is rnadeon the basisof a previousapproximation. compute(hopefully)betterand betterapproximations. edly,or iteratively,to successively For suchcases.the error is often estimatedas the dil'lerencebetweenthe previousandpreto Thus,percentrelativeerror is determinedaccorcling sentapproximations. - previousapproximation presentapproximation l\OC/t presentapproxrmatlon
( 45 )
This and other approachesfbr expressingerrors is elaboratedon in subsequentchapters. The signsof Eqs.(4.2) through(4.5) may be either positiveor negative.lf the approx' is greaterthanthecurim:rtionis greaterthanthe truevalue(or thepreviousapproximation rent approximation),the error is negative:if the approximationis lessthan the true value, may be lessthanzero, the error is positive.Also, for Eqs.(4.3) to (4.-5),the denonrinator
pro
can
EXAMPLE 4.1
Err
Pro ries
Thu esti
^0.5 c .1
: '
with term fied t
JOIU
resu I
Thui tima
or fc
Thir
4.I ERRORS
which can also lead to a negativeerror. Often, when performing computations.we may not be concemedwith the sign of the eror but areinterestedin whetherthe absolutevalueof the percentrelativeerror is lower than a prespecifiedtoleruncee.. Theretbre,it is often useful to employ the absolutevalue of Eq. (4.5). For suchcases,the computationis repeateduntil (4.6)
I r , ,I < € . ,
This relationshipis refered Io as a stop1'ting criterion.If it is satisfied. our resultis assumed to be within the prespecifiedacceptablelevel o.. Note that for the remainderof this text, we almostalways employ absolutevalueswhen using relativeerrors. It is alsoconvenientto relatetheseerrorsto the numberof significantfiguresin the approximation.It can be shown(Scarborough. 1966)thatif the followingcriterionis met,we can be assuredthatthe resultis correctto at leastrr significantfigures. (4.7)
e.,:(0.5 xl02-"1%o
EXAMPLE 4 E r r o rE s t i m o t e fso r l t e r o t i v eM e t h o d s Problem Stotement. In mathematics, functionscan otien be represented by infinite series.For exanrple,the exponentialfunctioncan be computedusing
e':l+.r+-'
,\'l 2
.rl
F:+.. -l!
1
.r"
(E4.l.l)
nt
'Ihus,
as more terms are addedin sequence,the apploxirnationbecomesa betterand better c'slimateof the true valueof e'. Equation(E4.L l) is calleda Moclourinserit,,s a.rput,sion. Startingwith the shnplestversion.e' : l, add termsone at a time in orderto estimate e" 5.After eachnew term is added,computethe true and approximatepercentrelativeerrors with Eqs.(4.3)and (4.-5), respectively. Notethatthe truevalueis e" 5 - |.648121. . . . Add termsuntil the absolutevalueof the approximateen'orestimatec,, falls below a prespecified errorcriterions.rcontbrmingto threesignil'icant figures. S o l u t i o n . F i r s t , E q .( . 4 . 1 ) c a n b e e m p l o y e d t o d e t e r m i n e t h e e r r o r c r i t e r i o n t h a t e n s u r e s a resultthat is correctto at leastthreesignificantfigures: €. : (0.-5x 102 t)"/n:0.05a/,,
Thus,we will add termsto the seriesuntil e,,falls belowthis level. The first cstimateis simplyequalto Eq. (E4.| . l) with a singleterm.Thus,the first est i r t i a t e i s e q utaclll . T h c s e c o n d e s t i l n a l . e i s t h e n g e n e l a t e d b y a d d i n g t h e s e c o n d t e r m a s i n et:
l*x
or fbr,r : 0.5 e o 5- I * 0 . 5 :
1.5
This represents a true percentrelativeenor of tEq. (a.3)l
.^_l ' _ |
r . 6 " 1 8 7-2 1t . . sI I x 1001,=9.02(/t t.64812t I
ERRORS ROUNDOFF AND TRUNCATION
84
Equation(4.5) can be usedto determinean approximateestimateof the error, as in
I b
t15-ll
:.1.1.3(Z r., : l-j_=_-l x 100(/o I
l.)
|
S
n
by Becauseen is not lessthanthe requiredvalueof e,, we would continuethe computation adding anotherterm,x212r.,and repeatingthe error calculations.The processis continued until le,,l < e,. The entire computationcan be summarizedas Ierms
I 2 3 4 5 6
Result 5 625 645833333 648437544 6486979)7
393 942 I44 a t75 a at72 000142
3 33 769 I27 0 t58 00t58
andthe errorfalls below e, : 0.05olo, Thus, after six termsare includcd,the approxirnate computationis terminated.Howcver,noticethat,ratherthan threesignificantfigures,the tbr this case,both Eqs.(4.5) and(4.1) areconresultis accurateto five! This is because, servative.That is, they ensurethat the resultis at leastas good as they specify.Although, this is not alwaysthe casclbr Eq. (4.-5),it is true mostof thc time.
4.2
ROUNDOFFERRORS Rountlo.ff'errur,s arisebecausedigital computerscannotrcpresentsomequantitiesexactly. and scicntificproblernsolvingbecausethey canleadto They are irnportantto engineering erroneousresults.In ccrtaincases,they can actuallylcad to a calculationgoing unstable are saidto be ill-conditioned, and yiefdingobviouslyerroneousresults.Suchcalculations thataredifTicultto detect. Worsestill, thcy can lcad to subtlerdiscrepancies Thcreare two major facetsof nrundoll'crrorsinvolvedin numericalcalculations: 1. 2.
D,
nl di nl tu
numben. Digital computershavesizeandprecisionlimits on theirability to represent are highly sensitiveto roundofTerrors.This canreCertainnumericalmanipulations as well as fiorn the way in which computsult liorn both rnathematical considerations e'rsperl'ormarithmeticoperilti()ns.
4.2.1 Computer Number Representotion Numerical roundofTerrorsare directly relatedto the mannerin which numbersarestoled in a computer.The fundamentalunit wherebyinfbrmation is representedis calleda word, This is an entity that consistscf a string of binary digil.l, or bits. Numbers are typically storedin one or more words.To understandhow this is accomplished,we must first revierry somematerialrelatedto numbersystems.
TI aft
w( ou 0c ele or po (0
Int res put the mai sen
The inF
repr mair
FIGI Thet mog
rT
lll
IL
I Sign
4.2 ROUNDOFF ERRORS
85
Anumber systemis merely a conventionfor representingquantities.Becausewe have l0 fingers and l0 toes,the numbersystemthat we are most familiar with is the decimal,or base-10,number system.A baseis the number used as the ref-erence fbr constructingthe s y s t e m . T h e b a s e - l 0 s y s t e m u s els0tdhieg i t s - O , 1 , 2 , 3 , 4 , 5 , 6 , 1 , 8 , a n d 9 - t o r e p r e s e n t numbers.By themselves,thesedigits are satisfactoryfor countingfrom 0 to 9. For largerquantities,combinations of thesebasicdigits are used,with the positionor place vuluespecifyingthe magnitude.The rightmostdigit in a whole numberrepresents a numberfrom 0 to 9. The seconddigit fiom the right representsa multiple of 10.The third digit from the right represents a multipleof 100and so on. For example,if we have the number8642.9,thenwe haveeightgroupsof 1000,six groupsof 100,four groupsof 10, two groupsof l, and nine groupsof 0. l, or ( 8 x 1 0 3 ) + ( 6 x 1 0 2 ) + ( , 1x l 0 r ) + ( 2 x
100)+(9x l0 ry:9642.9
This type of representationis calledpositionol notatktn. Now, becausethe decimalsystemis so familiar,it is not commonlyrealizedthal there Forexample,if humanbeingshappenedto havceightfingersand toeswe arealternatives. would undoubtedlyhavedevelopedan ot'tul,t>rba.se-8, representation. In the samesense, our fiiend the computeris like a two-fingcrcdanimalwho is limited to two states-either 0 or l. This relatesto the fact that the prirnarylogic unitsof digital computcrsare on/off electroniccomponents.Hence,numberson the computerare rcpresented with a birtary, system.Just as with the decirnalsystcm,quantitiescan be represented or ba"^e-2, using p o s i t i o n anl o t a t i o n F . o r e x a m p l c ,t h c b i n a r y n u r n b e rI 0 l . l i s e q u i v a l e ntto ( l x 2 r ) + ( 0 x 2 l ) * ( l x 2 " )+ ( l x 2 - - t ): 4 + 0 + I + 0 . 5 : 5 . 5 i n t h c d e c i m as ly s t e m . Integer Representotion. Now that we havereviewedhow base-10numberscan be representedin binarylorm, it is simplcto concciveof'how integersarerepresented on a cornputer.The most straightforwardapproach,callcd thc ,signadnngnitrule metfutd,employs the first bit of a word to indicatethe sign,with a 0 fbr positivcanda I firr negative.The remainingbits arc usedto storeths number.Forexample,thc integervalueof 173is represented i n b i n a r ya s l 0 l 0 l l 0 l : ( l 0 l 0 l l 0 l ) , : 2 1+ 2 s+ 2 r + 2 2+ 2 t t : 1 2 8+ 3 2 + I l + 4 + I : ( 1 7 3 ) , , , Therefore,thebinaryequivalentof - 173wouldbe storedon a l6-bit computer,asdepicted in Fig.4.2. If sucha schemeis cmployed,thereclearlyis a lirnitedrangeof integersthat can be represented. Again assurrring a l6-bit word size,if one bit is usedtbr the sign,the l5 remainingbits can represent binaryintegersliom 0 to I I I I I I I I I I I I I I I . The upperlimit can
FIGURE 4.2 T h eb i n o r yr e p r e s e n t o l i oonf t h e d e c i m c li n t e g e r l / 3 o n o l 6 b i t c o m p u t e u r s i n gt h e s i g n e d m o g n i l u d em e t h o d
t 1
Y
I
Sign
0
0
0
0
0
0
01110 Magnitude
1
0
1
0
1
86
ROUNDOFF AND TRUNCATION ERRORS be convertedto adecimalinreger,asin (1 * 2'o)+(1 x 2'3)+. . . + (l x 2') * (l x 20;= 32,161. Note that this valuecan be simply evaluaredas 2rs - l. Thus,a l6-bit computer word can storedecimalintegersrangingfrom -32,761 to 32,767. In addition, becausezero is already defined as 0000000000000000,it is redundant to use the number 1000000000000000 to define a "minus zero." Therefore,it is conventionally employedto representan additionalnegativenurrrber:-32.768, and the rangeis frorn -32.768 to 32,761. For an n-bit word,the rangewould be from -2" to 2" - l. Thus, 32-bit integerswould rangefrom -2,141.183.648 to +2,141,483,641. Note that,althoughit providesa nice way to illustrateour point, the signedmagnitude method is not actually usedto representintegersfor conventionalcomputers.A prefened approachcalled the 2s complemenltechniquedirectly incorporatesthe sign into the number's magnituderatherthanproviding a separatebit to representplus or minus.Regardless, the rangeof numbersis still the sameas for the signedmagnitudemethoddescribedabove, The foregoingservesto illustratehow all digital computersare limited in their capability to representintegers.That is, numbersaboveor below the rangecannotbe represented. A moreseriouslimitationis encountered in the storageandrnanipulation of fractit'rnal quantitiesas describednext. Flooting-PointRepresentotion. Fractionalquantitiesare typically represented in computersusing floating-point.format. In this approach,which is very much like scientific notation,the numberis expressed as *sxb" wheres : the significand,b: the baseof the numbersystembeing used,and e : the cxponent. Priorto beingexpressed in this forrn,the nurnbelis normaliz.ed by movingthedecirnal placeoverso thatonly onesignificantdigit is to thelefi of thedecirnalpoint.This is doneso cornpLlterrtlcmoryis not wustedcln storinguselessnonsignificantzeros.For example,a value like 0.00567ticould be represented in a wastefulmanneras 0.00-5671t x l0('.However,nort.ttalization would yield -5.678,. l0 I which eliminatesthe uselesszeroes. Beforedescribingthe base-2implementation usedon computers,we will first explore the fbndamentalirnplications of suchfloating-pointrepresentation. In particular,whatare the ramificationsof the fact that in order to be storedin the cornputer,both the mantissa and the exponentmustbe limited to a finite numberof bits?As in the nextexample,a nice way to do this is within the contextof our moretirmiliarbase-l0 decimalworld.
EXAMPLE 4.2
l m p l i c o t i o n so f F l o o t i n g - P o i nRte p r e s e n t o t i o n ProblemStotement. Supposethatwe hada hypothetical base-I0 computerwith a 5-digit word size.Assumethatonedigit is usedfbr the sign,two for the exponent,and two forthe mantissa.For simplicity,assumethatone of theexponentdigitsis usedfirr its sign,leaving a singledigit for its magnitude. Solution. A generalrepresentation of the numberfollowing normalizationwcluldbe s 1 d 1 . d x2 l O r o ' d )
tS
i"
:p
i
rq
F o
V
e) 0
of
pl TI or
VA
w( ca
* Th
fio<
4.2 ROUNDOFF ERRORS
)r nt nis 1S,
de ed mrSS,
ve. riled. an-
)mtific
. the :imal ne so rle, a {owplore at are ntissa a nice
87
w h e r e . r o a n d -. st h, e s i g n s ,d , : t h e r n a g n i t u d e o t hf e e x p o n e n t . a d n rda n d d , : t h e m a g nitudeof the significanddigits. Now, let's play with this system.First, what is the largestpossiblepositive quantity that can be represented? Clearly,it would correspondto both signs being positive and all nragnitudedigits setto the largestpossiblevaluein base-10,that is. 9: Largestvalue: +9.9 x l0+Y So the largestpossiblenumberwould be a little lessthan l0 billion. Althoughthis might seemlike a big nunrber,it's reallynot that big. For example,this computerwould be incapableof representing a commonlyusedconstantlike Avogadro'snumber16.022x l0rr.1. In the samesensc,the smallestpossiblepositivenumberwould be '' S m a l l e svt a l u e: * 1 . 0 x l 0 Again, althoughthis value rnighf seempretty snrall,you could nol use it trl representa q u a n t i t yl i k e P l a n c k ' sc o n s t a n(t6 . 6 2 6x l ( ) - r r J . s ) . Similarnegativevaluescouldalsobe developed. The resultingrangesaredisplayedin Fig.4.3. Largepositiveand negativenumbersthat fall outsidethe rangewould causean overflowerrcr. ln a similar sense,for very small quantitiesthereis a "hole" at zero.and very smallquantitieswould usuallybe convertedto zero. Recognizethat the exponentoverwhelrningly determinestheserangelimitations.For exarlple,if we increasethe manfissaby onedigit. the maxirnumvalueincreases slightlyto 9.99 x l0'. In contrast, a one-digitincrease in theexponentraisesthemaximurnby 90 orders of magnitudeto 9.9 x l0e'r! When it comesto precision,however,thesituationis reversed. Whereasthe signil'icand playsa minor role in detiningtherange,it hasa profoundetl'ecton specifyingthe precision. This is dramaticallyillustratedfbr this exanrplewherewe have limited the significandto only 2 digits.As in Fig. 4..1,just as thereis a "hole" at zero,thercarealso"holes"between values. Forexample.a simplerationalnumberwith a finitenurnberof cligitslike 2-5 : 0.0-l125 would haveto bc storedas -1.I x l0- r or 0.03I . Thus.'t rountlofferror is introduced.Fclrthis case,it represcnts a relativeerrorof 0 . 0 3 1 2 --5 0 . 0 3|
0 . 0 31 2 5
: 0.00t1
FIGURE 4.3 5-digit for the eaving
be
T h e n u m b e rl i n es h o w i n gt h e p o s s i b er o n g e sc o r r e s p o n d i ntgo t h e h y p o t h e t i c ob o s e l O f l o o t i n gp o i n ls c h e m ed e s c r i b e di n E x o m p l e4 2 . Minimum
l,,\l 9 . 9x 1 0 s Overflowl--{
Smallest
- 1 . 0x 1 0 s
1 . 0x 1 0 s
,Underflow|-{Overflow " H o l e "a t z e r o
Maximum
9 . 9x 1 0 s
88
ROUNDOFF AND TRUNCATION ERRORS 0.01
0.980.99 1
1. 1
1.2
FIGURE4.4 A smollporiionof thenumberlinecorresponding to thehypotheticol boselO flootingpoinl s c h e m de e s c r i b eidn E x o m p l4e. 2 . I h e n u m b e ri sn d i c o fveo l u e tsh o tc o n b e r e p r e s e n t e d All otherquonlilies follingrnthe"holes"belvreen ihesevolueswouldexhibitsome exoctly. t'rr
rounootterror.
While we could storea numberlike 0.03125 exactlyby expandingthe digits of the significand,quantitieswith infinite digits must alwaysbe approximated. For example, a c o m m o n l y u s e d c o n s t a n t s u c h a s ; r ( : 3 . 1 4 1 - 5 9 . . .h) a wvoeutlocbl e r e p r c s e n t e d a xs 3l 0 . lr ) or 3. l. For this case,it represented a relativeerrorof
- 93 .r 3.1415 3 .l 4 I . 5 9
I
(
F T p
"I
fl
b
: 0 . 0312
e/
F Although adding significanddigits can improve the approximation,such quantitieswill alwayshavesomeroundofTerrorwhen storedin a computer. Anotherrnoresubtleef'fbctof'f'loating-point representation is illustratedby Fig.4.4. Notice how the intervalbetweennumbersincreases as we move befweenordersof magnitude.Iror nurnberswith an exponentof - I (that is, between0.1 and I ;, the spacing is 0.01. Once we crossover into the rangefiom I to 10,the spacingincreasesto 0. l. This meansthat the roundofl-errorof a nurnberwill be proportionalto its rnagnitude.In addrtion, it nreansthat the relativeerrorwill havean upperbound.For tlrisexample.themaxinrum relativeerror would be 0.05.This value is calledthe nuc'hineepsilon(or machine precision).
As illustratedin Exanrple1.2,thefact thatboththeexponentand significandarefinite meansthaf thereareboth rangeand precisionlimits on lloating-pointrepresentation. Now, let us examinehow floating-pointquilntitiesare actuallyrepresented in a real computer usingbase-2or binarynumbers. First,let's look at normalization. Sincebinarynumbersconsistexclusivelyof 0s and ls, a bonusoccurswhen they are nornralized. That is, the bit to the left clf the binarypoint will alwaysbe onel This meansthat this leadingbit doesnot have to be stored.Hence, nonzerobinaryfloating-pointnumberscan be expressed as X(l-tf)x2'' wheref': thennntisso(i.e.,the fractionalpartof the significand). For example.if wenormalizedthebinarynumbell l 0 l. l, theresultwouldbe l. l0 l l x (2)'r or ( l + 0. l0 l l ) x 2-3.
p S
R tl b
u tl tI
T
P "l
4.2 ROUNDOFF ERRORS
89
Signed exponen
Mantissa
1 1b i t s
52 bits
Sign (1 brt)
FIGURE 4.5 Themonner inwhicho flooting-point number isstored inon B-byte wordin IEEE double prec s on formol.
Thus.althoughthe originalnumberhasfive signilicantbits,we only haveto storethe four fractionalbits:0. l0l I . By default, MATLAB has adoptedthe IEEE double-precision.fbnncrt in which eight bytes(64 bits) are usedto representfloating-pointnumbers.As in Fig. 4.5, one bit is reservecj fitr the nurnber'ssign.In a similarspiritto the way in which integersare stored,the expollL'nt and its sign are storcdin ll bits. Finally.-52bits are set asidefor the mantissa. Howevcr.becauscof normalization, 53 significandbits can be stored. Now,just as in Exarnple.i.2,this nreansthatthenunrberswill havea lirnitedrangeand precision.However,becausethe IEEE firrnratuseslnany more bits. the resultingnumber systemcan be usedtbr practicalpurposes. Ronge. In a tashionsimilarto the way in which integersare stored,the ll bits usedfor the exponenttranslatcsinto a rangefrom - I 022 to | 023.The largestpositivenumbercan be reprcscnted in binaryas l - a r g e sv ta l u e : + l . l l l l
... llll
x ?+10]'t
wherethe52 bits in the nrantissa areall l. Sinccthesignificandis approximately 2 (it is ac- l.i9ii x 10108. tually 2 - 2 t'), the largestvalueis therelbre2r0-rr In a similarfashion, the smallestpositivenumbercan be reprcsentcd as S r r a l l e svt a l u e: + 1 . 0 0 0 0 . . . 0 0 0 0 * 2 - t t \ 1 2 T h i s v a l u ec a nb e t r a n s l a t eidn t oa b a s e - 1 0 v a l u eo f 7 r { ) 1-1 2 . 2 2 5 1x l 0 r t ) 8 . P r e c i s i o n . T h e 5 2 b i t su s e df b r t h c m a n t i s scao l r e s p o ntdo a b o u tl 5 t o l 6 b a s e - l 0d i g i t s . Thus,z would be expressed as -'
formaL
Iong
"pl .1ll S
-
I. I4I592ri53a891.) Note thatthe machineepsilonis 2 sr : 2.2204x l0 16.
90
ROUNDOFF AND TRUNCATION ERRORS
4.
MATLAB has a numberof built-in functionsrelatedto its internalnumberrepresentation. For example,the rea Zmaxfunction displaysthe largestpositivereal number:
th
CI
nr >>
format
>>
realmax
Anq
long
=
t.
9
-o9l-I
lo02J.oe-Jtru
Numbersoccurringin computationsthat exceedthis valuecreatean overflow.In MATLAB they are set to infinity, inf . The te-aLmitTfunctionclisplaysthe smallestpositivereal number:
rt
L n
? . . 2 2 . 50 7 3 8 5 8 5 4 i - 2O i e - 3 0 8
>>
IL
n
>> realmirr
Nrlmbers that are smaller than this value create an underflow and, in MATLAB, zero. Finally, the e.p-sfunction displays the machine epsilon:
\4
are set t0
eps
d
S
o n il
2 . ?.2 A ,!.4 6 0 4 9 2 5 0 3 I I e - C I tj
4.2.2 Arithmetic Monipulotions of Computer Numbers Aside frornthe lirnitationsof a cclmputer's numbersysteul,the actualarithmeticmanipulations involving thesenunrberscan also resultin roundofTerror.To understandhow this occurs.let's look at how the computerperformssimpleadditionanclsubtraction. Becauseof their familiarity,normalizedbase-10numberswill be ernployedto illustratethe effectof roundofTerrorson simpleadditionand subtraction. Other numberbases would behavein a sirrrilarfashion.?l simplify the cliscussion, we will ernploya hypothetical decimalcomputerwith a 4-digit mantissaand a l-digit exponent. When two floating-pointnumbersare addecl,the numbersare first expressedso that they havethe sane exponents. For example,if we want to add | .5-57+ 0.0434l, thecomputer would expressthe nurnbersas 0. 1557x l0r + 0.00+3-ll x l()r.'fhen the nrantissas are addedto give 0. 16004I x t0r . Now. because this hypothetical computeronly carries a 4 - d i g i t m a n t i s s a . t h e e x c e s s n u m b e r o f d i g i t s g e t c h o p p e d o f f a n d t h e r e s u l lt 0i sl .0 . l 6 0 0 x Notice how the last two digits of the secondnumber(41) that were shifiedto the right have essentiallybeenlost fiom the computation. Subtractionis perfbrmedidenticallyto additionexceptthatthe sign of the subtrahend is rcversed.Forexample.supposethat we are subtracting 26.86frorn 36.41.That is, 0 . - 1 6 4 x1 1 0 2 -0.2686 x 102 0.095-5x l0' For this casethe resultmust be nornralizedbecausethe leadingzero is unnecessary. S0 we mustshift thedecirnalone placeto theright to give0.9-550x l0' : 9.-550.Noticethat
I I
(
4.2 ROUNDOFF ERRORS
9l
the zero addedto the end of the mantissais not significantbut is merely appendedto fill the empty spacecreatedby the shift. Even more dramaticresultswould be obtainedwhen the numbersare very closeas in 0.1642 x 103 -0.7641 x 103 0.0001x lOj which would be convertedto 0. 1000 x 100: 0. 1000.Thus, for this case,threenonsignificant zerosare appended. The subtractingof two nearly equal numbersis called subtractivecancellation.lt ts the classicexampleof how the mannerin which computershandlemathematicscan leadto numericalproblems.Other calculationsthat can causeproblemsinclude: Lorge Computotions. Certainmethodsrequireextremelylargenumbersof arithmetic manipulationsto arrive at their final results.In addition,thesecomputationsareoften interdependent.That is, the later calculationsare dependenton the resultsof earlierones.Consequently, eventhoughan individualroundofferrorcould be small,the cumulativeeffect A very simplecaseinvolvessumoverthe courseof a largecomputationcanbe significant. ming a roundbase-l0 numberthatis not roundin base-2.SupposethatthefollowingM-file is constructed: function -
sout = sumdemo()
n.
for i - 1:10000 s = s + 0.0001; end sout = s; When this tunction is executed.the result is >> formaL long >> sumdemo ans = 0.9999999999999r The f ormat long command lets us see the l5 significant-digit representationused by MATLAB. You would expect that sum would be equal to l. However, although 0.000 I is a nice round number in base-10, it cannot be expressedexactly in base-2. Thus, the sum comes out to be slightly diff'erent than l. We should note that MAILAB has features that are designed to minimize such e:'rors. For example, suppose that you form a vector as in >> format longr >> s = [0:0.0001:1]; For this case, rather than being equal to 0.99999999999991 , the last entry will be exactly one as verified by >> s (10001) 1
92
ROUNDOFF AND TRUNCAIIONERRORS
w e a d da s m a l l n u m b e r , 0 . 0 0 1t0o,a A d d i n g o L o r g eo n d o S m o l l N u m b e r . S u p p o s e largenumber,4000,usinga hypothetical computerwith the4-digit nrantissa andthe l-digir exponent.After modifying the smallernumberso that its exponentrnatchesthe larger, 0.4000x l0a 0.0000001x lOa 0.4000001x l0a which is choppedto 0.4000 x 104. Thus, we rright as well have not performeclthe addition! This type of errorcan occurin the computationof an infiniteselies.The initial terms in suchseriesareollen relativelylargein comparisonwith the laterterms.Thus.aftera few termshavebeenadded,we are in the situationof addinga smallquantityto a largequantity. One way to mitigatethis typeof erroris to sum the seriesin revelseorder.In thisway, to the accurnulatecl surn. eachnew ternrwill he of comparablernagnitude Smeoring. Smearingoccurswheneverthe individualterms in a summationare larger thanthe sunrmationitself.One casewherethis occursis in a seriesol'rnixedsicns. Inner Products. As shouldbe clearfrom the lastsections,someinfiniteseriesureparticthe calculationof seriesis not one o1'themore ularly proneto roundolTerror.Fortunately, A lll more ubiquitousmanipulationis thecalcommonoperationsin numericalnrethods. c u l u t i o rttr f i n n e rl ) r o d u e (uss i l l
: .trr,vr *,r:.v:*''' * x,,-)',, E.tr.t' linearalgebraic This operationis very comrron,particularlyin thesolutionol-simultaneous Suchsummations areproneto roundoll error.Cc'rnsequently. to equations. it is ofiendesirable in doubleprecisionns is doneautorraticallyin MATLAB. computesuchsumrnations
Trwrccttion(rrors are those that result fiom using an approximationin place ol- an exact procedure. For example.rn Chap.I we approximatedtheclerivative rnathematical of velocj u m p e rb y a f i n i t e - d i f f b r e n e ce q u a t i o no l ' t h ef i r r n r[ E q .( l . l l ) l ity of a bungee dt
Lt
u(ri+r) u(t1) t;"t I -
ze
pr fu
do rel
po
4.3 TRUNCATIONERRORS
du-A,u
FI
T
(48)
idt
gu wt no
ti
A truncationerrorwas introducedinto the nurnericalscllutionbecause the difl'erence equation only approximates thetruevalueof thederivative(recallFig. 1.3).To gain insightinto thepropertiesof sucherrors,we now turn to a rnathematical tormulaticln thatis usedwidely fashion-the Taylol series. in numericaln'rethods to expressfurrctionsin an approxinrate
4.3.1 The Toylor Series Taylor'stheoremand its associated formula,theTaylorseries,is of greatvaluein thestudy of nurnericalrnethods.[n essence.the Tay'lorthertremstatesthat any smoothtirnctioncan be approxirnatedas a polynomial.The Tay-lor.renes then providesa meansto expressthis in a form thatcanbe usedto comeup with practicalresults. ideamathematically
Th
.f ar ber ilar
fac on tior
leg
93
ERRORS 4.3 TRUNCATION
Zero order Jk,n):fk)
\:[:*':d::-s
/(.r,.1)= ^xt - r'(,,th
\.'3"^
r;+t : 1
ri=0 h
FIGURE 4.6
T h e o p p r o x i m o l i o on { / ( - r ) : O l - r a- O l 5 - r r O 5 . r r - a 2 5 x * l 2 o t r : zero order,firstorder,ond secondorder Toylor seriesexponsions.
l by
A usel'ulway to gain insightinto the Taylor scriesis to build it term by terrn.A good problcrnc()ntexttbr this cxerciseis to predicta functionvalucat one point in ternrsof the firnctionvalueand its dcrivativesat anothcrpoint. Supposethat you ale blindfbldedancltakcn to a locationon thc side of a hill facing downslopc1Fig.4.6).We'll call your horizontallocationJriand your vcrticaldistancewith respectto thc baseof the hill l(.r;). You are givcn the task of pledictingthe heightat a p o s i t i o n, r ; 11 ,w h i c h i s a d i s t a n c /cr a w a yf r o m y o u . At first, you areplacedon a platfbrmthat is completelyhorizontalso thatyou haveno idea that the hill is slopingclownaway fiorn yon. At this point,what would be your best guessat the heightat x;11'?If you think aboutit (remcmbcryou haveno ideawhatsoever what'sin front of you), the bestguesswould be the sameheightas whereyou're standing now! You c
G.9)
This rclationship, which is calledthe7.ero-onler uppntximatiorz, indicatesthat the valueof I at the new point is the sameas the valucat the old point.This resultmakesintuitivesense becauseif r, andr;11 erecloseto eachother,it is likely thatthe new valueis probablysimi l a r t o t h eo l d v a l n e . is, in Equation(4.9; proviclesa perf'ectestinrateif the functionbeing approxirnated f'act,a constant.For our problcm,you would be right only if you happenedto be standing on a pcrfectlyflat plateau.However,if thc functionchangesat all over the interval,additionaltermsof theTaylorseriesarerequiredto providea betterestimate. So now you are allowcdto get ofTthe plattbrrnand standon the hill surfacewith one leg positionecl in tiont of you anclthe otherbehind.You immediatelysensethat the front
94
ERRORS AND TRUNCATION ROUNDOFF
^2
foot is lower than the back foot. In fact. you're allowedto obtain a quantitativeestimateof the slope by measuringthe differencein elevationand dividing it by the distancebetween your feet. With this additional information. you're clearly in a better position to predict the you use the slopeestimateto projecta straightline out t0 heightat .f (xi+).In essence, .r;.,1.You can expressthis predictionmathematicallyby
No1 sig acc
(4.10) f (.,ri+r)= "f(.ri) *.f'(.u)h This is calleda.firsr-order approximationbecausethe additionalllrst-orderterm consistsof is a slope/'(.r;) multipliedby /r. the distancebetween.;r;and x;11.Thus, the expression now in the form of a straightline that is capableofpredicting an increaseor decreaseofthe functionbetweenJ; and-t;11. Although Eq. (,1.l0) can predict a change,it is only exactfor a straight-line, or linear, trend.To get a betterprediction,we needto add more terms to our equation.So now you First, you measurethe are allowed to standon the hill surfaceand take two measurements. slopebehindyou by keepingone foot plantedat x; and moving the otherone back a dist a n c e A , L . L e t ' s c a l l t h i s s l of lp, e Qi).Thenyoumeasuretheslopeinfrontofyoubykeeping one fbot planted at x; and moving the other one forward Ax. Let's call this slope .fi(.r). You immediatelyrecognizethat the slopebehindis milder than the one in front, Clearly the drop in height is "accelerating"in l'rontof you. Thus, the odds are that .f (;rr)is even lower than your previouslinear prediction. term to your equation As you rnightexpecf,you're now going to add a second-order and make it into a parabola.The Taylor seriesprovidesthe correctway to do this as in . l ' G i + ) = . t ' ' G i ) +/ ' ( r , , , *
J"lr'l
wht and
app ide
non
SOIC
con den yie
inn eno req Thi
SOIT
(o.t')
,'
(n-
L :
To makeuseof this fbrmula,you needan estimateof the secondderivative.You canusethe lasttwo slopesyou determinedto estitlateit as . f" ( r i + ) 7
.fiQt) - .fi,Qi)
JQ con
CITC
(4.r2)
Ax
Thus, the secondderivativeis merelya derivativeof a derivative;in this case,the rateof changeof the slope. Before proceeding,let's look carefullyat Eq. (4.I l). Recognizethat all the values subscriptedr representvaluesthat you have estimated.That is, they are numbers.Conse' quently,the only unknownsare the valuesat the predictionpositionx;a1.Consequently,it is a quadraticequationof the tbrm -r . f ( h ) = a 2 h 2I a 1 h a 1 y Taylorseriesapproximatesthefunctionwith a second' Thus,we can seethatthesecond-order orderpolynomial. Clearly,we could keep addingmore derivativesto capturemore of the function'scut' vature.Thus,we arive at the completeTaylorseriesexpansion
l - j J . n ' * " f ' ' - 1 t ' l r , ' + " * f " ) t t i t r ' + R , ,{ 4 . 1 3 1 f ( r , . r ): f ( r ; + / ' ( . rn; +
-----c--
wer ber
wht The this
plv
bas wil ter
oft lou onl trat
ERRORS 4.3 TRUNCATION Note that becauseEq. (4.13)is an infinite series.an equalsign replaccsthe approximate sign that was used in Eqs. (4.9) through (4.I l). A remainclerterrn is also includedto accountfor all terms fror.nn * I to infinitv: / t / / + ll l i \
R , ,- J -
l'-7,,,+l
g.l1)
wherethe subscriptn connotesthat this is the rernainderfbr the nth-olderapproximation and,.eis a valueof -t'thatlies somewhcrebetween.t';lifldr;+ r . Thus,we can now seewhy the Taylortheoremstatesthat any smoothfunctioncan be approximated as a polynornialand that thc Taylor seriesprovidesa meansto expressthis ideamathematically. Tiiylor seriesexpansionwill be exactfirr arrrth-ordcr polyIn general,the nth-orcler and sinuand continuousfunctions,suchas exponentials norlial. For otherclifl'ererrtiablc soids,a finite nunrbero1'termswill not yicld an exactestimate.Eachadditionalterm will r ill be s o m ci m p r o v e m e nht ,o w e v c rs l i g h t ,t o t h ea p p r o x i r n a t i oTnh. i s b e h a v i ow contribute in Example.1.3.Only if an infinitenurrbcrof terrnsalc addcdwill thc serics demonstrateci y i e l da n e x a c tr c s u l t . is that, Althoughthe tbregoingis true,the practicalvalueof Taylorscriesexpansions i n r n o s tc a s e st,h ei n c l u s i o no l ' o n l ya f ' e r vt e r n . rws i l l r e s u l ti n a n a p p r o x i r n a t i ot hna ti s c l o s e ol how many tL'nxsare cnollghto the true valuc lilr placl.icalpurposcs.The assessmcnt requiredto get "closeenclugh"is bascdon the remaindertcrrnof the expansion(Eq. 4.l4). This relationshiphastwo major drawbacks.First.{ is not known cxactlybut melely lics s o m e w h e rbc e t w e e nr ; a n d . \ ; 1t . S c c o n dt.o c v a l u a t eE q . ( 4 .l - l ) . w c n c c dt o d e t e r m i n tch c ( r r+ l ) t h d e r i v a t i v eo l ' / ( r ) . T o d o t h i s ,w e n e e dt o k n o w . / ( , r ) . H o w c v e r ,i f r v e k n c w /(,r), llrere would be no neeclto pcrlirrrnthc Tirylor scrics expansionin the present contexf! . q . ( 4 .1 . 1 )i s s t i l l u s e f u lt i r r g a i n i n gi n s i g h ti r r t ot r u n c a t i o n D e s p i t et h i s d i l c n r n r aE errols.This is becausewe rlo havecontrnlovcr lhc term ir in thc cquation.In otherwords. we canchoosehow lar awa-vfiorn.r'wc wiurlto evaluatel (r ), andwe cilltcontrolIhenttntb e r < r f ' r e r nw r sc i n c l u d er n t h ee x p a n s r o rCr .o n s e q u e n i yE, q . ( . 1 , 1 ) i s u s u a l l ye x p t e s . s e d a . s R , ,: O ( l t " ' ' ) e ( 1 , ' * ' ) m e a r t st h a tt h e t r u n c a t i o cnr r o r i s o f t h c o r d e ro l / t " * i . w h e r ct h e n o m e n c l a t u rO That is, the ellol is proportionalto thc stepsize/r raiscdto the (rr * l)th power'.Although this approximationimplics nothingregardingthe magnitudco1'thederivativcsthat multiply firr+t, it is extremclyLrsefulin.judgingthe comparativeerror of nurncricalrnethods Fur cxample,if the crror is O(fi), halvingthe stepsizc basedon Taylor scriesexparrsions. w i l l h a l v ct h ee r m r rO. n t h c o t h e rh a n d .i f t h ec r l r r ri s 0 ( f i r ) .h a l v i n gt h r -s t e ps i z ew i l l q u a r ter the error. by the addition In general,we can usuallyassumethatthe truncationerroris decreased sn'rall.the first- anclother of tcrnrsto the Taylor series.In many cases,il'fi is sLrfficiently high percentof the error'.Thus, ternrsusuallyaccountlbr a dispropoltionately lower'-order This propertyis illusapproxinration. only a t-ewtermsare requiredto obtainan adequate tratedb_\, the following exanrple.
4.
ERRORS AND TRUNCATION ROUNDOFF
96 E X A M P L E4 . 3
A p p r o x i m o t i o no f o F u n c t i o nw i t h o T o y l o rS e r i e sE x p o n s i o n : 0 to 6 to approxirnate Problem Stotement. Use Taylor seriesexpansionswith n ,f(-r):cosxat'rr+t:rl3onthebasisofthevalueof/(x)anditsderivativesat * x i : t l 4 ' N o t et h a t t h i sm e a n st h a t h : r 1 3 n l 4 : T l 1 2 '
4 B si
the correct Solution. Our knowledgeof the true tunction meansthat we can determine va|uef(r13):0.5.Thezero-orderapproxinrationis[Eq.(4.9)l
t ( i ) = ' " - ( i ) : o' o ' r 0 6 7 8 l
er se
which representsa percentrelativeerror tlf
.',: lu_L9#Sl'or* :rt o*
It sll
term where.f'(x) : -sinxl we adtJthe first clerivative approximatit'rn, For the fir-st-order
AI
+ l) : 0 . s 2 r e 8 6 6 s s 'r\/r1/ -\ -= . . , / " ) - , i n f\ 44/)\(1- 2 \-rl
(jfl
we add the secondderivaapproximatitln, which has le,l:4.400/o. For the seconcl-ttr
/r\
/r\/r\
t(i):c.'s(;)--'"|';)\i)-
cos(]r/4(l\':0.4()1154491
2
\r2l
---"'
ir.ran improvedestiwith le,l:0.44g%,. Thus, the inclusionof additionalterms results nrate.Thepfocesscanbectrntinueclandtheresultslistedasin Order n
0 I 2 3 4 5 6
fr"(x) cos,r -sin -r cos.t s i n. r COS -Y
-sirt -r -cos -r
ca ap
l'0r/t1 o 7 a 7l 0 6 / 8 l 0 5 2 19 B 6 e ' - 5 9 o 497754491 c )4 9 9 8 6 9 1 4 7 0 50000/551 0 500000304 0 499999988
le,l 414 440 a.449 ) 262x lA l 5l x lO r 608 x lO 5 244xlO"
for a polynornial' Notice that the derivatrvesnevergo to zero as would be the case estimate'Howevet' Therefbre,eachadditionalterm resultsin some improvementin the For thiscase'bythe alsonoticehow mostof the improvementcomeswith the initial terms' meansthat which tinre we have acldedthe third-orderterm, the error is reducedto 0.0267a' of the true value.Consequently,althoughthe additionof more we have attained99.974o/a becomesnegligible' terms will reducethe enor turther,the irr.rprovement
* Gr
ERRORS 4.3 TRUNCATION
97
4.3.2 The Remoinderfor the Toylor SeriesExponsion Before demonstratinghow the Taylor seriesis actuallyusedto estimatenumericalerrors, we must explainwhy we includedthe ar-{ument f in Eq. (4.14).To do this, we will use a simple,visuallybasedexplanation. Supposethat we truncatedthe Taylor seriesexpansion[Eq. (4. I 3)] after the zero-order term to yield . / ( . r , + r )] . / ( - t i ) A visual depictionof this zero-orderpredictionis shown in Fig. 4.7. The remainder,or error. of this prediction,which is also shown in the iliustration,consistsof the iniinite seriesof terms that were truncated .
f"tv
t
R r:r, / ' ( . \ it h+ + h 2 + '
f{l'tt
t
.l^''h3+...
It is obviouslyin.onu"Ji.nt to deallith the remainderin this inflnite seriesformat.One simplificationmight be to truncatethe remainderitself,as in R 1 1= . [ ' ( x 1 ) l t
(4.1s)
Although,as statedin the previoussection,lower-orderderivativesusuallyaccountfor a greatershaleof the remainderthan the lrigher-order terms,this resultis still inexactbeis impliedby the causeof the neglectedsecond-andhigher-olderterms.This "inexactness" a p p r o x i m a teeq u a l i t ys y m b o l( : ) e m p l o y e di n E q . ( 4 . l 5 ) . into an equivalence is An alternativesimpliticationthattransfbrmsthe approximation basedon a grnphicalinsight.As in Fig. 4.8, the derivativemenn-valuethertremstatesthat
FIGURE 4.7 GrophicoLdepiclicn cf o , r o r . , . t d - T o . L l s e i p s 6 r c f l . r ' r o ' ) . : r , dr e n o r . d e t
l (xl
Ll. )t, 1e
at re
98
R O U N D O FA F N D T R U N C A T I OE NR R O R S
n R o1
,f{x}
Nr
Eq
h
Tht the we tlor
FIGURE 4.8 G r o p h i c o ld e p i c t i o no [ t h e d e r ] v o t i v m e e o n - v o l uleh e o r e m
if a functionl(r) and its first derivativearecontinuousover an intervalfrorn.:r, to.ri+1, then thereexistsat leaslone point on the functionthat hasa slope,designated by /'(€). thatis parallelto the line joining,l{.r,)and,l(,r,*,).The parameter { marksthe r valuewherethis slopeoccurs(Fig. 4.8).A physicalillustrationof this theoremis that,il'you travelbetween two pointswith an averagevelocity,therewill be at leastone momentduringthecourseof the trip when you will be moving at that averagevelocity. By invokingthis theorern,it is simpletu realizethat,as illustratedin Fig.4.8.theslope .l'(6) is equalto the riseR,,dividedby the run h, or R., ,
f'(t): +
whichcanb. ilurr.ng.oto -{ive Ro-- .t'''(.t)h
(416)
versions f'hus,we havederivedthe zero-order versionof Eq. (4.14).The higher-order aremerelya logicalexfension ol'thereasoning usedto deriveEq. (4.16).Thefirst-order version is I"(F) R1 : :--------7,
Thu tion shou expe
4.3
Equ
ence
r4l7r
Li
For this case,the value of { confbrmsto the r- valuecorrespondingto the secondderivative thatmakesEq. (4.l7) exact.Similarhigher-order versionscanbe developed from Eq.(a.la)
4.3.3 Using lhe Toylor Seriesto EstimqteTruncotionErrors AlthoughtheTaylorserieswill be extremelyusefulin estimatingtruncationenorsthrough' out this book, it may not be clear to you how the expansioncan actuallybe applied to
wher, matic and i
99
ERRORS 4.3 TRUNCATION
numericalmethods.In fact, we have alreadydone so in our exampleof the bungeejumper. Recall that the objectiveof both Examplesl. I and I .2 was to predictvelocity as a function of time. That is, we were interestedin deterrniningu(/). As specifiedby Eq.(4. 13), u(l) can be expandedin a Taylor series: , 1 1 ' *- , I , ) * { f r , ( / i + r:) u ( t ) * u ' { / 1 LI
, r , * , - t i ) z+ " ' - F
R,,
Now let us truncatethe seriesafter the first derivativeterm: u ( t i + r ): u ( t i ) f u ' ( / r ) ( t r * -r / i ) * R r
( 4 .l 8 )
Equation(4.l8) can be solvedfbr t . .|
rl I I
R1
u(ri+r)-u(ti)
-
{i+l -
-
ri+l -
{i
ri
( 4 r. 9 )
Truncatiorl e[ot
First-ordcr appfoxrnra(ron
The first part of Eq.(4.19,)is exactlythe samerelationshipthat was usedto approximate thederivafiveinExamp1 l e. 2 [ E q . ( l . l l ) l . H o w e v e r , b e c a u s e o f t h e T a y l o r s e r i e s a p p r o a c h , with this approximawe havenow obtainedan estimateof the trtrncationerror associated tion of the derivative.Using Eqs.(4.l4) and (4.l9) yields rRr ti+t -
li
:
u"(6).
-
-(ti+t 2!
ti)
or R' ti+t- Ii
:O(tirr-l,)
T h u s ,t h e e s t i m a t e o f t h e d e r i v a t i v e[ E q .( l . l l ) o r t h e l l r s t p a l t o f E q . ( 4 .l 9 ) ] h a sa t r u n c a tion error of order ti+r - ti. In other words, the error of our derivativeapproximation shouldbe proportionalto the stepsize.Consequently, if we halvethe stepsize,we would expectto halvethe errorof the derivative. 4.3.4
Numericql
Differentiotion
Equation(4.f 9) is given a formal label in numericalmethods-it is calleda.finitedifJ'erence.lt can be representedgenerallyas f
'(.ri\ :
f
'(ri) :
.f\ri+t)
-
/(.fi)
Ji+t -r,
. l ( . r i + r )- . f ( r i )
f
O(-ri+t - ri)
+ o( h)
(1.20)
(4.21)
where /r is called the step size-that is, the length of the interval over which the approximation is rnade,.ri+t- x;. It is termeda "forward" differencebecauseit utilizesdataat i and i * I to estimatethe derivative(Fig.4.9a).
too
ERRORS ROUNDOFF AND TRUNCATION
S
e d tt e5 b1 al br
Bc
CX
Trr
wht
Cer mat
to yi
whic
tc) FIGURE 4.9 finite-difference depiclion Grophicol of (o)forword, ond{c)centered {b)bockword, c n n r o < i m n t i o n so f r h e f i r q rd e r i r o t i v e
Noti appr the tr the r war( tral r
4.3 TRUNCATION ERRORS
rol
'I'his
forward differenceis but one of many that can be developedfrom the Taylor seriesto approximatederivativesnumerically.For example,backwardand centereddifference approximationsof the llrst derivative can be developedin a fashion similar to the derivationof Eq. (4.19). The former utilizesvaluesat r;-1 and.r; (Fig. 4.9b), whereas the latter uses valuesthat are equally spacedaround the point at which the derivative is estimated(Fig. 4.9c).More accurateapproximationsof the first derivativecan be developed by including higher-ordertermsof the Taylor series.Finally, all the foregoingversionscan also be developedfor second.third, and higherderivatives.The following sectionsprovide brief summariesillustratinghow someof thesecasesare derived. Bockword DifferenceApproximotion of the FirstDerivotive. The Taylorseriescan be expandedbackwardto calculatea previousvalueon the basisof a presentvalr.re. as in f h f t x i - r t : J ' ( x t t- l ' ( x i f t + ::'t
(4.22)
Truncatingthis equationafter tlre first derivativeand rearrangingyields
1 ' G i )=
J (ri) - .f(xi-r)
(4.23)
wlrerethe error is O(/z).SeeFig.4.9b for a graphicalrepresentation. Centered DifferenceApproximotion of the First Derivotive. A third way to approximatethe first derivativeis to subtractEq.(4.22)from the fbrwardTaylorseriesexpansion:
J ' ( x i - t- ) . l ' ( r i*) l ' ( . r*i + f f n ' + ' . '
(1.21)
to yield J'^l(*'\ rt * . l ( x i + r ): . f ( x i ) - t 2 l ' ( x , r , * whichcanbesolvcdfor . l ( - r i + r )- . f G i t ) .l''(x) :
2h
t't".\ -
* " f ( . r i + r ) l ( . r i _ r )2h
.f'r)(",),, b
o(t)
(4.25)
Equation (4.25) is a centercdfinire tlifJbrencerepresentationof the first derivative. Notice that the truncationerror is of the order of ft2in contrastto the forward and backward approximationsthat were of the order of /2.Consequently,the Taylor seriesanalysisyields the practical information that the centereddifferenceis a more accuraterepresentationof fhe derivative(Fig. 4.9c). For exarnple,if we halve the stepsize using a forward or backward difference,we would approximatelyhalve the truncationerror, whereasfor the central difference,the error would be quartered.
r02 E X A M P L E4 . 4
ERRORS AND TRUNCATION ROUNDOFF F i n i t e - D i f f e r e n cAep p r o x i m o t i o n so f D e r i v o t i v e s problem Stotement. Use forward anclbackwarclclifferenceapproximationsof o1/r1and derivativeof a centereddifferenceapproximation of O(h2)to estimatethe first - 0 . l . r a - 0 . l 5 . t r - 0 . 5 . r r- 0 ' 2 5 ' t- t 1 ' 2
J (,r):
usingh:0-25. Note thatthe ar -r : 0.5 usinga stepsizeft : 0.5. Repeatthe comprrtation clerivativecan be calculateddirectly its - 0 . 4 ' t 3 - 0 . 4 5 x 2- 1 ' 0 x- 0 ' 2 5
f'(r):
andcanbeusedtocomputethetruevalueas/,(0.5):-o.9|25. So|ution. For h:0.5, ri t:0
.t; : 0.5 r r i + r:
1.0
the functioncanbe employedto determine
f (xi-t): l'2 .f @) :0.925 l'(-ri+t):0'2
(a'21)J' Thesevaluescan be usedto computethe forwarddifference[Eq' o'2 - 0'925 -l : '15 l e , l: 5 8 ' 9 % l ' ' ( 0 . - 5 )= * the backwarddifferencelEq. (a.23)1. /'(05)
- 0'925- l'2:-0.55 0.5
le,l:3g"7o/o
IEq. (a'2-5)l' and the centeredclifl'erence 0.2 1.1
.l''(0.5)= ff
--1.0
le,l:9.6a/<,
Forft : 0.25, .l(ri-r): l'10351563 J'Qi): 0'gZS . l ( r i * r ): 0 ' 6 3 6 3 2 l83 whichcanbe useclto computethefbrwarddif'ference' 3 - 0.925 0.6363281 -r;I :0.25 ,r; : 0.5 r ; 1 1: 0 . 7 5
l''(0.5)=
:-l.l-55
0.25
T
EI
lt,l:26.5"/o
the backwarddiff'erence'
0 . 9 2 5- 1 . 1 0 3 5 1 5 :6 3-0.1 11 = l''(o.s) 0.25
le,l : 21.'|Eo
diff-erence' andthecentered 5 5 6 3 -0.934 0 . 6 3 6 3 2 8 -1 31 .1 0 3 1 : f /' ( o O S{ \\ : = 0.5
V , \: 2 ' 4 % '
4.4 I
T rh
o ca E si c(
r 03
4,4 IOIAL NUMERICAL ERROR
For both step sizes,the centereddifferenceapproximationis more accuratethan forward or backwarddifl'erences. Also, as predictedby the Taylor seriesanalysis,halving the stepsize approximatelyhalvesthe error of the backwardand forward diff-erences and quarters the error of the centereddifference.
Finite-Difference Approximotionsof Higher Derivotives. Besidesfirst derivatives, the Taylorseriesexpansioncan be usedto derivenumericalestirnates of higherderivatives. To d o t h i s , w e w r i t e a f b r w a r d T a y l o r s e r i e s e x p a n s i o n f o r . f ( . . ri ni +t e) r m s o f . / ( - r i ) : . / ' ( r , + r ): . l ' ( r i ) I
J ' l x i ) t 2 h l+
'")r'''
,r,rrt*
"
(4.26)
Equation(4.24)canbe multipliedby 2 and subtracted from Eq. (4.26)to give . l ' Q i + ) - 2 . 1 ' ( r ; * 1 ):
. [ ( r i ) + . 7 ' " 1 t 1 1+h.2. .
which can be solvedfbr ./"(r,):
I ( r i + : ) - 2 . f ( . r i + t )* . / ' ( . r i ) h2
+ o(h)
(1.27)
This relationshipis called thc second.font,unl.finite diflerenca.S i r n i l a rm a n i p u l a t i o ncsa n bc cmployedto derivea backwirrdversion ,.,
. / { . t , ) - 2 . / ' ( - t r, ) * . 1 ( r i 3 )
.1"(r,):ff+o(h)
-
and a centeredversion . l ( . t , r ) - 2 / ( . t il I l ( . r i r )
.,/. ", ,\ . r i ) : f f + o f t 2 )
-
As wasthccascwith thefirst-derivativc approximations, thecentered caseis moreaccurate. Noticealsothatthecentercdvcrsioncanbe altemativelvcxnrsssed as l ( , r i + r ) - . 1 ' ( . r i )_ I G i ) - . / ' ( x i t ) hh f ' "( r i ) 1 Thus,just as the secondderivativcis a derivativeofa derivative,the secondfinite dilfbrenceapproximationis a differcnceof two first finite differcncesIrecallEq. (a.l2)1.
4,4 TOTATNUMERICAT ERROR The total nnnerit:aLerror is the summationof the truncationand ror-rndofTerrors. In general, the only way to minimizeroundofTerrorsis to increasethe numberof significantfigures of the computer.Further,we havenotedthat roundoff error may int:reasedue to subtractive cancellation or dueto an increasein the numberof computations in an analysis.In contrast, Example4.4 demonstratedthat the truncationerror can be reducedby decreasingthe step size.Becausea decreasein stepsizecan lead to subtractivecancellationor t.oarl increasein computations.the truncartion errors aredecreasedas the roundofTerrorsareincreasetl.
r04
ERRORS ROUNDOFF AND TRUNCATION
Point of diminishing returns
o J
Log step size
F I G U R 4E. I O errorthotsomelimes A croohlcodeoiction of thelrodeoffbetweenroundoff ond iruncotion relurns is coiresintoploy in thecourseol o numericomethodThepointof diminishing of step-size reduclion. shown,whereroundoff errorbeginsto negolelhe benefits
t i t
Therefore,we are facedby the following dilemma:Thc strategyfbr decreasing one of the othercomponent.ln a cornputation, componentof the totalerrorleadsto an increase we couldconceivablyclecrease thc stcpsizeto minimizetruncationerrorsonly to discover that in doing so, the roundofferror beginsto dominateths solutionand the total error grows!Thus,our remedybecomesour problem(Fig. 4. l0). One challengethat we laceis stepsize1bra particularcomputation. Wc'would like to choose to determinean appropriate the amountof calcr,rlations and roundoffcrrorswithout incur a largestepsizeto decrease ring the penaltyof a largetruncationerror.If the total error is as shown in Fig. 4. 10,the challengeis to identily the point of dinrirrishingreturnswhere roundofTerror beginsto negatethe benefltsof step-sizereduction. WhenusingMATLAB, suchsituations of its l5- to I6arerelativelyuncommonbccause they sometimes digit precision.Nevcrtheless, do occurandsuggcsta sortof "numericaluncertaintyprinciple"thatplacesan absolutelimit on theaccuracythat rnaybe obtainedusing certaincomputerized numericalmethods.We erplore sucha caseirr thc lollowing section.
EXAMPLE 4.5
R
P
c
P b R
4.4.1 Error Anolysis of Numericol Differentiotion As describedin Sec.4.3.4,a centereddifl'erence approximation of the first derivativecan be writtenas (Eq. 4.25)
l
ire
,f'(.r,): True value
J(ri+tl
-.1 (.ti r)
2h Finite-difference approximation
./'(t)(6), .' ---116
f
fr
(4.28)
d
Truncation error
h, H
Thus. if the two functionvaluesin the numeratorof the finite-difference approximation have no roundoff error,the only error is due to truncation.
D E
-I
r05
ERROR 4.4 IOIAL NUMERICAL
However, becausewe are using digital computers,the function values do include roundoff error as in -t ei-t .f(xi t) : f @i- ) , f ( - r i + r ): . f ( x i + ) I e i + r where the ;'r a.e the rounded function values and the e's are the associatedroundoff errors.Substitutingthesevaluesinto Eq. (4.28)gives /(.ri t)
Pi t -
, t',rl
?t. .l lxit:
f(*,,r1A,
2h
True value
Finite-difference approximation
RoundofT error
-
,/'l'(€),-: " 6
Truncation error
We can seethat the total error of the finite-differenceapproximationconsistsof a roundoff with stepsize. error that increaseswith stepsize and a truncationerror that decreases error has an roundoff of the Assuming that the absolutevalue of each component 2e. Further, €iwill be €i+rupperboundof e, themaximumpossiblevalueof thedifl'erence boundon M. An upper value of assumethtrtthe third derivativehasa maximumabsolute as the absolutevalue of the total error can therefbrebe represcnted t-
T o t uel r r o r :
e h2M | ., _ . [ 1 r , r, ) . / ( x r r ) < _ + _ h6 l./'t_r.r, n I
G.29\
An optimal step size can be determinedby diffbrentiatingEq. (4.29),settingthe result equalto zeroand solvingfbr ('1.30)
E X A M P4L.E5
R o u n d o f fo n d T r u n c o t i o nE r r o r si n N u m e r i c o lD i f f e r e n t i o t i o n Problem Stotement. In Example4.4, we useda centereddiffbrenceapproximationof O(h]) to estimatethe first derivativeof the fbllowing functionat,r : 0.-5, -l t ' G ) : - 0 . l , r a - 0 . l 5 , r r - 0 . 5 1 2- o . 2 5 x 1 . 2 Peribrm the samecomputationstartingwith ft : l. Then progressivelydivide the stepsize how roundoffbecomesdominantasthe stepsizeis reduced. by a f'actorof I 0 to demonstrate (4.30). Recallthatthe truevalueof the derivativeis -0.9 125. Eq. your results to Relate and plot the Solution. We can developthe following M-tlle to perfbrmthe computations as arguments: derivative its analytical function and pass the we both that results.Notice funcLion
di ffex
( func,
cLfunc, x, n)
1ong
format df t r.ue=df unc (x ) ; h-1; H ( I ) =h; D(1) = ( func (x+h) -func E (1 ) =abs (dftrue-D(1)
(x-h) ) ;
) / (2*h)
;
r0 6
AND TRUNCATION ERRORS ROUNDOFF i - )
i rr
. n
h = h , 1 r j; H(r)=5; D(r1= (func(x+h) (.)
o.l(il
func(x |
uF
h) ) / (2*h\ ;
));
end L)' E' I ' ; fprincl(' finite diflerenr-e srt-rp siz.: t1.u. ('%14.10f lprintf %]5.14f \16.lif\n',L) ; l ogLog (H, E) , xl abel ( 'SLcp Si zc' ) ,y Label ( 'Error' )
1 , =[ l |
!,r1-1e('Fla'! IormaL
rt
l , . r . r . o r \ / c r . i r , l . . rS L e p
errol'\n');
Stz.')
sholf
The M-flle can thenbc run usingthe tbllowingcommallcls: .>
f f=G(x)
L l. 1 * x ^ . 1
->
c 1 f = ( ; (r x )
0.4*x^
i
0.15*r^3-0.!*x^2 0.45*_r^2 x (1.2b;
0.25*x+1
.2;
> : ' c l i f I r - - - . x (f f , r - 1 f ,0.'1 ,ll) :tr--p :;izc f inite c ii 1 f e t , - n c c t.r'uLr error I . 0 0 0 ( 10 0 0 f { l ( l I . 2 . t t 2!.r 0 0 ( .rr)( 10 0 0 t r r - 1 0 . I r i - )0 0 1 :0) ( r i l0 t l 0 0 0 . I t l f l r r ( l 0 L l 0 r ) ( l 1 . ) !. 1 r . , 0 0 ( ) 0 ( ) ( l ( ) l l 0 r ) | 0 . 0 ( j t ' r ( ) t l L l t ) u t . l l l t ) ( ) 0 . r _1l 0 0 0 0 0 0 0 0
0. !ll2 5J i00000011(r
ll.0il(l0l 5000000t) ) Ol!t)000t) 0. U(r0t0
0.0(l 10000{)(r0 A.')12 500llr00000I i l . ( r ( ) 0I ( l o L r ( ) i ) r l l ) . ! ) l : 5 ( ) L r l r { ll . j 9 ! r t r ' , 0.0000
10 0 r ) r J 0
0 . 0 0 0 0 0 10 0 | 0 ( l . [ )( ) { ) 0 0 U1 ( )( ] i ) 0 . 0 ( ) ( l l r 0 0 0 L( l o 0.0f)000000
t 0
0 . O ( r ( ) ( ) 0 0 0 ( l (I )
0 . { i ( | ) 1 ) { ) ( ) ( )3t )1 i , r 9 3
0. t l 21000rJ00 l l I !
0.00utJ000000112
0.91 2!t)000000b4.. 0 . r l L - l4 9 t j c ) ' )q 4 5 ( r l I 0 . q 1 2 5 O ( t r ) ( Il i 1 6 0 ' r
0.0r100000000054 0 . 0 r ) [ )( ) 0 t )0 U 0 1 , 4 9 / 0 . 1 1 0 1 1 6 6 3 0 ( ) 1l 3 6 1
a .912,500u t 9 98q44 U.9 I 2!00(l i 'rb(10!tl
0.0000000199894 0.0.J0000075ir006
Q., As clepicted in Fig.4.I I. thc rcsLrlts arcasexpcctcd. At l'irst,foundol'fis minimalandthe Hcnce,as in Eq. (4.29).the totalerrordropsbya r'stimatcis dclrninnted by truncationL-rror. t a c t o r o l ' I 0 0 e a c h t i n i e w c d i v i d csttheep b yI 0 . H o w ' c v e r , s t a r t i n g a t / z : 0 . 0 0 0 l , w e s e e roundol'f'error bcginto crcepin andcrodethc rateat whichthecrrordirninishes. A rninimum t'. crror is reachcdat /r: l0 Bcyondthis point.the crrol irrcrcascs as roundofTclominates. Becuu.sc w'eafc dcalingw'ithan easilyclittbrcntitblel'unction,wc czuralso investigate whctherthescresultsarcconsistent with Eq. t4.30).First,we canestimatcM by evaluating t h e l ' u n e l i o n 'tsh i r t ld c r i v l r l i v cu s -0.():2. M : I l ' t ' t ( 1' -..-. ,5I ) l - l - 2 . 2 1 1 0 .1. 5
1
B e c a u sM c A T L A B h a sa p r c c i s i o n 0 1 ' a b o ult5 t o l 6 b a s c - l 0 d i g i t s a , rrouge h s t i m a t e ot hf e upperboundorl rounclofl'wouldbc abollt€r:0.-5 x l0 r('.Sr-rbstituting thescvaluesinto E q . ( 4 . 3 0 )g i v e s l0r(')
:4.3 x l0 ('
w h i c h i s o n t h es a m eo r d e ra s t h er e s u l o t l I x l0
('ohtuined withMATLAB.
For Thr apt mu
pro eng
gra. equ you pos and first
usir erro lem,
r07
4.4 TOTATNUMERICAL ERROR
FIGURE 4.I I Plot of error versus step saze
100
1o-2
10 - 4
10 - 6 t!
10 - 8
1 0 -1 0
10 8
10-6
10 - 4 Step size
4.4.2 Control of Numericol Errors rd the ;bya /e see imum rtes. itigate uating
: of the es into
Forrrrostpracticalcases,we do not know tlreexacterrorassociated with numericalrnethods. The exception,of course,is whenwe know theexactsolution,which makesour numerical Theretbre, approxitnations unnecessary. for rnostengineering andscientificapplications we mustsettlefbr someestimateo1'theerrorin our calculations. and generalapproaches Thereare no systematic to evaluatingnumericalerrorslor all problems.In many ciiseserror estimatesale basedon the expericnceandjuclgnrento1'the engiueeror scientist. Although error analysisis to a certain extent an art, there are severalpractical programmingguidelineswe can suggest.First and fbremost,avoid subtractingtwo nearly equalnurnbers.Loss of significancealmostalwaysoccurswhen this is done.Sometimes you canrearrange or refbrmulatetheproblenrto avoidsubtractive cancellation. If this is not possible,you may wirnt to use extended-precision arithnretic.Furthermore.when adding numbers.it is bestto sortthe numbersand work with the srnallestnumbers andsubtracting first.Thrsavoidslossof significance. Beyond thesecomputationalhints, one can attemptto predict total numericalerrors using theoreticalformulations.The Taylor seriesis our primary tool fbr analysisof such errors.Predictionof total numericalerroris very complicatedforeven rnoderatelysizedproblemsandtendsto be pessirristic. Theretbre, for only srlall-scaletasks. it is usuallyattempted
t08
4.5
ERRORS AND TRUNCATION ROUNDOFF
PR
The tendencyis to pnsh forward with the numericalcomputationsand try to estimate the accuracyof your results.This can sometimesbe done by seeingif the resultssatisfy someconditionor equationas a check.Or it n'raybe possibleto substitutethe resultsback into the originalequationto checkthat it is actuallysatisfied. Finally you shouldbe preparedto pertbrm numericalexperimentsto increaseyour awarenessof cor.nputatiorral errorsand possibleill-conditionedproblols. Such experimentsmay involve repeatingthe computationswith a difl'erentstep size or methodand comparingthe results.We may employsensitivityanalysisto seehow our solutionchanges when we changernodel parametersor input values.We may want to try different numeriarc basedon dit'terentcomputacal algorithmsthat havedif'tbrenttheoreticalfoundations, propcl'ties and stabilitycharacteristics. tional strategies, or haveditferentconvergence When the resultsof numericalcomputationsare extrcmelycritical and may involve it is appropriateto takespecial ramiflcations, lossof humanlit'eor havescverec.conornic groupsto solvethesame precautions. This may involvethe ussof two or morc indepcndent problenrso that theirresultscanbe compared. Thc rolesof errorswill be a topicof cclncernanclanalysisirr all sectionsof thisbook. to spccificsections. Wc will leavethcseinvestigations
be
AND DATAUNCERTAINTY BI.UNDERS, MODELERRORS,
Ju
as ca ca w(
4.
Er AI
i, n . . tt
lle th hit es ifr Ltl\
Although the following sourcesol'error are not dircctly connectedwith most of thenuofa mericalmcthodsin this book, they can sometimcshavegreatimpacton the success modelingeflbrt. Thus, they must alwaysbe kepl in mind when applyingnunrcricaltechnic;uesin thc contextof real-worldproblcms.
ist lo As th
4.5.1 Blunders
ntc ol. er tn
We areall f'amiliarwith grosserrors,or blundcrs.In theearlyyearsol'conrputcrs, erroneous numericalresultscor-rldsornetimesbe attributcdto rnalfunctionsol'the cornputeritself, Today.thissourceol'ernlris highly unlikcly.andmostblundcrsrnustbc attributcdto humar impert'ection. Blunderscan occur at any stageof the rnat.hernltical rnodelingprocessand canconThey can be rvoicled<-rnly by soundknowledge tributeto all the otherconlponcnts of err
4.5.2 Model Errors Model error.srelatcto biasthat can be ascribedto incorrrpletemathu'matical rnodels.An exampleof a negligiblemoclelerroris thef-actthatNewton'ssecondlaw does not accountfbr relativisticeff'ects. This doesnot detractfiorrrthe adequacyof the solutionin Exanrplel.l
PROBTE
4.1 Convert thc lirllowi l0l l00l and| 10.(X)10 4.2 Convcrt the lirlloui 7 I 563 and 3. I zl.
4.3 For computcrs.thc thought ol' as lhc srnirllc givcs a numbcI grcrtcr th idtr ca1\.t drr r\o\'td .r.
S t e pl : S e t r : I . Step 2: Il' I + e is lessth Otherwiscgo to I Step 3: r' : e/2 Step4: Rcturrrto Stcp2 S t e p 5 :a - 1 x €
PROBLEMS
r 09
with the br-rngee becausetheseerrorsare rninimalon the timc and sDacescalesassociatcd jumper problern. However,suppose thatairresistance is not proportional to thesquareofthe fall velocity. as in Eq. ( 1.7),but is relatedto velocity and otherfactorsin a diffbrentway. If suchwere the case,boththeanalyticalandnumericalsolutionsobtainedin Chap.I wouldbe erroneous becauseofmodel error.You shouldbe cognizantofthis typeoferror andrealizcthat,ifyou are rcsults. workingwith a poorlyconceivedmodel,no numericalmethodwill provideadcquatc 4.5.3
Dofq Uncertointy
in thephysicaldataon which Errorssomctimes enterinto an analysisbecause of uncertainty modelby hava rnodelis based.For instance, suppose we wantedto testthe bungeejr"rmper jurrrpsandthenrneasuring his or hcr velocityalicr a spccring an individualrnakerepeated fied time interval.Uncertainty wouldundoubtcdly with thcscmcasllrcmcnts, ns bc associatcd would flll tasterduringsomejumps thanduringothcrs.Thcsccrrorscancxthe parachutist hibit bothinaccuracy andirnpr-ecision. If our instmments consistently underestirnate or overor hiascd,dcvicc.On thc otherhand, estimatethevelocity,wc arcdcalingwith an inaccuratc. if the measurements ol'prccision. areranclomlyhigh andlow, we arcdcalingwith a qr-rcstion Measurernent by sr-rnrnrarizing thc datawillr oneor morewellerrorscan bc quarrtil'icd choscns(atistics lhat convcyas much inlbrrnationas possibleregardingspecil'iccharacteristicsol' thc data.Thcscdcscriptivcstatisticsarc most olien selectedto represent( I) the l o c a t i o no f ' t h cc c n t c ro l ' t h cd i s t r i h u t i oonl ' ( h cd a t aa n d( 2 ) t h ed e g r e eo f s p r e a do l ' t h ed a t a . As such.theyprovidca rnsasurc ol'Lhcbiasandirnprccision, rcspcctivcly. Wc will retLrrn to (hctopicol'charactcrizing whenwe discussregression in PartFour. dalar-rncertainty Althoughyou mr-lst hc cognizanto1'blundcrs, r-rroclcl errors,anduncertainclata,the nurncricalrncthodsuscdfirr bLrilcling nroclels can bc stndicd,lilr the nrostpilrt,indepcndently thatwc havcnot lnadcgross ol'thcscerrors.Thcrelilrc,ftrrrlost of thishook,we will assr"rmc Undcr en'ors,we have a sounclnrodel,and wc are dcalirrgwith crror-ll'ccrncusurcnrcnts. thescconditir)ns, wc can studynunrcricalerrorswitl'toulcrlrlplicatingl'actors.
PROBTEMS Writc your own M l'ilcbascdon this al-uorithrn to dctclnrirrc thc nrachincepsilon.Validatcthc lcsLrlthy comparingit with t h c v a l u cc o r n p u t c w d i t h t h c b L r i l t - i ln' u r t c t i ort,t1 r r . 4 . 4 I n a l n s h i o r sr i n r i l a lt o P r o b . , 1 . 3c, l o v c k r py o u r o w n positivcrcal nunrhelusedin M l'ilcto dctclnrincthc srnallest MAf LAts. Bascyour algorithrnon the notionthatyoLrrcornol as thc smallcst nurnbcl that whcn addcd to onc grcatcrthan l. An algorithrnhuscdon this putcrwill hc unablcto rcliablydistinguishbctwccnz.crornd a nurnbcr . otc thal thc c a nh c d c v c l r r p c da s a q u a n t i t yt h a l i s s r n a l l c tl h a n t h i s n u n r b e rN rcsultyou obtain will clil'l'crll'onrthc valLrcconrpLrtcd with l:Sets: l. r r , r l i r i n . C h a l l c n g eq u c s t i o n :I n v c s t i g a t e t h c r c s u l t sb y gcneratcdby your' 2:Il'I * e is lesstharror eqrralto l. thcn go to Stcp.5. takingthc base-2 logalitlrrnol'thenun.rhcr go to Stcp3. Olherwisc c o d ca n c tl h o s co h t a i n c dw i t h r r . r I m i r ; . zl.5 Although it is not conrrnonlyuscd,MATLAB allows 3:t = t:12 1:Returnto Stcp2 nuntben lo bc cxpresscdin single prccisiolr.Each valuc 5 : t= 2 x t is storcdin :1 hytcs with I bit lor thc sign, 23 hits li)r thc Convcrt thc lirllowing basc-2 nurnbcrsto basc l0: l 0 0 rl n dI 1 0 . ( X ) 1 0 1 . Convert the lirllowing basc-l{ nurnhcls to basc l0: and3.1,1. Forcomputcrs, thc rnachinccpsilon c can also hc
tro
ERRORS ROUNDOFF AND TRUNCATION
mantissa,and 8 bits fbr the signedexponent.Determinethe smallcstand largcstpositivefloating-pointnumbcrsas u,ell as the machine epsilon lor single precision represcntation. Note that the exponcntsrangefrom - 126to 127. .1.6 For the hypotheticalbase-10computcrin Exarnplc4.2, provethatthe rnachincepsilonis 0.05. .1.7Thc derivativeot l(.r) = l/1 I - 3xr): is givenby
4.12 Use zero- throughthird-ordcrTaylor sericsexpanslons to pre'dic:t.l(3) firr I (;r) : 25-1r- 6-12+ 7x - 88
usinga bascpointat.r: l. Computethe truepercentrelative error c, for eachapproximation. zl.l.l ProvethatEq.(4.I I I is exactfbr all valucsof.r if'l(-r)= urz+hr+c. 6-r 4. l,l Usezero-throughfourth-ordcrTaylorsericsexpansrons (I - 3rt), to predictl(2) fbr f(-r) : In "r'usinga basepoint at,r = l. Do you cxpectto havc diff iculticsevaluatingtlris Ir-rnction Computc the true percenlrclativc cror r, firl each appsoxiat ,r : 0.5771 Try it using 3- and ,l-digit arithmcticwith mation.Discussthc rneaningoI thc results. ,1.15 Usc lirrward rnd backwarcldillirencc approximations chopping. ctfO(h) and a ccntercddiff'ercnceapprtrximationof O(ft2)to l.tt (a) Evaluatethc polynornial cstimalc thc l'irst dcrivativc ol' the firnctioncxamincdin I:,rr-7-r:+u,r-o.-15 Prob.4. 12.Evaluatothe clcrivativcat.r.: 2 usinga stcpsize h : 0.2. CompareyoLrrrcsultswith the true valueof the c>l' irt .r.: 1.37.Usc 3-digit arithrnclicwith r:hopping.Evaluatc Interprctyour resultson the hasisol'thc remaindcrivativc. thc perccntrclativc crror. ol'lhc term Til) ltrr serir'scxpiln)ii{)r). dcr (b) Rcpeat(a) hut cxplcssI'as .f.16 Usc a ccntcrcddillercnccapproxirnation of 0(h2)to in cstimatethe seconddcrivativcof'thc lunctionexamined ,r': ((,r - 7).r * 8).r 0.35 Prob.-1.12.Pcrlirrnrthc evirluationat .\': 2 usingslepsizes Evaluutcthe clror and cornparewith part (a). ol'/r - 0.25 and0. 125.Curnparcyour cstimatcswith thctrue '1.9 Thc lirllowing infinitc scricscan hc used to approxivaluc ol'thr' sccontldclivativc.Intcrprctyoul rcsultson the n l a t cd ' : basisol'thc rcrnaindcrtcnr ol'thc Taylol scricscxllunsion. 1 . 1 7 l l l . r l < I i t i s k n o w nt h a t -rl .t l .{ r"-Ii-tl-r l. I .lI n'. (a) Provc that this MaclaLtrirr \criL'\ rr\piln\iruris a spucial cascol'thc Taylor sclicscxpansiun(Eq. 4. l3) with .r,0 ancllr : .r. ( b ) U s c t h c 1 ' a y l o sl c r i c st o c s t i n r a t c . l ' ( . r )c: ' a t , r i * r : I lirr.r,: 0.2. Enrploythc zcro-.l'irst-,sccond-,and thirdortlcrvelsionsand contputcthc lr:rI lirr cachcasc-1.10The Maclaurinscricscxparrsion lirr cos-r is
ll
RcpcatProb.zl.l0 lirr this scricsli)r -r : 0. I . ,l.llJ To calculatca planct'sspacccoordirrates, wc havet0 s o l v ct h c l u n c t i o n l(r'):
t - I-0.5sin.^
Lct thc bascpoint bc u: xi : rf2
R r: t
i'
2.1 o\
Yea
to s
The sen time
Rootsond Optimizstion 2.1 ovERvtEw Yearsago, you leamed to use the quadratic formula
*b+4F-4ai (PT2.1) LU
to solve
:.:,,
(Yrz.2)
fk):qf+bx+c:o
The valuescalculatedwith Eq. (PT2.1) are called the "roots" of Eq. (PT2.2).They represenl the valuesolx that make Eq. tPT2.2'lequalto zero.For this reason.rootsare sometimescalled lhez.eros of the equation. Although the quadraticformula is handy for solving Eq. (W2.2), there are many other functions for which the root cannot be determinedso easily. Before the adventof digital computers,therewere a numberof ways to solve for the roots of suchequations.For some cases,the roots could be obtainedby direcf methods,as with Eq. (PT2.l). Althoughtherewere eQuations like this that could be solved directly.there were,many more that could nol. ln suchinstances. the onlv alternative is an approximatesolution technique. One method to obtain an approximate solution is to plot the function and determinl where it crossesthe r axis. This point, which representsthe r value for which/(x) : 0, is the root. Although graphicalmethods are useful for obtainingrough estimatesof roots, they are limited becauseof their lack of precision.An alternativeapproachis to use trial and error. This "technique"consistsof guessinga value of x and evaluating whether/(r) is zero.If not (as is almost always the case),anotherguessis made,and/("r) is again evaluated to determine whether the new value provides a betterestimateof the root. The processis repeateduntil a guessresultsin anfir) that is closeto zero.
ltl '
:,.;.
rt2
PART 2 ROOTSAND OPTIMIZATION
l'(;r) = o ,f"(r) < o f(x) = o
.l'('r) = 0 f"G) > 0
PT2.I FIGURE the differencebelween rootsond oplimo A funciionof o singlevoriobleillustrciing
Suchhaphazardmethodsareobviouslyinefficientand inadequatefor the requirements of engineeringpractice.Numerical methodsrepresentalternativesthat are also approximate but employ systematicstrategiesto home in on the true root. As elaboratedin the following pages,the combinationof thesesystematicmethodsand computersrrrakesthe problemsa simple and efficient task. solutionof most appliedroots-of-equations Besiclesroots,anotherfeatureof lunctionsof interestto engineersand scientistsareits minimum and maximum values.The detenninationof suchoptimal valuesis ref'en-ed to as you As learned in calculus, solLltions be obtirined analyticalty optimi.ation. such can by deat which function f'lat; is, its derivative is zero. terminingthevalue the is that where Although such analyticalsolutionsare sometimesfeasible,most practicaloptimizationproblemsrequire nunrerical,computer solutions.From a numerical standpoint,such numerical optimizationmethodsaresinrilarin spirit to the root locationmethoclswejust discussed. Thatis, guessing fbr The involve and location on function. fundamental difference both searching a a betweenthe two types of problemsis illustratedin Figure PT2.1. Root location involves searchingfor the locationwherethe functionequalszero.In contrast.optimizationinvolves searchingfbr the functiorr'sextremepoints.
2.2
PARTORGANIZATION The first two chaptersin this part are devotedto root location.Chapter5 focuseson bracketing methodsfor finding roots.Thesemethodsstartwith guessesthat bracket.or contain, the root and then systematicallyreducethe width of the bracket.Two specificmethodsare covered:bisectionand.falseposition. Graphicalmethodsare usedto provide visual insight into the techniques.Error formulationsare developedto help you determinehow much computationaleffort is requiredto estimatethe root to a prespecifiedlevel of precision. Chaltter6 coversopenruetltods.Thesemethodsalso involve systematictrial-and-error iterationsbut do not requirethat the initial guessesbracketthe root. We will discoverthat thesemethodsare usuallymore computationallyefficientthan bracketingmethodsbut that they do not always work. We illustrate severalopen methods including theJixed-point iteration, Nev,ton-Raph.rolr, and secantmethods.
2.2 PART ORGANIZATION
S L '
K-
n, re nt )h or at at nt
rr3
Following the descriptionof theseindividual open methods,we then discussa hybrid approachcalledBrent's root-findingmethodthat exhibits the reliability of the bracketing rnethodswhile exploiting the speedof the open methods.As such, it fbrms the basis fbr MATLAB's root-findingfunction,f zero. After illustratinghow f zero canbe usedfor engineeringand scientificproblemssolving, Chap. 6 endswith a brief discussionof special methodsdevotedto finding the rootsof polynontials.In pafticular,we describeMATLAB's excellentbuilt-in capabilitiesfor this task. Chapter Z dealswith optimization.First,we describetwo bracketingmethods,goldensectionsearchandparabolic interpolation,for finding the optima of a function of a single variable.Then,we discussa robust.hybrid approachthat combinesgolden-section search and quadraticinterpolation.This approach,which again is attributeclto Brent, fbrms the basis for MAILAB's one-dimensionalroot-finding function:fminbnrl. After describing and illustrating fmlnbnd, the last part of the chapterprovidesa brief descriptionof optimization of multidimensionalfunctions.The emphasisis on describingand illustratingthe use of MAILAB's capability in this area:the fminsrearch function. Finally, the chapter endswith an exampleof how MATLAB can be employedto solveoptimizationproblems in engineeringand science.
.\
Roots:Brocketing Methods
N et
rf
5.r R
A ar dr rh
CHAPTEROBJECTIVES Thc primaryobjectiveof thischapteris to acquaintyou with bracketingmcthodsfbr findingthe rootof a singlenonlinearequation.Specificobjectivesandtopicscoveredare ' ' . . ' .
what rootsproblemsnrc and wherethey occurin cngineeringand Uncierstaniling science. Knowing how to determinea root graphically. Llnderstanding the incrementalsearchmethodand its shortcomings. Knowing how to solvea rootsproblemwith the bisecl.ion rnethod. Knowing horvto cstimatetheerrorol'biscctionanclwhy it difl'crsliom error estimatesfbr othcrtypesclf root locationalgorithms. Understanding lalsepositiclnand how it difl'ersfiom biscction.
e d i c a ls t u d i e sh a v ec s t a b l i s h ctdh a t a b u n g e ej u n r p c r ' sc h a n c e so f s u s t a i n i nag signilicantvertebraeinjury increasesignificantlyit'the liee-1allvelocityexceeds 36 m/s afier4 s of fiee fall. Your bossat the bungec-jumping compirnywantsyou to detet'minethe ntassat which this criterion is exccedeclgiven a drag coeliicientof 0.2-5kg/rn. You know f}om your previousstudiesthat the fbllclwinganalyticalsolutioncanbe usedto prcdictfall velocityas a functionof tirne: lntf^t r ' ( r ) : , / i - r i r n h' , l : - , |
V t,r
\V rtt /
ra tic pa
mi pe an ca
rt_B-lE s.l Fund3
YOU'VE GOT A PROBTEM
rsrr
Try as you might, you cannotntanipulatethis equationto explicitly solvefbr rn-that is, you cannotisolatethe masson the lefi sideof the equation.
tl4
at pr
Fundomentol Principle
De Vc
Heotbolonce Mossbolonce
T^.
Energybolonce
Cc of Mc ofl Ch
Newton'slows of motron Kirchhoff s lows
orl Cu
Forcebolonce
AC(
tt5
5 . I R O O T SI N E N G I N E E R I NAGN D S C I E N C E
An alternativeway of looking at the probleminvolvessubtractingu(t) from both sides to give a new function:
:,1*,un(,F') -,,', r(m)
(5 . 2 1
Now we can seethat the answerto the problem is the value of ru that makesthe function equalto zero.Hence.we call thisa "roots"problem.This chapterwill introduceyou to how the computeris usedas a tool to obtainsuchsolutions.
5.I
R O O T SI N EN GIN E E R INAGN D S CIENCE Although they arise in other problem contexts,roots of equationsfrequentlyoccur in the principlesthatareroutinelyusedin areaof design.Table5. I listsa numberof tundamental designwork. As introducedin Chap. 1, mathematical equationsor modelsderivedfiom theseprinciplesare employedto predict dependentvariablesas a function of independent rurrr,b\t:. torii.r,g{i,:r,:'j,cr,:.'s.dpN.rm'-t--rs.\ote \hrt \r.',rlh'i.rsi.'Jrt dtpendtrt. r'*i,ables ret'lectthe stateor pertbrmanceof the syster.n. whereasthe pararnetersrepresentits propertresor composition. An examp\e of such a mode\ is the equationfor the bungee\umper's ve\ocity. If the pa.'-'.::':r..|i:.-.-'.:.'.):l:.:,'
.i\'..'i:..-\.^\'::-^\j-'-::-'..'.:.-'-\-:.:':.).r:'-
) S
1 -:.i..
parameters. That is, it is isolatedon one sideof the eqLralsign. However, as posed at the start of the chapter,supposethat we had to determinethe ma.ssfbr a jumper with a given drag coefficientto attaina pre.scribedvelocirytin a .settinre period.AlthoughEq. (5.1) providesa mathematical representation of the interrelationship it cannotbe solvedexplicitlyfbr nrass.In such amongthe modelvariablesarrdparameters, cases./r, is said to be irrrnlicit.
5. t
principlesusedin designproblems Fundomentol Dependenl Vorioble
D0iOriCe bolonc-. 00lonce
Independent Vorioble
tenpeteioie C o n c e n l r o l i o no r q u o n t i t y of moss M c a n i t u d e o n d d r t e c lo n
ine
CtC
pOS,it)rl
T i m eo n d p o s i l i o n T i m eo n d p o s i t i o n
. . ::] ewlonslorts ol )l of rchhoff s iorrys
p o t e n t l oe l nergy Acce/ercllion,veloci\', or ocotron Currenlsond vo/toges
Poromelers Itei,aa picpeiites a: ratai a: :i:ie': Ee')"4"/ C h e m i c o lb e h o v i o ro f m c l e r l o l ,m o s sl r o n s f e r , syslem(leomelry 5 l r e r r g t lor l r n o l e r i o ls. l r u c i u r cpl r o p e r le s , s y s l e r n
qeometry T r m eo n d p o s r t i o n Lme
Alc-rss of molerrol syslemgeomefry, dissipoiive potcmerers F l o r t r l r' -a l n r - n o r l r a q r"f
induclonce)
/ro
opollronra
tt6
5.3
ROOTS: BRACKETING METHODS This representsa real dilemma,becausemany designproblemsinvolve specifyrngthe propertiesol cornpositionof a system(as represented by it.spararneters) to ensurethatit perfornrsin a desiredmanner(as represented by its variables).Thus, theseproblemsoften reqr-rile the determination of implicit parameters. The solutionto thc dilemmais providedby numericalmethodstbr rootsof equations. To solvethe problemusingnumericalmethods,it is conventional to reexpress Eq. (5.1)by The subtractingthedcpendentvariableu frornboth sidesof the equationto give Eq. (-5.2). valueof'nithatrnakcs.f(m):0is,thercfbre,therootoftheequation.Thisvaluealsorepresentsthc rnassthat solVesthe clesignproblem. The lbllclwingpagesdealwith a varietyof numericaland graphicalmethodsfbr deterrniningrootsof relationships suclras Eq. (5.2).Thesetechniclues can be appliedto many in cnginecringand science. otherproblcn.rs conl'ronted ror-rtincly
5.2
pro
cal t
ans
whi par
GRAPHICATMETHODS
wh
A sinrplenrcthodlbrobtainingan estir.nate of the root of the ecluation ./(r) :0 is to make a plot ol'thc firnctionant'lobservcwhercitcrossesthc.raxis.This point,which represents t h e . rv a l u el b r w h i c h . l ( r l : 0 . p r o v i d c sa r o u g ha p p r o x i r . n a t ioofnt h er o o t .
E X A M P5 L .EI
Ho ma
TheGrophicolApprooch Problem Stotement. Use thc graphicalapproachto dcternrinethe nrassof the bungee jurrper with a drag coel'l'icicnt ol'0.25 kg/rn to havca velocityol'36 m/s after4 s of free 1 a l l .N o t c :T h e a r c c e l c r a t i oonl ' u r i r v i t vi s 9 . 8 1n r / s r .
un me ab pl(
s c t su p a p l o t o l ' E q .( 5 . 2 )v e r s u sm a s s : S o l u t i o n . T h c l i r l k r w i n gM A T L A B s c s s i o n
Cv
''
r'ril = nlll ''
(.).2,\; q
c ). l l 1 ;
I = 4; v '; ' 1 r.. 1 , , I f ) = r i q f L ( r 1 * m 1 r c, : c 1 ). * t . a n l r ( : ; q r 1 ( . J * c d . / n l r ) * L ) plot (rrr;r,11-r),qrr11
' ::'
-
ro ar ha
'\6;
!'
Fc tr rh
te
-1
pr u ri
-z
5.3 "50
I
I] e 150
tt7
METHODSAND INITIALGUESSES 5.3 BRACKETING
The functioncrossesthc llr axisbetweenlrl0 and l50 kg. Visualinspectionof the plot providesa roughestimateof the root of 145kg (about320 lb). The validity of the graphical estimatecan be checkedby substitutingit into Eq. (5.2)to yield >> sqrt (g*145/ccL)*r,anh(::qrt,(g*cd/1,i5)*t,)
v
0.0456 which is close to zero. It can also be checked by substitutingit into Eq. (S.l ) along with the parametcr values liorr this exarnple to give > > - s q r t , ( q * 1 4 f i l c o ) * t , a n h ( s c T r t -( q * c d / r 4 5 ) * t )
35.0456 which is close to the desired fall velocity ol-36 rr/s.
are ilf limited practicalvaluebeciiusethcy arc not vcly precise. Craphicaltechniques Howevcr,graphicalmethodscan be utilizedto obtainroughcstimatcsol'roots.Theseestiin this chapter. matescan be ernployedas startinggucsscslor numcricalmcthodsdiscussed ol'tlreroot,graphiculinterpretations areusefirlfirr Asidefiorn providingroughestimatcs understanding the plopertiesol'the lirnctionsand anticipatingthe pitlallsof the nurnerical rrethods.For cxample,Fig. -5.1showsa numberof'ways in which rootscan occur (or be by a lowcr bound,r1andan uppcrbound.r,,.Figurc-5.lb dcabsent)in an intervalprescribcd pictsthecasewherea singlcroot is brackctcdby ncgativeandpositivevalueso1'./(r). Howe v e r , F i g . 5 . l r l , w h e r/ (c- r r ) a n d . / ' ( - r , , ) a r e a l s o o n o p p o s i t e s itdhces.or af x i s , s h o w s t h r c c rootsoccurringwithin the interval.In general,if / (.rr)and ./ (-r,,)havcoppositcsigns,thcrc arean odd numberof rootsin the interval.As indicatcdby Fig.-5.la anclr', il' l (rr ) and.f (.r,,) havethe sarncsign,thereareeitherno rootsor an evennunrbcrol'rootsbctwecnthc values. arcusuallylrue,lherearecaseswherethcy do not holcl. Althoughthcsegcncralizalions F o r c x a r n p l et ,i r n c t i o n tsh a ta r ct a n g c n t i at lo t h c , r a x i s( F i g .- 5 . 2 aa) n dd i s c o n t i n u o ul 'su n c t i o n s( F i g .5 . 2 b )c a nv i o l a t et h e s cp r i n c i p l c sA. n c x a r n p l co l ' a I ' u n c t i o tnh a ti s t a n g e n t i at tl l t h ca x i si s t h ec u b i ce q u a t i o n. / ( - r ' ): ( r - 2 ) ( . r-' 2 ) ( x - 4 ) . N o t i c ct h a t , r: 2 t n a k c st w t r t e r m si n t h i s p o l y n o m i acl q u a lt o z e r o .M a t h e r n a t i c a l l y , . r : 2 i s c a l l e d' . tt r r u l t i l t l cn r t t . that arc Although they are bcyond the scopeof this book, thereare specialtcchniqr"rcs expresslydesignedto locaternultipleroots(Chapraand Canale.2002). The cxistenccof casesof thc typedepictcdin Fig.-5.2nrakesit difiicult to dcveloptirolproof computeralgorithmsguaranteed to locateall the rootsin an interval.Howevcr,whcn the rnethods describedin thc lirlkrwingsccusedin conjunctionwith graphicalapproaches, tions are extrerrelyusefulfor solvingrnanyproblernsconfl'ontedroutinelyby cnginccrs, scientists, andappliedrnathematicians.
METHODSAND INITIATGUESSES 5.3 BRACKETING If you hada rootsproblemin thedaysbefbrccomputing,you'd oftenbe told to use"trial and until the function error" to come up with the root.Thiit is, you'd repcatcdlymakeguesses was sufTiciently closeto zero.The processwasgrea(lylacilitatedby the adventof sofiware
tt8
ROOTS: BRACKETING METHODS
xl
xu
(bl
(d)
F I G U R E5 . I I lustrolion of crnumberof generolwoysthoio roolmoy prescribed by o lowerbound.r,ond occurrnon intervol on upperboundx,,. Poris(oJond (c)indicoiethctif both ,/ (rr) ond ./(r,,) hovethesomesign,eithertherewili be no rootsor therewill be on evennumberof rools w i t h i nt h ei n t e r v oP l o r t (sb )o n d ( d )i n d i c o tteh o tl f l h e f u n c l i ohno sd i f f e r e ns ti g n so i t h ee n d p o i n t si ,h e r ew i l l be on odd numberof roolsin theintervol.
FIGURE 5.2 lllustrotion of someexceptionslo lhe generolcoses depicted in Fig. 5. 1 (o) Multiplerootsfhot,occurwhen the functionis tongentic to the -r oxis. For thiscose, ol'h., 'nh lnc e"d
nninls nr'p rf nnno.ile
s;.n<
iho'c ore
for the inlervol. on even numberof oxis inlercepiions (b) Discontinuous funciionswhere end pointsof opposiie s i g ^ o ' o c [ e t o n e v e nn u r r b e r , or[o o ' s . , S p ei oc l s t r o ' e g i e s ^ro
,o^,
",ol
lnr nolarmrn,nn
Jho ,^^r.
[^,
thoa^
.-.o.
5.3 BRACKETING METHODSAND INITIALGUESSES
lr9
tools suchas spreadsheets. By allowing you to make many guessesrapidly, suchtools can actuallymake the trial-and-errorapproachattractivefbr someproblems. But, for many other problems,it is preferableto have methodsthat come up with the correctanswerautomatically.lnterestingly,as with trial and error,theseapproachesrequire an initial "guess" to get started.Then they systernaticallyhome in on the loot in an iterative fashion. The two major classesof methodsavailableare distinguished by the type of initial guess.They are . .
As the namc implies, thescare basedon two initial guessesthat Brttcketingmethod.s. "bracket" the root-that is. are on either sidc of the root. but thereis no Opertmethotls.Thesemethodscan involveone or moreinitial guesses, needfor them to bracketthe root.
For well-posedproblems,the bracketingmethodsalwayswrlrk but convergeslclwly (i.e.,they typically take more iterationsto home in on the answer).In contrast,the open rnethodsdo not alwayswork (i.e.,they can divergc),but when they do they usuallyconvergequicker. arerequired.Thesemay naturallyariseliom the physical In bothcases,initial guesses contextyou are analyzing.However,in othercases,good initial guessesmay not be obvious.In suchcases,automated approaches to obtainguesses would be useful.The following sectiondescribes one suchapproach,the incremental search. 5.3.1 Incremenlol Seorch When applyingthe graphicaltechniquein Example-5.1,you observedthat /'(.r) changed sign on oppositesidesof the root. In general,if ./(x) is real and continut'rus in the interval from x7 to,r,, and .l (,r7)and f (x,,) haveoppositcsigns,that is, / ( - r 1 ) / ' ( x , , )< 0
rre
(5.3)
thenthereis at leastone real root between.r1afld,r,,. Ircrenrentalseun:hmethodscapitalizeon this observationby locatingan intcrval where the functionchangessign.A potentialproblcm with an incrementalscarchis the cl.roice ol'the incrcmentlength.If the lengthis too srnall,the searchcan be very time consuming.On tho other hand, if the length is too great,there is a possibilitythat closely by the possibleexisspacedrootsmight be misscd(Fig. 5.3).The problemis compounded tcnceof multipleroots. An M-file can be developedrthatimplementsan incremental searchto locatethe roots of a firnctionf L.rncwithin the rangefrom xmin to xmax (Fig. 5.4).An optionalargument ns allowsthe userto specifythe numberof intervalswithin the range.lf ns is ornitted,it is automatically setto 50.A f or: loop is usedto stepthrougheachinterval.In the eventthat a sign changeoccurs!the upperand lower boundsarestoredin an arrayxb.
site lies ' This lirnction rs a rnoditledversionof an M-lilc originally prcscntcdby Recktenwald(2000).
5.2 EXAMPLE
FIGURE 5.3 theincremenlol lengthof lhe seorchprocedureis becouse Coseswhererootscouldbe missed o n d w o u i db e missedregordlessof the t o o l o r g e N o t et h o t h el o s tr o o to n l h er i g h it s m u l t i p e i n c r e m elneln g l h .
xb = incsearch(func,xmin,xmax,ns) function incremental search root fo.-aLor ? incsearch: xb - incsearch (func, xmin, xmax,ns) : % finds brackets of x that contain sign chattges % on an interval. of a function ? ? inpuL: furLc - name of funct-iorr % Y xni -, xmax cndpoitr(s oI inrerva] (default = 50) ? ns = number of subintervals % outpuL: xb(k,1) is the lower bound of Lhe kth % 2 xb (k,2 ) is tire upper bound of t-he kth If no brackef.s found, xb - Ll. % if
nargin
< 4,
ns = 50;
encl %rf ns blank
sigtn chatrge sign change
set
to
50
search Z Increm€ntal x = I i n s p a c e ( x m r - n ,x m a x , n s ) ; f - func (x) ; unless sign c|ranqe det ec t ed nb = 0r :{b - i I ; e.xb is: null for k = 1; length (x) -1 if sign(f (k)) ^-- sign(f (k+-L)) licheck for sign chanqe nb = nb + 1; . x{kl; xbrnl,l) xb(t'tb,2) = x (k+1); end end jf isempt),,(xb) ?displzr-v that no brackets disp ( 'no brackets forind' ) or increase ns' ) disp ( 'check rnterval disp('nr:mber disp (nb) end
ol
brackets:')
%drspLay number of
FIGURE5.4 seorch An M-file fo irnplemenlon incrernelrtcl
r20
were
f ound
brackets
5.3 BRACKETING METHODS AND INITIALGUESSES
t2l
EXAMPLE 5.2 Incrementql Seorch ProblemStotement. UsetheM-flle i ncsearch (Fig.5.a)to identifybracketswithin the interval[3, 6] fbr the function:
(s.4)
J G ) : s i n (l 0 . r ) * c o s ( 3 . r ) Solution. The MATLAB sessionusingthe detaultnumberof intervals(50) is >>
incscarch(@r
numbcr
sin(10*x)
possiblc
of
+cos (3*x),3,
6)
rooLs:
5 afnS 3 . 2 1 , 49
t.3061
3.3061
3.36'.t3
3. /34-t
3. t9\9
4.6531
4.'-t14j,
\.{:i2
l
a.6939
A plot ol'Eq. (5.4)alongwith thc root Iocationsis shownhere.
4.5
?E
J
5.5
Althoughfive sign changesare dctccted,becausethe subintervals are too wide, the function rnissespossiblerootsat .r j 4.25and-5.2.Thesepossiblemots look like they might be doLrblenrots.However,by usingthe zoom in tool, it is clearthat eachrepresents two real rootsthal are very closc together.The f'unctioncan be run againwith more subintervals with thc resultthat all nine sign changcsarc located > : . i n c s e a r c h ( t l x s i n ( ' lO * r ) + c o s ( 3 * x ) , 3 , number of 9 3 .2 424 3.3536
possiblc
rooLs:
3 .212.7 3.3939
b,100)
122
ROOTS: BRACKETING METHODS 3.-t213 4.2121 4.2424 4.5910 ,r.1515 5.1318 5 . 566J
3.t57ia 4 .2424 4.2',t21 4 . 1 ? , 13 5.1B1B 5.2r2). ':.591 A
2 The lbregoingexampleillustrates that brute-lbrcenrethodssuchas incremental search are not lbolproof.You would be wise to supplementsuchautomatictechniqueswith any otherinlbrnrationthatprovidesinsightinto the locationol'the roots.Suchintbrrnation can be fbunclby pkrttinglhe functionand throughLrnderstandinc the physicalproblernfrorn w h i c ht h ee q u a t i o no r i g i n a t e d .
5.4
FIG Ac f,oi
Ex plc let Th
BtSECT|ON The bisectionntctlu)dis a variationof the incremental searchrnethodin which the interval is alwaysdivideclin hali. If a functionchangessign over itn interval,the firnctionvalueat the rnidpointis evaluated. The locationof the root is thendeterminedas lying within the subintervalwherethe sign changeoccurs.The subintervalthen beconresthe intervalfix the next iteration.'flreprocessis lepeateduntil the root is known to the reqr"rired precision, A graphicaldepictionof the methodis provicledin Fig. 5.5.The lbllorvingexamplegoes throughthe actualcornputations involvedin the method.
E X A M P L E5 . 3
<1
T h e B i s e c i i o nM e t h o d ProblemStotement. Usebisectionto solvethe sarneproblenrapproached graphically in E x a r n p l5e. l . Solution. The first stepin bisectionis to guesstwo valuesof the unknown(in the present problenr.rr) thatgive valuesl'or t 0n') with dillelent sisns.Frornthe graphicalsolutionin
Nc lat
Nr
wl thr ,, t,
r23
5.4 BISECTION
.f(nt) 2 U
-2
-6 First iteration
I
V
"r/
xl
#
Second iteration
I t
xt
Third iteration
)(f
x,,
xt.\r
xu
ffi
I
V
Fo_.t
Fourth iteration
FIGURE 5.5 A grophicoldepiclionof the bisectionmethod Thisploi correspondsto the firsifour iterotions trom Er.omple5 3
Example5. l, wc can seethatthe t'unctionchangessignbetweenvaluesclf-50and 200.Thc p l o t o b v i o u s l y s u g g e s t s b e t t e r i n igt u i ael s s e s , s al y4 0 a n d I 5 0 , b u t t i l r i l l u s t r a t i ' u ' t ' p u r p o s r - ' \ lct's assumewe clon'thave the bencfitof the plcltand havc madeconservativeguesses. Thcrcfble,the initial estimatcof the rtx>t.r,.lies at thc midpointof the interval .50+ l(x)
_
| -)<
./. Notethattlreexactvalueof tlrerootis 142.1316. Thismeans thatthc valueof 125calculatedherehasa trueDercent errorof relative - r2-sI ,.,-l 142.1316 lx100?4:12.434/,
r,/r-
i
112.7316 I
Next wc computethe productof the functionvalueat the lower boundandat the midpoint: I ( 5 0 )I ( 1 2 - 5:1 - 4 . 5 1 9 ( - 0 . 4 0 9 1: 1 . 3 7 1 which is greaterthanzero,and henceno sign changeoccursbetweenthe lower boundand the midpoint.Consequently, in the upperinterval betweerr125and theroot mustbe locatecl 200.Therefbre,we createa new intervalby redefiningthe lower boundas 125.
124
BRACKETING METHODS ROOTS: At this point,the new intervalextendsfronrxl : rnatecau thenbe calculatedas
l2-5to r, :200. A revisedl'ootesti-
125+ 20t) 2 r v h i c h r e p r e s e n t s a t r u e p e r c e n t e rlreo,lr:o1f3 . 8 5 % . T h e p r o c e s s c a n b e r e p e a t e d t o o b tain refineilestinrates. For example, - 0 . 4 0 9 ( 0 . 3 . 5 9: ) - ( t . | 1 l . f ( t t s l . 11 16 2 . - 5:; Therelbre.the roo( is now irr the lower intervalbetween125and 162.-5. The upperbound is rede[inedas l 62.5.anclthe root cstinratetirr the third iterationis calculatedas
1 2 5* 1 6 2 . 5 : | 43.7-5 ) , i
The nrethodcan be repeated until which represents a percentrelativeelrur of a, : 0.101)()(,. the resultis accurafeenoughto satistyyour needs. We endedE,xarnple 5.3 with the statement thatthe rnethodcouldbe continuedto obtain of therrxrt.We rnustnow developan olr.jective criterionlirr decidingwhen a lel'inedr'stimate to terminatethe rnethod. rnightbe to end the calculationwhen the errrlrl'allsbelow some An initial suggestion p l c s p e ci e f d l e v e l .F o r i n s t a n c ei n. E x a r n p l e 5 . 3 .t h c t r L r e r e l a t i v ee r r o r d r o p p e fdr o m l 2 . 4 3 t
T] te
to T} on
Fl Er of
(55)
lo','l:lff]l(x)%'
whc.re.r]1""is the loo{ lirr the present iteration and -rf'l'lis the rool lionr the plevious itemt i o n . W h e n c , , b e c o r n c sl e s s t h a n a p r e s p e c i l ' i e d s t o p p i n gc r i t e r i o n e , , t h e c o n r p u t u t i o ni s Icillriilulc(1.
E r r o rE s t i m q t e fso r B i s e c t i o n
E X A M P L E5 . 4
P r o b l e mS t o t e m e n t . C o n t i n u eE x a r n p l e5 . 3 u n t i l t h e a p p l o x i r n a teer r o r t a l l s b e l o wa s ( o p p i n c r i t e r i o rot l ' € . ,: 0 . - 5 % . U s eE q . ( - 5 . - 5 t o) c o n r [ ] u t teh ee r r o r s . S o l u t i o n . T h e r e s u l t s o l ' t h e f i r s t t w o i t e r a t i o n s l b r E x a r r p l e 5 .l 32 w 5 ae nr ed l 6 2 . 5 . S u b s t i t u t i n gt h e s ev a l u e si n t o E q .1 - 5 . -y5i1e l d s ,
l u rl,:,
lrAl 5-r?sl #ll\l0o/,,:23.08c/o I6t.5 I
Fig exh
---__--*---+
5.4 BISECTION
125
Recallthat the true percentrelativeeror for the root estimateof 162.5was 13.85%.Therefore, le,,I is greaterthan le, I. This behavioris manifestedfor the other iterations: Iterolion
xt
50 125 125 125 1 3 43 7 5 t39 a625 l4t 4463 t 4 25 7 B l
In )n 1e +-1
xu
204 204 t 6 25 I43.7 5 143.75 1 4 37 5 1 4 37 5 1 4 37 5
xl
125 t 6 25 1 4 37 5 1 3 43 7 5 r3 9 0 6 2 5 14t 4063 |42 578t |43t64)
le,l ("/"1 2 30 B 1 34 4 698 337 )66 o82 o4l
le,l l7"l 1 24 3 t385 a7l 586 258 093 0il 030
Thus aftereight iterationsle,,l finally falls below €.,:0.5c/o, and the computationcan be terminated. Theseresultsare summarizedin Fig. 5.6.The "ragged"natureof the true error is due to the fact that,fbr bisection,the true root can lie anywherewithin the bracketinginterval. The true and approximateerrors are far apart when the interval happensto be centered on the true root. They are close when the true root firlls at either end of the interval. F I G U R E5 . 6 Errors for lhe bisection meihod.Trueond opproximcte errorsore piottedversus thenumber of iterolions.
CS
be VC
he all
Approximateerror, le,,l o
1n
.z o
rais
(t) rl or
Although the approximateerror does not provide an exact estimateof the true error, Fig. -5.6suggests that la,,l capturesthe generaldownwardtrendof le,l. In addition,the plot exhibitsthe extremelyattractivecharacteristic that le,,l is alwaysgreaterthan le,l. Thus,
126
ROOTS: BRACKETING METHODS when le,,I falls below s' the computationcould be terminatedwith confidencethat theroot is known to be at leastas accurateas the prespecifiedacceptablelevel. While it is dangerousto draw generalconclusionsfrom a single example,it can be demonstratedthat le,,l will alwaysbe greaterthan le,l for bisection.This is due to the fact that eachtime an approximateroot is locatedusingbisectiofloS.r,.: (xt * x) /2, we know that the trlle root lies somewherewithin an interval of A-r : xu - xt. Therelbre,the root must lie within *Ax/2 of our estimate.For instance,when Example5.4 was terminated, we could rnakethe definitive staternentthat
x, : 143.1641 +
- 142.5781 1.13.7500 : I ' 1 3 . l 6 4+l 0 . 5 8 5 9
In essence, Eq. (5.5) providesan upperboundon the true error.For this boundto be exceeded,the true root would haveto fall outsidethe bracketinginterval,which by definition could nevetoccur fbr bisection.Otherroollocatingtechniques do not alwaysbehave as nicely.Although bisectionis generallyslowerthan other methods,the neatness of its error analysisis a positivefeaturethat nrakesit attractivefor certainengineeringand scientificapplications. Another benefitof the bisectionmethodis that the numberof iterationsrequiredto attain an absoluteenor can be computedu priori-that is, before startingthe computation. This can be seenby recognizingthat beforestartingthe technique,the absoluteerroris
sll:r!l-rro:AxO where the superscriptdesignatestlre iteration.Hence,beforestartingthe methodwe areat the "zero iteration."After the first iteration.the errorbecomes , "2
A.r"
F'-_
Becauseeachsucceedingiterationhalvesthe error,a generalfbrmula relatingthe errorand t h e n u r n b eor l ' i t e r a t i o nrs i s E (" - -
A-rt' .\D L
If C,,.,1 is the desirederror,this ecluation can be solvedfbri
:,"r,(#)
log(L.r\tf E,,.,1) log 2
(5.6)
Let'stesttheformula.For Example-5.4,theinitialintervalwasA.ro : 200 - 50 : 150. After eisht iterations. the absoluteerrorwAS tr-
- 142.518t1 1r43.7s00 =
0.58-59
Wecansubstitr-rte thesevaluesintoEq.(5.6)to give n: log:(150/0.5859):8 I MATLAB providesthe I oq2 function to evaluatethc base-2logarithmdirectly.lf the pocket calculatoror computerlanguageyou are usinl doesnot includethe base-2logarithm as an intrinsic function.this etluation showsa handy way to computc it. ln general,log,,(,r)= log(_r)/log(b).
-.------
5.4 BISECTION
127
Thus, if we knew beforehandthat an error of lessthan 0.5859was acceptable,the formula tells us that eight iterationswould yield the desiredresult. Although we haveemphasizedthe useof relativeenors for obviousreasons,therewill be caseswhere (usually through knowledgeof the problem context) you will be able to specifyan absoluteerror.For thesecases,bisectionalongwith Eq. (5.6)can providea useful root locationalgorithm. 5.4.1 MATTAB M-file! bisecr An M-file to implementbisectionis displayedin Fig. 5.7. It is passedthe function(func) alongwith lower(x1) andupper(xu) guesses. ln addition,anoptionalstoppingcriterion(es) F I G U R E5 . 7 A n M J r l et o i m p l e m etnhteb i s e c t i om nethod. f unct ion Iroot, ea, it.er I -bi secr (f unc, xl, xu, es, maxit, varargin) % bis;ect: r:oot Iocation zer.oes ? . .. ) : lroot,ea, iterl=bisec1,(func,xl,xu,es,maxit,p1,p2, % uses bisect-ion rnethod to find the root of func ? inpuL: func : name of function % ts xl , x r louer orrd .pl,Fr gues:-s = 0.0001%) es = desired relative % erroi: (default (defaul t = s0) % maxiL = maximum allowabl e iterations pl,p?,, ... = additional parameters used by func % % output: ? r oot ,. rea L root ? ea = approximate : rlumbef of ? rter
relative er:ror iterations
(t)
if nargin<3,error('at least 3 input argurnents requi red'),end test - tunc (xl.,varargin{ : } ) *func(xu,varargin{ : }) ; i f test>0 / €rror ( 'no si gn change' ) , end if nargin<1 I i sempty(es), es-0.0001.;end :lf nargin<5 i lsempty(maxit), maxit=50;end itcr- - 0; xr = xl; (l) while xr:old = xr'; (x1 +.xu)/2; xr'iLer = iLer + 1; * 1 . 0 0 ;e n d i f xr -= 0, ea = abs ( (xr , xrold) /xr) test = func (xl,varargin{: } ) *func (xr,varargin{: } ); if
< 0
test XU
=
XI;
-
xr;
=
0;
elseif xf
test
>'0
el se ea end i f
ea
<=
end l:OOt
=
Xl-;
es
iter
>=
maxit,break,end
r28
ROOTS: BRACKETING METHODS
5
and maximum iterations(maxit) can be entered.The function first checkswhetherthere are sufficient argumentsand if the initial guessesbracket a sign change.If not, an error message is displayedandthe functionis terminated. It alsoassignsdetaultvaluesif maxit a n d e s a r e n o t s u p p l i e dT. h e n a w h i l e . . . b r e a k l o o p i s e r n p l o y e dt o i m p l e m e ntth e bisectionalgorithmuntil the approximateerror falls below es or the iterationsexceed m a x it . We can employ this functionto solvethe problemposedat the beginningof the chapter. Recall that you need to determinethe massat which a bungeejumper's fiee-ftrll velocity exceeds36 rrls after4 s of free fall given a drag coefficientof 0.25 kg/rn.Thus, you havet0 find the root of 9.81m
f (m): \,1025 ,"n (
@,, -) - 3 6
In Example5.1 we generated a plot of this functionversusrltlssandestimatedthattheroot fell between140and 150kg. The bi sect f unctionfrom Fig. 5.7 can be usedto determine the root as *tarrh(sqrt, sqrL (9.81*m/0.25) ea i ter] =brsect ( fm, 40,200 )
>>
fm-G(rn)
>>
[mass
mclSS
(9.81*11.25lm)
*4)
Fr
36;
Fo
=
142.13'11 5.3450e
0Olr
pla q ' " ico
2.r Thus,a resultof nr : 142.1311kg is obtainedafter2l iterationswith an approximate relativeerrorol'€,,:0.0000-534-50/o.wecansubstitutetherootbackintothel'unctiontoverify that it yieldsarvaluecloseto zero:
is bis
EXAMPLE 5.5 Th
>:' fm(mass)
Prc
cal 4.6089c
007
So xu
5.5
FArSEPOSTTTON
Fir
Falseposition(alsocalledthe linearinterpolation method)is anotherwell-knownbracketing method.It is very similarto bisectionwith the exceptionthatit usesa diffbrentstrategy to come up with its new root estimate.Ratherthanbisectingthe interval,it locatestheroot . h e i n t e r s e c t i oonl ' t h i sl i n ew i t h b y j o i n i n g . l ( r r ) a n d . l ( , r , , )w i t h a s t r a i g hlti n e ( F i g . 5 . 8 ) T the r axis represents an improvedestimateof the root.Thus,the shapeof the functioninfluencesthe new root estimate.Using similartriangles,the intersection of the straightline with ther axis can be estimatedas (seeChapraand Canale,2002,for details), f (.r,,)(x1- r,,) .f Gt) - .f (.Y,,)
(5.7)
wh Se
5.5 FALSE POSITION
129
lre ror ir the )ed
ter. rity 3to
root rine 6;
F I G U R E5 . 8 F o l s ep o s i t i o n
This is the.fhl,sc-po,sition.litrmula. The vah-re of ,r,,computedwith Eq. (-5.7)then rep l a c c sw h i c h c v eor f t h et w o i n i t i a lg u c s s c s , . o r /r . r , , ,y i e l d sa l i r n c t i o nv a l L r w e i t h t h es a m e sign as l (.r,). In this way the valuesof .r7andr,, alwaysbracketthc true root.The proccss is repeateduntil the loot is estimatedadequately. Thc algorithmis identicaltclthe one for ( F i g .5 . 7 . w bisection 1 i t h t h r :c x c e p t i o n t h a rE q . ( 5 . 7 )i s u s e d .
relaerifY EXAMPLE 5 5
The Folse-Position Method Problem Stotement. Use lalse position1o solvc the sameproblernapproachecl graphic a l l ya n dw i t h b i s e c t i o ni n E x a n r p l e5s. 1 a n c -l 5 . 3 . S o l u t i o n . A s i n E x a m p l e- 5 . 3 ,i n i t i a t ct h e c o m p u t a t i o nw i t h l l u e s s e so f . r 1- 5 0 a n d .r,,: 200. F i r s ti t c l u t i o n :
rcketategy 3 root ) with )n inlt line
.rt : 50 'r,, : 200 .\,:200 -
./ (.ri) : -'4.519381 .f (.r,,) 0.860291 r(50- 2(n) 0.rJ6029
: l 76 . 2 1 l1 - 4 . 5 1 9 3 8 1- 0 . 8 6 0 2 9 1
which hasa true rclativeerrorof 23.5clo Seconditeration: (5.7)
. l ( . r r )f 6 , ) : - 2 . 5 9 2 1 3 2
t30
ROOTS: BRACKETING METHODS Therefbre,the root lies in the first subinterval,and r,. becomesthe upperlimit for the next iteration.r,, : 116.2713.
.f (rt) : -4.519381 .r, : 176.2113 t'Q,) :0.566114 0.566t74(50- t] 6.2113) : . r , : 1 1 6 . 2 7 7 3 -1519381 - 0.566174 t 62.3828 ,.-{n
which has true and approximaterelativeerrorsof 13.16a/c and 8.-56olc, respectively. Additional iterationscan be performedto refinethe estimates of the root.
Although firlse positionolien perfbrmsbetterthan bisection,there are other cases where it does not. As in the fbllowing exarnple,thereare certaincaseswhere bisection yieldssupericlrresults. E X A M P L E5 . 6
A C o s e W h e r e B i s e c t i o nl s P r e f e r o b l et o F o l s eP o s i t i o n ProblemStotement. Use bisectionand lalseposition(o locatethe root of 'l(r) :'r"'-l b e t w e e n- r : 0 r n d 1 . 3 . S o l u t i o n . U s i n sb i s e c t i o nt .h e r e s u l t sc a nb e s u n r m a r i z eads Iterotion l
2 ,J
4 5
xt
u
O (r5 0 975 o 975 o 975
X,,
3 ,J J
137 5 45625
r0 6.5 4 975 I 1375 ) 05625 I 0t5625
e" ("/"1
e, (o/ol
r 0 00 333 1 43 77 4A
35 25 t3B 56 6
Thus, alier l'ive iterations,the true error is reducedto lessthan 27r,.Frlr falseposition, a very diffbrentoutcomeis obtained: herotion I 2 3 4 5
xt
0 a a943O 0 tBtz6 o 26287 0 3 3 8 II
3 3 3 3 3
.r,
e. (Y"l
0 09430 a tBl76 o 26287 0338Ir a 40788
48I 309 223 l7 l
e,(7"1 906 BIB 737 662 592
After five iterations,the true error hasonly beenreducedto about59%. Insightinto theseresultscan be gainedby exarlininga plot of the l'unction.As in Fig. -5.9.thecurve violatesthe prerniseon which fhlsepositionwas based-that is, if .l(.rr) is muchcloserto
POSITION 5.5 FATSE
r3l
he next
r cases section
(;; ", 35 2.5 r3 8 56 t6
; r t l o n ,a
e, (7"1 906
Br 8 737 66.2 5q2
ght into .e curve loser to
F I G U R E5 . 9 P o to f . F ( x ): x l r t
n ethod { t h ef o l s ep o s i i i om ce 1 , i l l u s t r o t isnlgo wc o n v e r g e n o
zerothanf(x"), thentheroot is closerto x1thanto x,, (recallFig. 5.8).Becauseofthe shape of the presentfunction,the oppositeis true.
regardingroclt-location illustratesthatblanketgeneralizations The fbregoingexarrrple positiorris often supeas lalse method such possible. a Although rnethodsare usuallynot Therefbre, general conclusion. that violate this rior to bisection,thereare invariably cases the root substituting (5.5), checked by be the resultsshouldalways in additionto usingEq. zero. to the result is close whether estimateinto the originalequationand determining method:its oneThe examplealso illustratesa major weaknessof the false-position points will tend to bracketing of the one That is, as iterationsare proceeding, sidedness. significant fbr with particularly functions stay fixed. This can lead to pool'convergence, (Chapra and elsewhere available curvature.Possibleremediesfor this shortcomingare Canale.2002).
r32 'il
ROOTS: BRACKETING METHODS
{
'
"ii'i*,iry'ii"",.lf,iei{rl:
GASES AND RAINWATER GREENHOUSE ,"' ,,,,SY Bcckground. It is well documentedthat the atmosphericlevels of severalso-called "greenhouse"gaseshave been increasingover the past 50 years.For example,Fig.5.10 shows data for the pafiial pressureof carbon dioxide (COr) collected at Mauna Loa, Hawaii from 1958through2003.The trendin the datacanbe nicely fit with a quadraticpolynomial,3 p c o , : 0 . 0 1 1 8 2 5 ( r - 1 9 8 0 . 5 )+2 1 . 3 5 6 9 7 5-( t 1 9 8 0 . 5+) 3 3 9 where pgo, - COz partial pressure(ppm). The data indicate that levels have increased a little over 19Voover the period from 315 to 376 ppm. One questionthat we can addressis how this trend is affecting the pH of rainwater, Outsideof urban and industrialareas,it is well documentedthat carbondioxide is theprimary determinant of the pH of the rain. pH is the measureof the activity of hydrogenions and, therefore,its acidity or alkalinity.For dilute aqueoussolutions,it can be computedas pH:
-log,n[H+]
6.s)
w ul
tr
st
l(
rh
where [H+] is the molar concentrationof hydrogenions. The following five equationsgoverx the chemistryof rainwater:
fc
^ , ,I H * J" l H C O i I K r' - 1 0 " ' Knpco.
(5.9)
a
5r is pc
Sr
1960
1980
1990
2000
2010
Er
FIGURE 5.IO Averogeonnuol portiolpressures o{ otmosphericcorbon dioxide (ppm)meosuredol MounoLoo, Howoii.
A
tr AI I In Part Four, we will learn how to determinesuchpolynomials.
fa
5.6 CASESTUDY
continued
-
1ll:-
tH+llcol2l
tHcotl
(5.10)
K,,,: [H-][OH-]
( 5 . lr)
, r' : f u # +
(5.r2)
106
tHcotl11coJ2l
0: tHcoil + 2tco;21+toH-l - [H+]
( s .3 1)
where Ks : Henry's constant,and K1, K2, and K,, areequilibrium coefficients.The five bicarbonate,tCO;21 : carbonate. unknowns utec7 : total inorganiccarbon,tHCOtl: : hydroxyl ion. Notice how the partialpressureof CO2 [H*] : hydrogenion, and tOH-l showsup in Eqs.(5.9)and (5.12). Use these equations to compute the pH of rainwater given that K n = l 0 - 1 4 6 ,K 1 : l Q - 63 , K z - 1 0 - t o : . a n dK , : l Q - l a . C o m p a r et h er e s u l t si n 1 9 5 8w h e n the pgo. was 3 l5 and in 2003 when it was 375 ppm. When selectinga numericalmethod for your computation,considerthe following: . .
You know with certainty that the pH of rain in pristine areasalways falls between2 and 12. You also know that pH can only be measuredto two placesof decimalprecision.
Solution. There are a variety of ways to solve this systemof five equations.One way is to eliminate unknowns by combining them to producea single function that only dependson [H+1.To do this, first solveEqs.(5.9)and (5.10)for
tHcotl:-fu, Kupco,
(5.r4)
{#F rcor2l:
(5.I s)
S u b s t i t u tE e q . ( 5 . 1 4 )i n t o ( 5 . 1 5 )
: tco;21 ffi*rnro"
( 5 .I 6 )
alongwith Eq. (5.11)into Eq. (5.13)to give Equations(5.14)and(5.16)canbe substituted
o:
#*,
'z##I KuPcoz. - [H+] Ku.co'+ #
(s.l7)
Although it might not be immediatelyapparent,this result is a third-orderpolynomial in [H+]. Thus, its root can be usedto computethe pH of the rainwater. Now we must decidewhich numericalmethodto employ to obtainthe solution.There are two reasonswhy bisectionwould be a good choice.First, the fact that the pH always falls within the rangefrom 2 to 12, providesus with two good initial guesses.Second,becausethe pH can only be measuredto two decimalplacesof precision,we will be satisfied
t34
ROOTS: BRACKETING METHODS
continued with an absolute error of Ea,d : t0.005. Remember that given an initial bracket andthe desirederror,we can computethe numberof iterationa priori. Substitutingthe presentvaluesinto Eq. (5.6) gives r l . ' .- L ) . . ' ,: '
l:i.rrl-1-l.1ll)t); lri:'.'ll r,lr
'
rr-ll,t.'
I ( r . t l L , ' ri l
Eleven iterationsof bisectionwill producethe desiredprecision. Befbre implementingbisection,we must first expressEq. (5.17) as a function.Becauseit is relativelycomplicated,we will storeit as an M-file: lii|aij f.L.-l 1 l . . i l. r r ) ll
li)
I
i.l
,. irrrl I ',.:,;1,.
1l)l (ljil,iiir.
l
iLr. i;i';,t
ILI
i i
i i;
t,;
L,l i.,l;
ll'.{,*lll*llj*1.')l
lrl
)*l'i
(L,r,1li)ll.ll^i,l'i,ir
",ll
ll;
We can then use the M-file from Fig. 5.7 to obtain the solution.Notice how we have set the value of the desiredrelativeerror (r., : I x l0-8) at a very low level so thattheiteration limit (maxit) is reachedfirst so that exactly I I iterationsare implemented lrijlli,ir
I ilI't
,i
',I
|r1i1l,'',i.',
tt,rll
r,)
,
,1r ,- {).lr'(l
l-ilrli,l-l
ll
Thus, the pH is computedas 5.6279with a relativeerror of 0.0868%.We can be confident that the roundedresultof 5.63 is correctto two decimalplaces.This can be verifiedby performing anotherrun with more iterations.For example,settingmaxi t to 50 yields ll,lll'r iill"
,,
l 'i . i, rit i
'r. li,'lLl, tl
rr
I llrr
t,l
],. For 2003,theresultis
rl
il
l';.
r irilill,
,,'.,
,
i,
ll,)
PROBTE
5.1 Use bisectionto de so that an 65-kg bungee 4.5 s of freelall. Note:Tl Start with initial guesse ate until the approximat 5.2 Develop your own J ion to Fig. 5.7. Howev iterationsand Eq. (5.5) criterion. Make sureto r next highest integer.Ter using 8 ,,,7: 0.0001. 5.3 RepeatProb. 5. I, t obtain your solution. 5.4 Develop an M-file b y s o l v i n gP r o b . 5 . l .
r35
PROBLEMS
lii ;L;,1,ri{ , :ll ri. i'
tt'_,,,'
continued ll .0E74 i r ^,-
-
tL
Interestingly,the resultsindicatethat the l9%orise in atmosphericCO2 levels has produced only a 0.67Vodrop in pH. Although this is cenainly true, rememberthat the pH representsa logarithmic scaleas definedby Eq. (5.8).Consequently,a unit drop in pH representsa l0-fold increasein the hydrogen ion. The concentrationcan be computed as [H*l : lO-pHand its percentchangecan be calculatedas. I
clil:l
I
1112.,
I
r
. \
lr
p ] ]1 9 5 8 ) * 1 0 0
:
( r .o q l ( l Therefbre,the hydrogenion concentrationhas increasedabout.97o. There is quite a lot of controversyrelatedto the meaningof the greenhouse gastrends. Most of this debatefbcuseson whetherthe increasesare contributingto global warming. However,regardless of the ultimateimplications,it is soberingto realizethatsomethingas largeas our atmospherehaschangedso much over a relativelyshorttime period.This case study illustrateshow nunrericalmethodsand MATLAB can be employedto analyzeand interpretsuchtrends.Over the coming years,engineersand scientistscan hopefully usesuch tools to gain increasedunderstanding of suchphenomenaand help rationalizethe debate over their ramifications.
PROBTEMS l l U s eb i s c c t i o nt o d c t c r r n i n c t h e d l a g c o c l ' l ' i c i c n (n c c d c c l m t h aatn6 5 - k gb u n r c c . j u n r p c r h u s a v c l o c i t y o l ' 3 5 n r / s a l t c r '
soffrcelall. Notc:Thc accclcration ol'gravitvis 9.8I nr/sl initialgLrcsscs Stafiwith of',ri :0.2 and i,, :0.3 anclitcrthcapproxilnatc unlil rclativccrror lirllsbcl
5.5 A bcantis loadcdas shownin Fig. P5.-5.Usc thc biscction rncthoclto solvc lir thc 1-rosition insiclcthc bcantwhcrc thclc is no rnolllcn(.
100 lb/ft
8,.,r= 0'000I Prob.5.1, but nsc thc falsc-positionIncthodto Repeat yoursolution. Develop an M-filc fbl thc lnlsc-position rnethod.Tcstit solvine Prob.5. I
F I G U R EP 5 . 5
1 0 0t b
r36
ROOTS: BRACKETING METHODS
5.6 (a) Dctermine the roots of .l (x) : - 14 - 20r * l9-{2 - 3i3 graphically.In adclition,determinethe flrst root of the lunction with (b) bisectionand (c) falscposition.For (b) and (c) use initial -guesses of ;r7: -l and.rr :0 and a stoppingcriterion of I o/c. j 5.7 l,ocatethe first nontrivialfu()tof sin(.r) : .r w hcre,rrs in radians.Use a graphicaltcchniqueand biscctionwith the initial intervallrom 0.5 to l. Perfblm the computationuntil e,, is lessthan o, : 2c/,. 5.tJ Detcrminethe positivc rcal rrrtr(of ln(,rl) : ().7 (a) gfaphically, (b) using threc iterations of the biseetion nrethod,with initial gucssesof -r1: 0.5 and -{r,: 2, and (c) using threciterationsof thc talse-position rnethod,with the sameinitial gucssesas in (b). 5.9 Thc salurationconccntrationof clissolvedoxygcn in licshwatcrcan be calculatcdwith thc etluation
tnr,1:
t3().3441 +lj]I01"]o' 1,,
6.64230Ux 107 -l-
r,i
t i . ( r 21 9 4 9x 1 0 "
T,l
l.2z1.ltl00 x l0r(l
r,l
P
case,the width and the cross-scctionalareacirnbe related to dcpth -r'by
I*-
B: 3*.r,
,tr-
ano
A , .: 3 r , *
_f-
2
Solve lbr the criticaldepthusing (a) the graphicalmethod, (b) bisection.and (c) falsc position.For (b) and (c) use initialgucsscstll'ri : 0.-5iind-r,,: 2.-5,anclitcrateuntilthe approximatc errorljrllsbelorvl% or thc numhcrof iterations exccccls10.Discr.rss yonr results. 5.ll Thc Michaelis-Mcntcn rnocleldcscribcsthc kineticsof cnr)/nlemccliatcd rstctiolls: tlS tlr
.S "1,+S
whcrc .! = suhstrateconccntration (rnolcs/L),tl,r = Inrxi. rlum uptakcrltc (rnolcs/L/d).anclA. : thc hall-saturarion constan[,which is thc s{.{bstratc lcveI at rvhichuptakeishalf ol Lhcrnaximumlrnolcs/Ll. Il' thc initial substritlc levelat / : 0 is So,this dil'f'crcntial cquiltioncan bc solvcdfbr
I
\a
FIGURE P5.T3
.S: Sir- tt,,,/* ft, In(Sg/S)
Devclopan M-l'ilc to gcncratca plot ol'S vcrsusr lbr the whcrc o.7 : thc saturationconccntration ot'dissolvcdoxycasc whctc .Sp: l0 ntolcs/L, 1,, : ().J lrolcs/L/d,and r): gen in licshwatcrat I atnr (ntg L and 4, : ahsolutc *., : 2 rlttlcs/L. lcmpcrature(K). Rcmcmbcrthat f, : T +213.1 5. wherc 5.12 A rcvcrsilrlcchcnticalrcaction ("C).Accordingto thiscquation,satLlrilti()n r : tcmperature dccrcascswith incrcasingtempcraturc.For typical rratural 24+B-c u'atcrsin tcnipcrateclimatc.s, thecquationcan hc usc.cl to dclclrnincthatoxygcnconccntration rangcsli'orl 14.62| ntg/l, can bc churactcrizcd by thc cqLrilibriunt lclationship at 0 "C to 6.4I 3 nig/L at 40 ''C.Gir,cn a valucol oxygcnconK: ccntration,this lirlrnr"rla and the biscctiontncthodcan hc uscdto solvc lirr tcmpcratulcin "C. (i l'cptcscnts lhc rrorlcnclaltrrc lhc conccntrationof (a) Il'thc initial gucsscsalc sct as 0 and 40 "C, how rlany r,,'hcre constitucntl. Supposclhat wc rlcl'inca variublc.{asrepre. biscctionitcrationswould bc rcquircdtu dctcrniincternscntingthc nurnbcrol'rnolcsol'C thatarc ploduccd.Conser. pcralurcto an absolulccrror of'0.05'C'l vutior)ol' nlasscirt bc uscd kr lclirrrnulatcthc cquilibrium on (a),dcvclopandtcsta biscctionM-l'ilcl'unelion {b) Basecl r c l a t i o n s h iaps to dcternrine7 as a lunctionol a givcn oxygcncor)ccn( t , . . r*r. I ) trction.Tc.styour lunclionlirro,T : lJ. l0lnd l2 rng/LA'Checkyoul results. Q ' , , . n 2 x ) ) ( t 1 , . 0- . r ) -5.10Watcr is llowing in a trapezoidalchannclat a ratc ol' whcrc thc subscript0 dcsicnatcsthc initial concentrution Q : 20 mr/s.The critical clepth.r'tirr suchI channelrnust ol cachconstitucnt.Il K - 0.016,(,.o: 42, (i,.0: 28,and satisfvthe cquation (,.0: -1,dctcrntincthc valucol'.r'. (a) Obtainthc solLrtion graphically. O2 0:I Lt r An. (b) On thc basisol (a), soh,clbl theroorwith inirialguesu ^,^1 of'-t,: 0 rnd -r,,: 20 to r, :0.5%,. Choosccitherbisec tvherc g : 9.81 rll/s1. ,4,.: the cross-scctionillarca (rrr). tion or' lulsc position to ohtaiu your solution.Justif,v and R : thc width ol'thc channel lt the surtacc (m). Fol this your choicc.
5.13 Figure P5.l3a show early increasing distributc i n g e l a s t i cc u r v c i s r s t c F i g 'v : - t l L - r - r s + l ' l20EIt.
Use bisectionto dctcrminc (thatis, thc valueof _rwhcn valueinto Eq. (P5.l.l) ro dcr d e f l e c t i o nU . s ct h c l i r l l q y * p u t a t i o nL:: 6 ( X ) c m , E : ond u.r,,: 2.5 kN/crn. 5.14 You huy a $25,000 1 down at $5,500 pcr ycar Ic you paying'?The firrmular paymentsA, nr-rmbcr ol'yea A:
i(l,L;\t]
"' P-" (t+i),_l
5.15 Many ficlds ol'engine estimates.F-orexamplc, trar it necessaryto determinc s trends of a city and adjacer urbanareais dccliningwith
PuQ):Pu.n^^e-k"'aI
r37
PROBLEMS rvhilethe suburbanpopulationis growing,as in rD \ . i l t l t
P rrt -
I + L P , . m a x / P o- l l s - t ' r
$'hcrc P,,,,,,.,*, tu, P,,u,"*,P,,,and k, : ernpiricallyderivcd pavalucsof ralnctcrs.Determincthe timc and con'esponding 'nt'c2()%largerthan the city. P,,(t)andP,(/) when thc sLrburbs The pararnctcrvalues arc P,,.n,",: 75,000, k,, : 0.045/yr, P,.u,iu: 100.000people, P,.,'.,x: 100'000 people, P0 : 10.000people,and t, : 0.08/yr.To obtain your solutions, mcthods. usc (a) graphical,and (b) false-position -i.16 Thc rcsistivily p ol' doped silicon is based on thc chargcr7on an clcctron,thc clectrondcnsity n, anclthc electrorrrrobility ll. Thc electrondcnsity is givcn in terms of thc dopingdcnsityN iinclthc intrinsiccarricrdensityn,.The clcctronrnobilityis describcdby the tempcrature I thc rcfThe crcncc tempcraturcI,,, and thc referenccmobility pr1y. cqLlntions rcquircdto computcthe rcsistivityare
(a)
(.r=0,t=0)
I
iIGURE P5.I 3
qnlr \4',ncfc
5'13 Figure P5.l3rrshowsir unilirlnrbcarnsubjcclto a lindistlihutcdloacl.Thc ccluatiorr lirr thc rcsultatlyrncreasing hgelastic cun'cis (sccFig. P5.l3r) r = -l_l.t 120F. IL
..t + 21-r.rr Lr r )
(p5.13)
,,:j(r* : DctcrnrincN, givcn 7ir : 300 K. f : 1000 K. 1111 ( Vr s ) r ,4 : 1 . 7 x l 0 r ' )C . n , : 6 . 2 1 x l 0 ' ) c r n3 , l3-50crn and a dcsircdlt : 6.5 x l0t' V s cm/C. Usc (a) biscctionand (b) the lllsc positionnrcthod. -5.l7 A totalchargcB is unilirrrnlydistrihutedarounda ringshapcclconductorwith rldius a. A chlrgc q is l()catcdat a The lirrcc -r lhrrn thc ccntcrol'thc ring ( Fig. P-5.17). drstancc cxcrtcdon thc chargcby the ring is given by
Usebisectron ttrdctcrr.ninc thc noint ol'rnaxirnurn dcf'lcctitrrr valueol'.rwhcrctlt'ld.r'- 0). Then sr.rbstitulc ts,thc this intoEq.tP5.l3)to dctcrmincthc valucol'thcrnaxirnurn . Uscthclirllowingprrarnctcrvalucsin y()r.r'corlr ion:L - (r(X)crn.E : -50.(XX) kN/crn'. 1 : 30.000cm*. I u0.t 'r : ub=25kN/ctn. 1n co (.rl + a2yr,': Youbuy a $2-5.000 ;riccc ol' cqr:ipnrcntlil' nothing at$5.500Dcl'voar lor' 6 vcars.What intcrcst ratc arc w h c r c c , , : t 1 . 8 5x 1 0 ' t C t ( N m l y . F i n d t h c c l i s t a n c c x w h c r c 5 pupaying'l Thc lirrnrularclatingpfcscntworth P. ittrttual t h c l o r c c i s 1 . 2 5 N i l ' 1 1a n d Q a r c 2 x l 0 C l i ) r a r i n g w i t h a r n . r a d i u s o l ' 0 . 9 pyments A. nurnber ol'ycirrsrr,lnd intercstratci is
i(l+ l)" A =P i l + l ) , r- I rcquireaccuratcpopulation Manylieldsof'cnginccring Forcxarnplc. transportation cngincclsrnightlind necessary to dctcrnrincscparatclythc populationgrttwth of a city andad.lace nt suhurh.Thc populatiorto{' (hc tcr isdecliningwith timc accorcling area P , ,u , " * t ,
l"'
F I G U R EP 5 . 1 7 *
Prr.rrin
r38
ROOTS: BRACKETING METHODS
5.lti For tluid flow in pipcs, friction is dcscribedby a dimerrsionlessnumber.the F unn i ng .fr i c't i on .fot:to r .1.The Fanning liiction factor is dependcnton a nurnberof paramctcrs related to the sizc of the pipe and the fluid, which can all be representcdby anotlrcrdimcnsionlcssquantity.the Revnoldsttumher Re. A firrmula that prcdicts.l given Rc is the r.,onKurnuul etlutttion: Ir-\ ::4toc,,(nc.
Jf
l)
heatof dry air r1,kJ/(kg K) to tcmperaturc(K): r ; , : 0 . 9 9 4 0 3 + 1 . 6 1| x l 0 1 I + 9 . 7 2 1 5x l 0 8 r : - 9 . - 5 8 3 8x l 0 r r T 3+ 1 . 9 5 2 0x l 0 l ' 1 r l
R
Detcrminethetcmperaturc heat thatconesponds to a specilic o i l . l k J / ( k gK ) . 5.20 The upwardvclocity of a rockct can be computcd by thc tbllowinglblnrula:
u.+
Typical valucstbr thc Reynoldsnumbcrtbr turbulentI'low are 10,000to 500,000anclfbr thc Fanningfiiction tactor arc f ) . 0 0 It o 0 . 0 1 .D e v c l o p al i r n c t i o nt h i l tu s e sb i s c c t i o tno s o l v c lor./ given a uscr-sr.rpplicd valuc of Rc bctween2.500and 1,000,000. Designthe lunctionso that it cnsuresthatthe absolutecrror in thc'rcsulti, 6,,.,,. 0.00000-5. 5.19 Mcchanicalcnginccrs,as wcll as mostothercnginecrs, uscthernrodynamics cxtcnsivclyin thcirwork.Thc lirllowing polynornialcan bc uscd to rclatc thc zelu-prcssurc spccilic
l' :
,.{ lfl --
llln
ttto - qt
-
,qI
whereu : upwardvclocity,u : thc vclocityat whichfuelis : thc initial massofthe expcllcdrelrtivc tt>thc rosket./??o rockctat tinlc/ :0, q : thc l'uclconsumption riite,irndg = ths dorvnwat'daccelera(ionol' gravitv (assumcd constant= t).lJI rn/sr).If r.r: 2000 m/s, rt,, : 150.(X)0 kg. and4 = 2700 kg/s, computethc tirne at which u = 7-50rn/s.(Hint:t is somcwhcrcbctwccnl0 and.50s.)Dcternrincycurrresults0 thutit is w'ithinl% ol'thc trr,re value. Chcckyoul answer.
;^
F
Senl
valu time Hov quic a sir ing I
il
Roots:Open Methods
CHAPTEROBJECTIVES The primary objectiveof this chapteris to acquaintyou with open methodslilr finding the root of a singlenonlinearequation.Specificobjectivesandtopicscovereclare . o . ' . .
Recognizingthe diff'erence betweenbracketingarrdopenmethodslbr root location. the fixed-pointiterationrlethodand how you can cvaluatcits Understanding c o n v e r g e n ccch a l u c l c r i iscts . Knowing how to solvea rootsproblernwith the Ncwton-Raphson methodand the conceptof quadraticconvergence. appreciating Knowing how to irnplementboththe secantand the modil'iedsecantrnetlrods. Knowing how to useMATLAB's f zero lunctionto estimateroots. Learninghow tclrnanipulate anddctcrminethe rootsof polynomialswitlr MATLAB.
or the brackctingrnethodsin Chap.5, the root is locatedwithin an intcrvalprescribed by a lower and an upperbound.Repeated applicationof thesemcthodsalwaysresults in closerestimatesof lhe true valuc ol'thc root. Suchmethodsare saiil lo be conver(Fig. 6. la). progresses ,genlbecausethey moveclosertclthe truthas the cornputation In contrast,the open ntethotlsdescribedin this chapterrequireonly a singlestarting valueor two startingvaluesthat do not necessarily bracketthe root.As such,they sometimes divergeor move away from the true root as the computationprogresses(Fig. 6. lb). However,when the open methodsconverge1Fig.6.lc) they usuallydo so ntuch more quickly thanthe bracketingmethods.We will beginour discussion with of opentechniques a simpleapproachthat is usefulfor illustratingtheirgeneralfbrm and alsoltr demonstrating the conceptofconvergence.
r39
t40
ROOTS: OPENMETHODS
EXAMPLE 6. |J
ri
r,
^1
^u
# xt
-lx
lH -tt-ru
lol
FIGURE6.I Grophicol deplcfionof lhe fundcmentoldjfferencebetweenthe (o) brocketingond (b) ond {c) open rnelhodsfor root loccrlionln (crl,which is bisection,the rooi is consircrinecl wilhinlheinfef vol pr-^scribed by.r1 ond -r,,.]n controst,for ihe open methoddepictedin (b) ond (c),whLchs Newlon Rophson,cr lorrnulois used fo proiectfrorn,r; io -r;, I in cn itercrtive foslrion.Thusthe thr:shope of the frnc-rion methodccrneilher(bJclivergeor (c) convergerop,id1y, dependingc-,rr c n d t h e v o l u eo f t h - ^i n i l i o l( t L r e s s
TI
6.I
SIMPLEFIXED-POINTITERATION As just mentioned,opennrcthodsemploya fbrnrulato predictthe root.Sucha fbrmulaca be developedfbr sirnple.li,red-poitrt itcration(or, as it is alsocalled,one-pointiterirtionor s u c c e s s i vseu b s t i t u t i o nb)y r c a r r a n g i ntgh c ' f ' u n c t i o n . l ( . r: 0 ) s o t h a t . ri s o n t h el e f t - h a n d sideof the equation: -r:.q(x)
(61)
This tlansfbrnlationcan be accourpli.shed eitherby algebraicmanipulafionor by simpJy addingr 1oboth sidesof the originalequation. The utility of Eq. (6. l) is that it providesa tbrmula to predicta new valueof .rasa f u n c t i o no f a n o l d v a l u eo f r . T h u s ,g i v e na n i n i t i a lg u e s sa t t h e r o o t - r i , E Q . ( 6 l.) c a n h usedto computea new estimate.r;11eSexprcssed by the iterativefbrnrula ri+t : g(ri)
(6.:l
pr pr
sit
gri be alt
Tt an
t4l
6 . 1 S I M P LF EI X E D P O I N TI T E R A T I O N
As with many other iterativeformulasin this book, the approximateerror for this equation can be determinedusing the error estimator:
'":lfflrcou"
(6.3)
lterotion EXAMPLE 6.1 SimpleFixed-Point Problem Stotement.Usesimplefixed-point iteration to locate therootof /(r) : e '\ - x. Solution. The function can be separated directly and expressedin the form of Eq. (6.2) as ,fi+l
:
C-"
:0, this iterativeequationcan be appliedto computc: Startingwith an initial gr-ress of -r11
rt U I 2 3 4 -5 6 7 B 9
r0
0 0000 r 0000 4 3675 0 6922 0 5005 o 6462 a 5454 4 5796 0 560r o 5/II 4 5649
le ol, "/"
r00 000 t7l B2B 46 854 38 309 )7447 | 157 5 903 3 4Bt t93I t r09
le,l, "h t 00 000 76322 35 I 3-5 22 450 t) 755 6 894 3 835 2 tsg ) 239 o 7a5 0 399
le,l,/le'1,-t 4 763 a 46A a 628 0 533 0 586 0 556 o 573 4 564 0 569 4 566
Thus. eachiterationbrinssthe estimatccloscrto the true valueof the root: 0.56714329.
Noticethat the true percentrelativcerrorlbr eachiterationof Examplc6. I is roughly proportional(by a firctorof about0.-5to 0.6) to thc errorfiom the previousiteration.This propcrty,calledIinearconvergen(e. is charactcristic ol'fixed-pointiteratron. Aside liom the "rate" of convergence, we mustcommcntat this point aboutthe "possibility" ol'convergence.The conceptsof convcrgenceand divergencecan be depictcd graphically.Rccallthat in Scction5.2, we grapheda functionto visualizeits structureand bchavior.Suchan approachis ernployedin Fig. 6.2a for the f unction.l (r ) : e ' - ,r. An a l t c r r t u t i v r ' g r u p h i ca ap lp r o u c h i s t o s e p a r u ttch cc q u a l i o ni n l o t w o c o m p o n e npt a r t s a . sin
.ftG): JzG) Thenthetwoequations ,vr- .ll(.r)
(6.4)
.t'::./:(-r)
(6..5 )
and
142
OPENMETHODS ROOTS:
F I G U R E6 . 2 'h' :-:^:-Jr theroot'l fordererminins methods, i;;il;;;e r v v uu " E ' "' srophiccrl l9l !1ll::' i u n c t i o n s i h " . o * p o n " n t o f l n i e r s e c t i o n t h e lhe, Ot ,".tot ooiniwhereit crosses ""li;
6.1
FIG Gro ilnrn
osc canbeplottedseparately(Fig.(l'2b).The-rvalucscorrespondingtotheintersectionsof the rclotsof /('r) : 0' thesefunctionsrepresent and divergence 'l'he twcl-curvemethodcan now be usedto illustratethe convergence =r equiltions'\'l pair of a as of fixed-pointiteration.First, Eq. (6.l) can he rcexpressed with case was the As separately' ancl.y2:g(r). Thesetwo equalionscan then be plotted Eqs.(6.4)and(6'5),therootsoll'(x):0correspondtotheabscissavalueattheintersec' shapeslbr.v2 : g('r) ate tion of the two cufves. The function .)'t : 't and tbur diflbrent plottedin Fig. 6.3. theconesponding For the first case(Fig. 6.3a),the initial guessof r11is usedto
iteri
roo Fig
Cha por slop
r 43
ITERATION 6.I SIMPLE FIXED-POINT
J0
(cl
(d)
FIGURE6.3 of simpefixedpoinl G r o p h i c o ld e p i c t i o no f ( o )o n d ( b ) c o n v e r g e n coen d ( c )c n d i d ) d i v e r g e n c e iterotion.Grophs (o) ond (c) ore co led monotonepotlernswhereos(b) ond (c) ore colled o s c il o l i n go r s p i r o p o t t e r n sN. o t e t h o tc o n v e r g e n coec c u r sw h e n g ' ( . r ) l - < 1 .
iterationis equivalentto the equation 12: s(,rt, The solutionin Fig. 6.3a is convergentbecausethe estimatesof -r move closerto the root with eachiteration.The sameis true for Fig. 6.3b. However,this is not the casefbr Fig. 6.3c and d, wherethe iterationsdivergefrom the root. A theoreticalderivationcan be usedto gain insightinto the process.As describedin Chapraand Canale(2002), it can be shown that the error for any iterationis linearly proportional to the error from the previous iterationmultiplied by the absolutevalue of the slopeof g: Ei+r :
g'G)Ei
ROOTS: OPENMETHODS
144
Consequer-rtly, with eachiteration.For lg'l > I theenors if lg'l < I. the errorsdecrease grow. Notice also that if the derivativeis positive,the enors will be positive,andhencethe errorswill havethe samesign (Fig.6.3aandc). If the derivativeis negative,theenorswill changesign on eachiteration(Fig. 6.3b andd).
6.2
NEWTON.RAPHSON Perhapsthe most widely usedof all root-locatingformulasis IlteNewtott-Raphsott method (Fig. 6.a). If the initial guessat the root is .r;, a tangentcan be extendedfrom the point The point wherethis tangentcrossesther axisusuallyrepresents an improved Iri, I (,,.;)1. e s l i m a t co l ' t h er o o t . The Newton-Raphsonmethod can be derived on the basisof this geometricalinterpretation.As in Fig. 6.4, the first derivativeat.r is equivalentto the slope: /('r't-o
/'('r't:
Ji -'ft+l
f
which can be rearranced to vield -
' r r + l: ' ,
.l Gi)
( .l
(66)
f o,
which is calledtlreNcwton-Raph.son lbnnulu. E X A M P L E6 . 2
N e w t o n - R o o h s oM n ethod Problem Siotement. Use thc Newton-Raphson rnethodto estimatethc root of /(x) = ? ' - .tremployingan initial guessol'.r1;- 0. Solution. The I'irstderivativeo1'thefunctioncan bc cvaluatcdas /'tt) :
-<"
T
I
which can be substituted alongwith the originall'unctioninto Eq. (6.6)to give '\','1 -
\',
it tt
t/ , z, .r1 ^. e-',i-l
Startingwith an initial gucssof ,re : 0, this iterativeequationcan be appliedto compute le,l, "/" 0 I 2 3 4
0 0 -500000000 056631t003 o 56/143165 4 . 5 6 71 4 3 2 ( ) 0
IOO llB a )47 0 0000220
.10 8
Thus, the approachrapidly convergeson the true root. Notice that the true percentrelative el'rorat each iteratiorrclecreases much faslerthan il doesin simple fixed-nointiteration (comparewith Example6. l).
EXAMPLE 6.3
P R
6.2 NEWTON-RAPHSON
t45
l(r)
F I G U R E6 . 4 Grophicoldepiction of theNewton-Rcrphson method.A tongentto ihefunction of ,r; fthotis, of ihe rootot .r;11. /'(.r)l is exlropooteddown to the,r oxislo provideon estimote
As with otherroot-locationmcthods,Eq. (6.3)can be usedas a terminationcrite.rion. ln addition,a thcorcticalanalysis(ChapraandCanale,2002)providesinsightregardingthe rateof'convergence as expressed by (6.7)
Thus,thc crror shouldbc roughlyproportionalto the squareol'the prcviouserror.In other words, the numbcr of' significantfiguresof accuracyapproxirnately doubleswith each iteraticrn. This behavioris calledqruulrilic (onvcrgcn(eandis oneo1'the rra.jorreasonsfbr thc popularityol'thc rnethod. Although thc Ncwton-Raphson method is ofien very efl-icient,thcre irrc situations whereit pcrlbrmspoorly.A specialcase-multiple roots-is discussed elscwhcre(Chapra and Canale,2002). However,cven whcn dcalingwith simple roots,difficulliescan also a r i s ea . s i n t h ef b l l o w i n se x a m o l .e E X A M P6L.E 3
A S l o w l y C o n v e r g i n gF u n c t i o nw i i h N e w t o n ' R o p h s o n Problem Stotement. Determinelhe positiveroot of .l(l) : ,rr0Raphsor.r methodand an initial guessof .r : 0.-5. Solution. The Newton-Raphson tbrmulafbr this caseis .t .i r - rr : . t l
-t -. tl { ) -
|
l,)rI:
which can be usedto compute
I using the Newton-
146
ROOTS: OPENMETHODS
xi
0 l 2 3 4
0.5 5 16 5 46.485 4t 8365 37.65285
4A 4l 42
0 0 2 3I 6 aaoa24
leol, "/"
99.432
2130 o 22g 0 002
Thus, afterthe first poor prediction,the techniqueis convergingon the true root of I, but at a very slow rate. is Why doesthis happen?As shownin Fig. 6.5,a simpleplot of the first few iterations helpfulin providinginsight.Noticehow thelirst guessis in a regionwherethe slopeisneat zero.Thus, the first iterationflings the solutiontar away from the initial guessto a new where./(x)has an extremelyhigh value.The solutionthcn plodsalong value(,r: -51.6-5) fbr over 40 iterationsuntil convcrgingon the root with adequate accuracy.
F I G U R E6 . 5 methodfor o cosewilh s ow convergence The Grophicoldepiction of theNewtonRophson shoots thesolution for fromtherool Thereofter, insetshowshow o neorzeroslopeinitiolly thesolulion veryslowlyconverqes on lhe root l(x) 2E+17
1E+17
FI
T Aside from slow convergencedue to the natureof the function, other difficultiescan arise,as illustratedin Fig. 6.6. For example,Fig.6.6a depictsthe casewherean inflection
6.2 NEWTON-RAPHSON
F I G U R E6 . 6 poorconvergence methodexhibits FourcoseswheretheNewton-Rophson
r48
ROOTS: OPENMETHODS ' point (i.e..f (x) : 0) occursirr the vicinity of a root. Notice that iterationsbeginningat.re progressivelydiverget'rorntheloot. Fig.6.6b illustratesthetendencyof theNewton-Raphson techniqueto oscillatearounda localmaximumor minimunl.Suchoscillationsmaypersist, or, asin Fig. 6.6b,a near-zeroslopeis reachedwhereuponthesolutionis sentf'arfrom thearea of interest.Figure6.6c showshow an initial guessthatis closeto one root canjump to a location severalrootsaway.This tendencyto move awayfronr thearcaof interestis dueto thefactthat near-zeroslopesareencountered. Obviously,a zeroslope[./'(.r) : 0l is a rcaldisasterbecauseit causesdivisionby zeroin thetr-ewton-Raphson fornrulaIEq.(6.6)1.As in Fig.6,6d, it meansthatthesolutionshootsoff horizontallyandneverhitsthex axis. Thus,thereis no generalconvergence criterionfor Newton-Raphson. Its convergence dependson the natureof the lunction and on the accuracyof the initial guess.Theonly remedyis to havean initial guessthat is "sufficiently" closeto the root. And for somefunctions,no guesswill workl Good guesses areusuallypredicated on knowledgeof thephysical problemsettingoron devicessuchas graphsthat provideinsightinto thebehaviorof the solution.It also suggeststhat good cornputersoliware shouldhe designedto recognize slow convergence or divergence. 6.2.1
I ATLAB M-file! newtraph
An algorithmlbr the Newton-Raphson methodcanbe easilydeveloped(Fig.6.7).Notethat theprogrammusthaveaccessto thefunction(f unc) andits first derivative(ofunc).These can be simply accomplished by the inclusionof user-dcfined functionsto compute these quantities.Alternatively.as in the algorithmin Fig. 6.7, they can be passedto thefunction as arguments. Alier the M-file is enterecland saved.it can be invokedto solve tbr root. Forexamole. lor the simplcfunction.r2 - 9, thc root cun he dcterminecl as in : ' j ' n e \ ^ , / L r a I J n ( G ( x ) - " , ^ : l ! 1, ( 4 ( x )
2,"x.,5)
Ft An
.' Newton-RophsoB n u n g e eJ u m p e rP r o b l e m
E X A M P L E6 4 '
Problem Stotement. Usc the M-file functiontiom Fig. 6.7 to detcrntinethe rnass ofthc bungeejumper with a dragcoefTicient of 0.25kg/rnto havea velocityol-36rn/safter4sof fice fall. The acceleration of grlvity is 9.81 nr/sr. Solution. Thc firnctionto bc evaluatedis .f ltnl
:rff""n(
grii, nl
) /
- r,'(l;
(E6.41)
To apply thc Newton-Raphsonmethod,the derivativeof this function rnustbeevaluirtedwith resDectto the unknown.n:
'+#:irF-""^(,H') -fir r..rrt(#;)
6.3 SE As the ma
6.3 SECANT MEIHODS
f u n c c i o n I r o o u , e a , i r e r J- n e w L ' d p h r f u n - , d i r n c , y r , F S ,m a x ' L , v o ' d r g i n ) % newtraph: Newton-Raphson root Location zeroes % lroot, ea, it:erl=newtraph (f unc, df unc, xr, es,maxit,pl ,p2, . . . ) : ? uses Newton Raphson nethod to find the root of func % input: % func = name of function ? df unc = nane of derir,zat irze of f unct lon ? xr = init ia1 guess - 0.0001?) (default % es = desired relative error (default = 50) % maxit = maximum allowable iterations - additional p1,p2,.., parameLers usted by function % % output: % root = real root (?) % ea - appr.orimate relative error = number of iterations % iter if if if itoz
nargin<3, error ( 'at least 3 input arguments required' nargin<4 isernpt)'(es),es=0.0001;end nargin<5 | isempty (maxit ), naxit,-50 ; end -
) , end
O.
(1) while = xr ; xrold xr = xr - f u;rc(xr)/df unc (xr) ; . 1, iLer ir= maxit, break, end end root : xr;
end
FIGURE 6.7 A n M l i l e t o i m p e m e n if h e N e w l o n R c p h s o nm e t h o d
W e s h o u l dn r e n t i o nt l r a ta l t h o u g ht h i sd e r i v a t i v ci s n o t d i f T i c u l[to e v a l u a t ei n p r i n c i p l ei,t involvcsa bit of concentration andeflbrt to arriveat thc l-inalrcsult. The two fbrntulascan now be used in cilnjunctionwith the function n, wtr nLih [o evaluatcthe root: >> y = Gn sqrt ('j.81*m/0.25)*l,anh(sqrl, (9.81*0.25lm)*4) l6; > > d y = G n I / 2 * : - r c 1 r l1, 9 . ' E I / ( m * 0 . 2 5 ) ) * t a n h ( ( 9 . 8 - l * 0 . 2 5 l m ) ^ (1 / 2 , 1 * 4 ) 9 . r 3 1/ ( 2 * p ; * s c c h ( s q r l ( ( 1. B l * 0 . 2 5 l m ) * 4 ) ^ 2 , ; r " > n c w L r . r p h ( - { , d y , 1 , 1t l , 0 . 0 0 0 0 1 ) \.\2.'tj]6
6.3 SECANTMETHODS As in Example6.4, a potentialproblemin implementingthe Newton-Raphson methodis the evaluationof the clerivative. Although this is not inconvenientfbr polynomialsand many other tunctions,thereare certainfunctionswhosederivativesmay be difficult or
rso
ROOTS:OPEN METHODS
6.
inconvenientto evaluate.For thesecases,the derivativecan be approximatedby a backward finite divided difference:
Se
f (xi-t) - .f(ri) xi-t
- xi
This approximation can be substitutedinto Eq. (6.6) to yield the following equation: xi+t : xi -
/(xi)(xi
t
-
xi)
./(xi-r) - .f(x,1
Th
(6.8)
Equation(6.8) is the fbrmula for the secantmethod.Notice that the approachrequirestwo initial estimatesof x. However,because.f (.r) is not requiredto changesignsbetweenthe estimates,it is not classifiedas a bracketingmethod. Rather than using two arbitrary values to estimatethe derivative,an alternativeapproach involves a fractionalperturbationof the independentvariableto estirnate./'(.r), .f
-./(,r,) ' { x i \ = .f (xi + 6.ri) dri
where 6 : a small perturbationfraction.This approximationcan be substituted into Eq. (6.6)to yield the following iterativeequation: ri+l
:
"tr
-
6x;.l'(x;) .f(xi+lxi)-
swa Eq.
(6.9) f(.ri)
der
We call this the modified secan.tnrcthod.As in the fbllowing example,it providesa nice meansto attain the efTiciencyof Newton-Raphsonwithout having to computederivatives,
E X A M P L E6 . 5
asi alw cha nifir rese met
M o d i f i e d S e c o n tM e t h o d Problem Stotement. Use the modified secantmethod to determinethe massof the bungeejumper with a drag coefficientof 0.25 kg/m to havea velocity of 36 m/s atter4 sof f r e e f a l l . N o t e : T h e a c c e l e r a t i ognr oa fv i t y i s g . 8 lm / s 2 . U s e a n i n i t i a l g u e s 5 so 0 fk g a n d a value of I 0 6 for the perturbationfraction. into Eq. (6.9)yields Solution. Insertingthe parameters First iteration: xrt : 50 .ro*dru:50.00005 Jl
- 5r)-
l'(xo) : -4.57938708 I l8 ,f (xo+ 6xo)- -4.579381
I 0-b(50)(-4.57938708) -4.57938rI r8 - (-4.57938708)
= 8 8 . 3 9 9 3 1 ( l:€ ,318 . l % ol e; o l : 4 3 . 4 V o )
6.4
Ml The but ofb fepr
whe Nott
whe
F U N C T I O Nf: z e r o 6.4 MATLAB
l5l
Seconditeration:
.f(.r-r): -1.69220111 ./(.rr+dxr) - -1.692203516
.tr : 88.39931 . r 1f d r 1 : 8 8 . 3 9 9 4 0 . r r : 8 8 . 3 9 9 3l :
10-6(88.3993 1)(- 1.6922017 t) -t.6922035r6 - (- 1.69220't1t)
je,| : 124.08910(
l3.l7c: lo,l :28.16a/")
The calculationcan be continuedto yield i 0 I
,2 3 4 5 ()
xi -500000 B B3 9 9 3 t24 ABS7 ) 4 A5 4 1 7 t 42 7A72 1 4 27 3 , 7 6 14 2 7 3 7 a
letl, "/" 64971 38064 I3064 t 53B o o2t 4 l x lO' 3 4 x l0 '
le.l, T" 43 438 28 762 II 746 I 5t7 a a2t 4 I x lO ''
The choiceof a propervaluefor 6 is not automatic.If J is too snrall,the methodcan be swampedby round-oft error causedby subtractivecancellationin the denominatorof Eq. (6.9).If it is too big, the techniquecan becorneinefTlcient arrdevendivergent.However. if chosencorrectly,it providesa nice alternativefor caseswhere evaluatingthe d e r i v a t i v ei s d i f f l c u l ta n dd e v e l o p i n tgw o i n i t i a lg u e s s eiss i n c o n v c n i e n t . generalsense,a univariatefunctionis rnerelyan entitythat returns Further,in its m<-rst a single value in return for valuessent to it. Perceivedin this scnsc.functionsare not alwayssimpletbrrnulaslike theone-lineequationssolvedin theprecedingexamplesin this chapter.For example,a f unctionmight consistof nranylinesof coclethatccluldtakea significantamountof executiontime to evaluate.In sornecases,the functionmight evenrepresentan independent computerprogram.For suchcases,the secantand modifledsecant methocls are valuable.
6.4 MATLAB FUNCTIONz fzero The rnethodswe havedescribedto this pointareeithcrreliablebut slow (bracketing) or f-ast but possiblyunreliable(open).The MMLAB f zerc-rfunctionplovidcsthe bestclualities of both.The f zero functionis designedto find the realroot of a singleequation.A sirnple representation of its syntaxis f z , e r o( f u n c t i o n , x a J where funcrjon is the nameof the functionbeingevaluated, and x0 is the initial guess. Note that two guessesthat bracketthe root can be passedas a vector: f zero (funcLion, lxa xl ) ) where x0 and x1 are guessesthat bracketa sign change.
r52
ROOTS: OPENMETHODS - 9 Hereis a simpleMATLAB sessionthatsolvesfor theroot of a simplequadratic:.f: -3 Clearlytwo rootsexistat and 3. To find the negativeroot: >> x = fzero(Glx)
x^2'.;,
4)
x3 If we warrt to find the positive root, use a guess that is near it: > > x = f z e r o ( ( 4( x )
-r^2 9,4)
x3 I f w e p u t i n a n i n i t i a l g u e s so f z e r o , i t l i n d s t h e n e g a t i v er o o t >> x -
f :ero(tj (x)
x^2
9,0)
F C
x-
1
3 If we wanted to ensure that we fbund the positive root, we could enter twil guessesasin
rh >:.
1, -
f2c.1-6(t;l(1)
x'2
!),|0
Ii)
IS
x3 an errormessage is displayed Also, if a signchangedoesnot occurbctweenthetwo guesses, o{ >.: x
-
fzcro((4(x)
x-"'),9,1-4
| . , r : . : I 1 r u, ir,1 'lhe' f unct-ion val ues; .r1- Lhc clif fcr
in
4l)
ill
pa us c , ' t v . r1 e ' n c l p o i f l l . : j l i l u : i t
srgrr.
The f zero lunctionis a combinationof the relitble bisectionrnethoclwith two t'aster interpolation. lrn'erscquudruticinteralgorithnrs:the secanllnethodanclinversequaclratic poltttionis sirnilarin spiritto the secantmethod.As in Fig. 6.8rr,the secantrnethodisbased l'he intelsection of thisstraight on computinga straightline thatgoesthroughlwo guesses. line with the,r'axisrepresents the new root estimate.The inversequadraticinterpolation that usesa similarstrategybut is basedclncomputinga quadraticequation(i.e.,a parabola) goesthroughthreepoints(Fig. 6.8b). The f zcro functionworks as filllows. lf a singleinitial guessis passed,it firstpersearch forms a searchto identily a sign change.This searchdifl'ersfrom lhe incremental in thatthe searchstartsat the singleinitial guessandthentakes describedin Section5.-1.1, increasirrgly bigger stepsin boththe positiveandnegativedirectionsuntil a signchangeis detectecl. Thereafter,the fast methods(secantand inversequadraticinterpolation)areusedunresultoccurs(e.g..the root estimatefalls outsiclethe bracket). lf abad lessan unacceptable root is obtainedwith oneof resulthappens,bisectionis inrplemented until an acceptable
wl et( thi
EXAMPTE 6.6 Th
Pr< ,r l' pr(
So
F U N C T I O Nf: z e r o 6.4 MATLAB
J(x)
(al
FIGURE 6.8 N.o i e t h o tt h e C o m p o r i s o no f ( o )t h e s e c o n tm e l h o do n d ( b ) i n v e r s eq u o d r o f i ci n t e r p o l o t l o n o p p r o o c hi n ( b ) i s c o l l e d" i n v e r s eb" e c o u s et h e q u c d r o t i cf u n c t i o ni s w r i l i e ni n . r ' r o l h etrh o n i n
thefastmethods.As might bc expected,bisectiontypicallydominatesat first but asthe root is approached, the techniqueshiftsto the tasterrrrcthods. A morecompletercplescntation of the f'zcrrosyntaxcan be writtenas Ix, fx]
i z , e r c :\ f u n c : Lj o n , x A , o l . t t i o n s , p l , p 2 , . . . J
-
where I xr , f ;<] . a vector containing the root x and the function evaluatcd at the root f-x, optron.s is a data structure created by the opt- lmseL function, and p1 , ;:2. . . are any parermetersthat the function requires. Notc that il'you dcsirc to pass in parametersbut not u s e t h e o p t , i o n s , p a s sa n e m p t y v e c t o r I I i n i t s p l a c c . The opt imsct functicln has the syntax optiors
-
o f i L r m s c t .( ' p . l r ' , ' , v c t l , , ' p a r
'
,v,tl ,, ...J
w h e r e t h e p a r a m e t e r p a r -h a s t h c v a l u e v a l , . A c c l m p l e t e l i s t i n g o f a l lt h e p o s s i b l e p a r a m eters can be obtained by rnerely entering opt- i rnsct, at the command prornpt. The parameters that are cornmonly used with the Izer o function are clisplay, Whensetto'ir-er'displaysadetailcdrecordol'all theiterltions. Lolx: A positive scalar that sets a termination tolerancs on x.
EXAMPLE 6.6 The r
toofld,,1
rl iir
F un c t i o n s
P r o b l e mS t o t e m e n t . R e c a l lt h a t i n E x a m p l e6 . 3 , w e f b u n dt h e p o s i t i v er o o t o f . l ' , x ) : rl0 - I using the Newton-Raphson mcthodwith an initial guessol'0.5. Solve the same p r o b l e mw i t h o p l i m s e t a n d f z e r o . Solution. An interactiveMATLAB sessioncan be implementedas fbllows: >>
opLions
= op1-imset ('display',
'ite
r');
154
ROOTS: OPENMETHODS Func-count L 2
5 6
f (x) -0.999023 -4.9q9267
r-L6
0.999454
0 .95254E 0.14 I.I4
-0.385007 -1
searcll
2.14122
search
l-he interrral
t 0.14, 1.141
4.47I
23 24 25 Looking
26 21 2B '2
x 0.5 c.485358 4.5r4r42 0.48 0.52
.Q
30 31 3',.,
for
a
zero
a .24\21 :, a.61 2636 O.90tr318 1.0:3itsJ 0.98q128 0.998894 1.00001
3i
34 l1 Zero
I found
ir'l-
0.998709 -0.9993s1 -0.998554
in
bisectron
If n( H U de let
bisecl-ior-r interpo
laf,ion
interpolaLion inl-erpolatlorr rnterlrolaLiorr
I.324 5e D11 0 4,
6
br.sect i on
7.6BlB5e-005 -1. B306ie 007
.
a
fr E
interpolati.orr
0.9P,1442. a .626056 o .2572t I -0.103511 ,0 . 01Laarl
--/.
n r
search
-1
it'-rv.rl:
v n
Procedure initial search search search search search
i nterpolat
i orr
i ni erpoIat
i on
L 141.
wl no
VA
eig we
0 Thus.after25 iterationsof searching,fzero finds a signchange.It thenusesinterpolationandbisectionuntil it getscloseenoughto theroot so that interpolation takesoverand rapidlyconvergeson the root. Supposethatwe would like to usea lessstringenttolerance. We canusethe opt imset functionltl set a low maxinrumtoleranceand a lessaccurateestimateol'the rootresults: >- opl-ions = optim:,;e1,('l-olx', 1e-l); = tzcro(G (x) x^10 1,0.5,options) >) lx,l.xl
Di
{.s
the
1 0009 0.0090
6.5
POTYNOMTATS Polynomialsare a specialtype of nonlinearalgebraicequationof the generalfbrnr . f r ( . r ): d 1 , t "* a 2 x ' ' - t+ . . ' + o u t . y 2| u , , . !t a , , a 1
ertJ the MI tior foll
r55
6.5 POLYNOMIALS
where n is the order of the polynomial, and the a's are constantcoefficients.In many (but not all) cases,the coefficientswill be real.For suchcases,the rootscan be real and/orcomplex. In general,an nth order polynomial will haven roots. Polynomialshave many applicationsin engineeringand science.For example,they are used extensivelyin curve fitting. However,one of their most interestingand powerful applicationsis in characterizing dynamicsystems-and,in particular,linear systems. andelectricalcircuits. Examplesincludereactors,mechanicaldevices,structures, 6.5.1 MATLAB Function: roots If you are dealing with a problem where you must determinea single real root of a polymethodcanhaveutility. nomial,the techniques suchasbisectionandthe Newton-Raphson However,in many cases,engineersdesireto determineall the roots,both real and complex. like bisectionand Newton-Raphson arenot availablefbr simpletechniques Unfbrtunately, determiningall the rootsof higher-orderpolynomials.However,MATLAB has an excellent built-in capability,the roots function,fbr this task. The roots functionhasthe svntax. x..
roots(c)
wherex is a columnvectorcontainingthe rootsand c is a row vectorcontainingthe polynomial'scoefficients. So hclwdoesthe roots functionwork? MATLAB is very good at finding the eigenvaluesof a matrix.Consequently, the approachis to recastthe root evaluationtask as an eigenvalueproblem.Becausewe will be describingeigenvalueproblemsIaterin the book, we will merelyprovidean overviewhere. Supposewe havea polynomial ( 6 . 1| )
. / t . r 5+ o 2 x 1+ , r . , r t + a + x z* o 5 x+ a 6 : 0 yields Dividing by rrsand rearranging 5 I
:
-
(Il -f
I
-
(l I -l' (ll
dl
I
(14
.
At
(l\
(Irr
Ul
Al
(6.12)
A specialmatrix can be constructed by usingthe coefflcientsfiom the right-handsideas 's the first row and with I and 0's writtenfbr the otherrows as shown: -o2fu1
-u3fd1
10000 0t000 00r00 00010
-aafo1
-a5fu1
-aolar
(6.13)
matix. It has the usefulpropEquation(6. l3) is calledthe polynomial'sr:ompanion are the rootsof the polynomial.Thus, the algorithmunderlying erty that its eigenvalues the roots functionconsistsof merely settingup the companionmatrix and then using MATLAB's powerful eigenvalueevaluationfunction to determinethe roots. Its applicafunctions,aredescribedin the tion, alongwith someotherrelatedpolynomialmanipulation following example.
r56
ROOTS: OPENMETHODS
6.(
We shouldnote that roots has an inversefunctionc-alledpo1y, which whenpassed the valuesof the roots.will returnthe polynornial'scoefficients. Its syntaxis
cie
''
''
l'1Y(r)
Be )'>
where r is a column vector containingthe roots and c is a row vectorcontainingthepoly. nonliai'scclefficients.
wit fbu
U s i n g M A T L A Bt o M o n i p u l o t eP o l y n o m i o l q s n d D e t e r m i n eT h e i r R o o t s
X
;
Problem Siotement. Use the following equationto explorehow MATLAB canbeemployedto ntanipulate polynomials:
2. 1. 1.
i I
(E6.7.1) r 1.2-5 / ! ( x ) : - r ' ' - 3 . 5 . r '*1 2 . ' / 5 x t 1 2 . 1 2 5 , 1 2- 3 . u 7 5 . + Note that this polynomialhasthreereal rcots:0.5. - 1.0,and 2: and one pair ol complex r o o t s :I + 0 . 5 1 .
E X A M P L E6 . 2 '
r I i '
We
Solution. Polynomialsare enteredinto MATLAI) by storingthe coefflcientsas a row \/ector.Ftir example,enteringthe following line storesthe coefficientsin the vectora: -'' a = f I 3.5 2.15 2.'l'2,\ i.875 1.251;
We
We can then proceedto manipulatethe polynomial.For exarnplewe can evaluate it at .r : I, by ryping
X:
:>
(a, I )
p c )l y v a l
, w i t ht h er e s u l tl ,( l ) 5 - 3 . 5 ( l ) 1+ 2 . 1 5 ( t t + r 2 . 1 2 5 i l ) -2 3 . 8 7 5 0 + ) 1 . 2 -:5 - 0 . 2 5 : t).:tb0o
Fin
We ciin createa quadraticpolyn
.5 ,... 1
i i
la)
1.0000
0.5()00
(r.5000
Note that the pc;1y function can bc useclto pellirrrn tlre satle task as in -:
b = poly([0.5
1])
D-
1.0000
0.5000
0.500t)
We can divide this polynomialinto the originalpolynornialby r:> [q,r]
= deconv(a,b)
with the resultbeinga quotient(a third-orderpolynontial.,1)and a rerrrainder(r) cl-
1.0000 0
-4.000r1 0
Bqe man of l' rang
calle calcr
157
6.6 CASESTUDY
Becausethe polynomial is a perfect divisor, the remainderpolynomial has zero coefficients.Now, therootsof the quotientpolynomialcan be determinedas >>
X
-
rootS(q)
with the expectedresultthat the remainingrootsof the original polynomialEq. (E6.7.I ) are tbund: 2.0000 r.0000 + 0.5000i r.0000 0.5000i We can now multiply c1by b to colrL-up with the originalpolynomial: ->
if
-
conv
(q,
b)
,l:
1 . f .0r t )( l
-3.5011 0
?,.15A()
2.i2')A
-1.8'l!.0
1.2500
We can thendetermineall thc rootsof the originalpolynomialby .:
x
-
I-.Jal,S(.1)
_\: . 2.0000 r. rJ000 1.000t) + 0.5000i 0-!0t)(li t.00rlrr 0.5000 Finally, we can return to thc original polynomialagainby usingthe poLy function: _::
J
-
l).Jr-\1(X)
-l.500tr
2,.1,,A0
2.1),\O
3.87!0
1.2500
P IP E F R IC T ION Determiningfluid flow throughpipes and tubeshas great relevancein Bockground. many areasofengineeringand science.ln engineering,typical applicationsincludethe flow of liquids and gasesthroughpipelinesandcooling systems.Scientistsareinterestedin topics througha plant'svascularsystem. rangingfrom flow in bloodvesselsto nutrienttransmission The resistanceto flow in such conduitsis parameterizedbya dimensionlessnumber called the.lil t:tionJ'actor For turbulenttlow, the Colebrookequ.otionprovidesa meansto calculatethe friction factor:
o: +* 2orog (rr. U1#)
(6.14)
t58
ROOTS: OPENMETHODS
continued where e : the roughness(m), D : diameter(m), and Re = the Reynoldsnumber: oVD Re - :__ LL
where p : the fluid's density {kg/m3t, V: its velocity (m/s), and & : dynamic viscosiry (N.sim2). In addition to appearingin Eq. (6.l4), the Reynoldsnumber also servesasthe criterion for whetherflow is turbulent(Re > 4000). In this casestudy,we will illustratehow the numericalmethodscoveredin this partof the book can be employedto determine/for air flow througha smooth,thin tube.Forthis case,theparametersarep - 1.23kglm3,! : 1.79x I 0r N . slmz,D : 0.005m, V = 40nr/s and e :0.0015 mm. Note that friction factorsrangefrom about0.008 to 0.08. In addition, an explicit formulationcalledthe Swamee-Juin equationprovidesan approximateestimate:
1.325
f-
(6.1t
['"(t;.#)]'
1
The Reynoldsnumbercan be computedas
Solution.
o VD 1.23(40)0.005 : 13'743 R e- : - - : r . z st l o This value along with the other parameterscan be substitutedinto Eq. (6. l4) to give
Er gc 0.
r /0.0000015 2.st \ g(J'l: -+2.0loel=-+=" \ 3 . 7 ( 0 . 0 0 s ) 1 3 . 7 4 3 Jlf ) J.f Before detennining the root, it is advisableto plot the function to estimateinitial guessesand to anticipatepossibledifficulties.This can be done easily with MATLAB: Ilt,, liI .t :
l.
lJ;'l
I. I
1(1,"r;l)
1 l . i ) ( lr l t i - , , ] { ); ,
().f)0It)/'I(){)0;
I l l t , * \ , t * l ) , / [ l L t; I(I)
f lrl,rl
1
r t L i t 1 .( I )
(rr, f {).t)0:r
t l*ilLl
( i , : , - , * : : , , 1 r (t i ) ) ) ; ill (,.:i | 1,.,'"1.t t,..'tti i , i , > : t , r l r , ' i ( ' 1 ' ) , _ vl , r f r r i ( ' r r ( i ) )
0.1)il ,),rrr
As in Fig. 6.9, the root is locatedat about0.03. Becausewe are suppliedinitial guesses(x/ : 0.008 andx, : 0.08),eitherof thebrack eting methodsfrom Chap. 5 could be used.For example,the bisect function developed in Fig. 5.7 gives a value of/: 0.0289678with a percentrelativeerror of error of 5.926x l0{ in 22 iterations.False position yields a result of similar precision in 26 iterations. Thus, althoughthey producethe corect result,they are somewhatinefficient.This would not be importantfor a singleapplication,but could becomeprohibitiveif many evaluations were made.
Hr di
Ar gu fo
r59
6.6 CASESTUDY
c o n tn i ued
ty ne
2
\2
of tis t/s tn, [e:
1
i\
i
i'\-iiii
- \ i\-\
i
1
i
r
- - - - - - L - - - - - - - - - - - - - - - - - - l
H.
-2
5)
i
_?L!
" 0 . 0 1 0.02 0.03 0.04 0.05 0.06 0.07 0.08 f
FIGURE 6.9
We could try to attain improvedperformanceby turning to an open method.Because Eq.(6.14) is relatively straightfbrwardto differentiate,the Newton-Raphsonmethod is a good candidate.For example,using an initial guessat the lower end of the range (ro : quickly: 0.008),the newtraph functiondevelopedin Fig. 6.7 converges
)(, r.1., l(
ir.ill-1
. '
'
rr
r . , , 1r , L i r l l ( ( i ,
), t{I
j
':,:irlt i ll
( l i r i rl
il
I
l.l r
'liril ,fi l iilt,
ilir l
: rl
However,when the initial guessis setat the upperend of the range(xn: 0.08),the routine diverges, ,,,t
:i'
t i.Llir,
As can be seenby inspectingFig. 6.9, this occursbecausethe function'sslopeat the initial guesscausesthe first iterationto jump to a negativevalue. Further runs demonstratethat for this case,convergenceonly occurswhen the initial guessis below about0.066.
t60
ROOTS: OPENMETHODS
' continued So we can see that although the Newton-Raphson is very efficient, it requires good initial guesses.For the Colebrook equation, a good strategy might be to employ the Swamee-Jainequation(Eq. 6.15) to providethe initial guessas in '
J l l , i - l . l . l ' , ' l 6 1 1( t : r /{ 1 . / " D ) + 5 . 7 4 . / R c ' 0 . 9 ) ^ 2
I :l,I 1 j. i J : . ) , .1r it l, l ': .rr / l L . l b l , , , ' Li I i
II
i . l , , 1 , : a t r . - r p )1- ,i 1 , c 1 r 1 , l ! i . l )
r ). t ) . t f l ' l r{ fi j I 0 1 ' ' I 4 4 i t . r rI I I l j l ) . .1i . r r , t( r) 0 r , l ) ! f )I { ) ir ,., , Aside from our homemade functions, we can also use MATLAB's built-in f zero function. However, just as with the Newton-Raphson method, divergence also occurs when f zero function is used with a single guess. However, in this case, guessesat the lower end of the range cause problems. For example, .
I / ( r I ( r ( r . 1 (, 1 . ( J l l l - l )
l : : ,j 1 i l j ( . J L ' t t : t r , :
. r i r o r l i i r ( 1 r ; r r , r i( l f L l r l l
, i r , r r t r J , . 'r r c , c i rr l s { r
i omJtl r:r
. r n i n l . t , f ' . / ' ( i lr : o r r l , a i n i l q , , r r . t u- r :i r r q
( l , ' L r r ,l r , ) r r \ . / LI rr l , , , r t I 'lilll
'
,
. r : .) 0 0 t l l i \
si.gn
J L i n c' l i o n , , r . 1 i l r r - , r . r r c o i . r n L c r e d
t.,.,.
:rccl r (j1r.
i :;
4 . 9 2 A ' . )B ? , 1 ) .: 14 ; l I - i . ) r , , ' r t - . h . r r l I l . 1 | r c t r l , s l a r . 1 -L r L c ' ti , ' , . L . L u e .
f
,:
!
i!, r irj
If the iterationsare displayedusing optimset (recall Exanple 6.6), it is revealedthata negativevalue occursduring the searchphasebeforea sign changeis detectedand theroutine aborts.However,for singleinitial guessesaboveabout0.016,the routine worksnicely, For example,for the guessof 0.08 that causedproblemsfor Newton-Raphson,f zero does just fine: '
,:rIl!t
I
I
t,
.(
-
'
.
f ) - ( i : )i ' r ) t ) / ' . ' I 0 I
',1 ,1 i 4
As a final note,let's seewhetherconvergenceis possiblefor simple fixed-point iteration, The easiestand most straightforwardversioninvolves solving for the first./in Eq. (6.14): f...
-
I
t
0.25
('"'(#.*#))'
(6.16)
r f
o n u tl b
6.6 CASESTUDY
l6l
!t= s6)
0.01
0.04
0.08 "(
FIGURI6.IO
The two-curvedisplayof this functiondepictedindicatesa surprisingresult(Fig. 6. 10). Recallthat fixed-pointiterationconvergeswhen the)2 curve hasa relativelyflat slope(i.e., lg'({)l < l). As indicatedby Fig. 6.10, the fact that the y, curve is quite flat in rhe range from/: 0.008 to 0.08 meansthat not only doesfixed-pointiterationconverge,but it convergesfairly rapidly! In fact, for initial guessesanywherebetween0.008 and 0.08, fixedpoint iteration yields predictionswith percentrelative errors less than 0.0087oin six or fewer iterations!Thus, this simple approachthat requiresonly one guessand no derivative estimatesperformsreally well for this particularcase. The take-homemessagefiom this casestudy is that even great,professionallydeveloped software like MMLAB is not always foolproof. Further,there is usually no single methodthat works bestfbr all problems.Sophisticatedusersunderstandthe strengthsand weaknesses of the availablenumericaltechniques.In addition,they understandenoughof the underlying theory so that they can effectively deal with situationswhere a method breaksdown.
ROOTS: OPENMETHODS
162
PROBTEMS 6.1 Enrployfixed-pointitcrationto locatethc root ot . l ( - r ): : s i n ( v 4 ) - . ^ Usc an initial-{ucssof -rir: 0.-5lncl iterateuntil t',, < 0.Ola/a. 6.2 Use (a) fixed-point itefltion antl (b) tlrc Nc'ur'lonRaphsonmcthod to cletcrminca t'clotoi ./ (.t) : -.rl + : -5.Pcrtirlntthe conrputation l.8r * 2.-5using-i'11 unlil e,, is lcssthane, :0.05%. Also checkyor-rll'inaliutswer. 6.3 Dctcrrnincthc hishcst rcal root ol ./ (.f) : 0.9-5.t:t5 . 9 , r r +1 0 . 9 . r - 6 : (a) Graphically. (b) LIsin-cthrr Ncwton-Raphsonmcthod (thrcc itcrations, 'r;:3'5)' (c) Using thc sccantnrclhod(thrcc itcrations,-r; I - 2..5 i l n d . f ;: 3 5 ) . (d) Using thc rnodificd sccant rncllrod (thrc'c itcnttiotrs, . t i : 3 . 5 ., J: 0 . 0 1 ) . (c) Dctcrnrinctll thc rootswith MATLAB. 6.4 Dctcrrninc the l()wcst positivc r(x)t ol' ./ (-r) : 8 s i n ( . t ) r' - I : (a) Craphically. (lt) Using the Nervton-Ruphson ttrctl.tod(thrcc itctirtiotts, . t i : 0 . 3) . (c) LIsing the sccantmcthod (thrce itcrations..r';.1 - 0.5 and,ti: 0.'1. (d) Using tlrc rnodilicd scculrt nlethorl (llvc itcllti()n\, , r ; : ( ) . 3 ., l : 0 . 0 1) . nlethodand (b) lhc nrocli6.5 Usc (a) the Ncwton-Raphsorr (3 : 0.05) to clctclnrirtc ticclsecantnrcthocl a r-ootol ./ (,r'): , t 5- 1 6 . 0 5 . r*ru u . 7 - 5 . r -r l t ) 2 . 0 1 7 5 . r* r l l 6 . 3 5 r + 3 1 . 6 u 7 5 usirrg an initial gucss ot' ,t : 0.-5t125ancl t, : 0.(\l(tr,. Explain-yourrcsults. 6.6 Dcvclt>pan M-filc lirr thc sccantnrcthod.Along with thc (wo initial gucsscs,passthc l'unctiorrils an argunrctrt. T c s ti t b y s o l v i n gP r o b . 6 . 3 . 6.7 Dcvelop an M-l'ilc tir thc modificclsccant nrcth()d. Along with the initial gucssand the perturbationfiirclion. passthL-I'Lrnction asrn argurncnt. Tcstit by solvingProb.6.3. Eq. (86.4.l) l() gct Eq. (86.21.2). 6.ll DiUcrcrrtiate 6.9 Enrploy the Ncwton-Rlphsonnlcthod to dctenninc-a r c a lr t x r tf o r . / ( . i ) : - I + 6 . r - , l . r l | 0 . 5 , t 1 . u s i n ga n r n i aud usegraphical tial gucssof (a).1.-5. and (b) 4.43.DiscLrss to cxplairtany pcculiariticsin y'our and analyticalr.ucthods rcstrlts. 6.10 Thc "di"iclcand avcragc"tlctlrocl,un old-timcInctho(l lbr approxirratingthc :iquarcroot ol'any positivcrrulttbcrzr, can be firrrlulatcclas .ti luf .t1 1
Provc that this firnnula is bascd on the Newton-Raphson algorithm. 6,ll (a) Apply the Ncwton-Raphsonrnethodto thet'unction .l(r): tanh(,r2- 9) to evaluatc its known real rootat ,r : -1.Usean initial guessof .t1;: 3.2 andtakea minimum of thrcc itcrations.(b) Did thc nrcthodexhibit convergence on(o its r:al root?Sketchthe plot with thc rcsultsfor each iterationlabclcd. - 0.284"rr polynomial / (,t) : 0.0074.ra + 6. 12 Thc 3.3-5.5.rr-12.183.r-1--5has a rcal ro<;tbctwcenl-5and20, u)cthodto thisfunc(ionusingan Apply thc Ncwton-Raphson i n i t i a ls u c s so f - r 1-; 1 6 .1 5 .E x p l a i ny o u r r c s u l t s . 6.13 In a chcmicalcnginccringproccss,wiitelvapor(H:0) thatn significant is hcatcdto sLrlTiciently high tcrnpcraturcs portion of thc watcr dissociatcs.o| splits irpart.to filnnoxygcn (O1)anclhydrogcn(H:): H:OP H: * lO: the If it is assLrnrcd thut this is the only rcactioninvolved, nrolctiaction-t of HlO thatclissociates canbc Rrpresentedby
.\' Tr\ ,/^r
,,
A:,
r -
f Y l
(P6.13.1)
1-.\'
whcr-cK is thc rcaction'sccluilibriumconstultandp, isthe totalprcssurcof thc rnixturc.ll'7r, : l.J atnrandK - 0.M, dctcrnrincthc valr.rc of ,r thatsatisl'ies Eq.(P6.I3.1.1. ccluation ol statcis givenby 6.14 Thc Rccllich-Kwong R7 ''-
t-lt
r:(t:I lt)JT
whcrc R : tlre uniycrs:rlgas constrntl:0-5ltt kJ(kgK)1, (kPa), (K), / : absolutepressurc 7 : afrsolute tenrpcrnturc lunclir : thc volurncol a kg ol'gas {rrr/kg). Thc parameten tr irntllt arc calculatcclby ,r -- 0.4)7
R r r ': ' 5 l),.
t. /r -- 0.0u66Rl),.
engi. whcrc 7r,.: 45tt0kPa arrd7, = l9l K. As a chcrnical fuel ncer,you arc askcdto dctcrnrint:thc ilmountof methane that can bc hcld in a 3-rnt tilnk a( a tcmperirture of -50'C with a prcssurcol'65.0(X)kPa.Usea rtxrtlocatingmethodof your choicc to calculatcl and the-ndctcrnrinethe mass of rncthilnccontaincdin the tank. cylindu 6.15 Thc volurncof liquid V in a hollow horizontal o1'r'adius land lcngthL is relatcdto thedepthof thcliquid/tby T
,,.
,.,
I
6.16 A catenary cable i points not in the sarr Fig. P6.l6a, it is subje weight. Thus, its weigh length along the cable u scction AB is depicted in the tension forces at the e cal force balances, the fol ofthe cable can be derive d'y
--::-
w
dxz
TA
,rc
Calculus can be employ height of the cable y as a Ta ./w \ y : _.o.nl\ */ .o
(a) Use a numerical me parameter ?".rgiven and Y6 : 6, suchthat -t:50. (b) Develop a plor of v ) 6.17 An oscillatingcurre bY I :9e-t cos(2tt), w valuesof/ such that 1 : 6.18 Figure P6.18showr tor, and a capacitor in par to expressthe impedance
-1
- , , - / i t ' ,} / r - h : l L V = l l r c , , s{ ( ' - 1 I L \T / J /r givenr : 2 nr,L : 5 rnr,andl': 8.5mr. Dctelntine
+(,c-
r 63
PROBLEMS
(b)
F I G U R EP 6 . I 6
cable is one which is hung between tw
FIGUREP6.I8
cablecan be derived,
#=;,FfS
whereZ: impedance(Q). and roris the angularlreguenc_v Frnd the ar that resultsrn an rmpedance ot /-i (/ usrngthe tzero lunction with initial guessesof I and 1000for the lirllowing parameters: R :225 A, C' - 0.6 , l0-r' F, and L : 0.-5H. (r.19 Real mechanicalsyslemsmay involve the deflection o l ' n o n l i n e a rs p r i n g s I. n F i g . P 6 . 1 9 ,a b l o c k o f m a s sr i i s
canbe employedto solvc this equationtbr the ofthecablel asa lunctionof distance,r: 7,1 - cosn ur Usea numerical methodto calculatea value lbr the parameter I,q given values for the parametersut : 12 and r,s= $, suchthat the cablehas a height of ,r': I 5 at r=50. Drvelopaplot of,y versus-rfbr,r : 50 to 100. Anoscillating currentin an electriccircuitis described l=9e-' cos(2rt).where/ is in seconds.Deterrrrine all oftsuchthatl:3 Figure P6.18showsa circuit with a resistor,an inducacaoacitor in oarallel.Kircfihofl-'sru/escan be used theimpedance o1'thesystemas I ll p:
+(.c-
@t
FIGUREP6.19
r64
ROOTS: OPENMETHODS
P
fixed-point iteration sch givenQ:5.S:0.0002 your schemeconvergesf< equal to zero. 6.25 Seeifyou can deve the friction factor base describedin Sec. 6.6. yor result for Reynolds numbe e/ D ranging from 0.0000 6.26 Use rhe Newton-Ra
f (x) : FIGUREP6.2O
releaseda distanceI above a nonlincarspring.The resistancefbrce F of the spring is given by F--\krt!
Employinitialguesses of ( results. 5.27 Given f (x) : -2xb - 1.5
FIGURE P6.2I
*k.d3i'1
infbrmation:(a) For hracketingmethods,initialguesses of0 and R will bracket a single root ti)r this cxample.(b) Fot 2krtl5l2 I open mcthods,an initial guessof R will alwaysconverge. {): + ; k 1 t l '"- n t g < l - t t t q l t tl 6.22 Pertbrm the identical MATLAII operationsas thost Solvc fbr z/, given thr: fbllowing parametervaluc-s:ft1 : in Example6.7 to manipulatcand iind all the rootsof the 5 0 . 0 0 0 g / s 2f .t z : 4 0 g / ( s 2 n r s ) .n : 9 0 8 . . ( : 9 . 8 1 m / s 2 , polynomial and /r - 0.4-5m. . l s ( x ) : ( - r | 2 ) ( . r* 5 ) ( x - l ) ( x - 4 ) ( . r- 7 ) 6.20 Aerospacc enginc'erssomctimes computc the trajcctories of projcctilcssuch as rockcts.A relateclproblemdcals 6.2-1 ln control systcrns analysis, transfer tunctionsae u,ith the traicctory of a thrown ball. Thc traje-ctoryo1'a ball devr'loped rclatethc dynamicsof asys. that mathematically thrown by a right ficlder is clefinedby thc (x,.1,)coordinatcs tem's input to its output. A transf'erfunction for a robotic as displayedin Fi-e.P6.20.Thc trdecbry can be modelecl as p o s i t i o n i nsgy s t c nils g i v c nh y
Conscrvation ofcnergl'can bc uscdttr show that
Rt v - (tan0o)-r- -j5r' Ltrai cos-u0
*.r'1y
F-indthe appropriateinitial angledo,if urr:20 m/s, and the distanceto the catcheris 3-5m. Note that thc throw lcavcsthc right ficlder's hand at an elcvationof 2 m and the catclrcr rcceivcsit at I m. 6.2| You arc designinga sphcricaltank (Fig. P6.2I ) to hold water fbr a small villagc in a devclopingcountry.The volume of liquid it can hold can bc computedas l /
-
n
,llR-rl
l t ' -
3 wherc'V: volunreIm]1,/r : depthof waterin tank Im], and R : t h e t a n kr a d i u sl m l . If R : -l m, what depth rnustthe tank be filled to so that it holds 30 m3? Use three iterationsof the most efllcient nurnericalmethodpossibleto determineyour answer.Determine the approximate relatir"e cror after cach iteration. Also, providejustification fbr your choice of method.Extra
C(.tt-
c(s)
rt+ l2.5sl*-50.5,r*66
= N(.t) r'u +
l 9 s r+ 1 2 2 s+22 9 6 s- l 1 9 2
where C(s) : system gain, C(s) : systemoutput,N(s)= systcminput, and s : Laplacetransfbrmcomplexfrequency, Use MATLAB to find thc roots of thc numeratorandde. nominatorand t:rctorthescinto the form G ( , r )=
( . s* a r ) ( . s + d 2 ) ( . r+ . r 3 ) ( s * D r ) ( s* b : ) ( . r* h ) ( s * l t t )
wherc zri andb1 : the rootsof thc nume-rator anddenomina. tor, fespectively. 6.2-l The Manning equationcan bc wriften fbr a rectangulat openchannelas (=^r :
- x " - n s ^1 4
- .,5(sH)''' rt(B i 2H Jtlt
where B : flow (m3/s),S : slope (mlm), H = depth(m), l and n : the Manning roughncsscoefficient. Develop
Usea rootlocationtechn this function.Performiter tive error falls below5Zo
r65
PROBLEMS fixed-point iterationscheme to solve this equation for H given 0 = 5, S : 0.0002.B :20, and r?: 0.03.Provethat your scheme convergesfor all initial guessesgreaterthan or equal t0 zero. 6,25Seeif youcandevelopa lbolproof function to compute fie frictionlactor based on the Colebrook equation as described in Sec.6.6. Yor.rrfunction should return a precise rcsult forReynoldsnumberranging from .1000to | 07and lbr e/Dranging fiom 0.00001to 0.05. 6,26Usethe Nervton-Raphson method to find the root of :'@_ x ) 2 .f(.r)-.-tr Employ initialgucsses of (a) 2, (b) 6, and (c) 8. Explainyour rcsults. 6J?Given - Z r b - l . - 5 . r 4* l o x * 2 f (t) =
0 JT
ie le
re S-
ic
v.
), a
Use arootlocationtechniqueto dctt'rminc(hemaxirnumof this function. Perlbrmiterationsuntil the approximaterelativeenorialls below 5%,.If you usc a br'ackctingmcthod,
use initial guessesof x1 - 0 and -r, : 1. If you use the Newton-Raphsonor the modified sccantnrethod,use an initial guessof;r; : l. Ifyou usethe sccantmethod,useinitial guessesof x; I : 0 and Xi : l. Assumingthxt convergenceis not an issue,choosethe techniquethat is bestsuited to this problem.Justify your choice. 6.211You must determinethe root of the following easily di fferentiablef unction: n <,. ?"-^ =
)-l-r
justify your choice,and Pick the bestnunrcricaltcchnique, use technique determine then thal to the root.Note that it is known thatfi:rrpositivcinitial guesscs, except all techniques fixed-point iterationwill cvcntually converge.Perfbrm iterrelativeerror fallsbclow 2olc.If ationsuntil the approximate you use a bracketingmethod,usc initial guessesof r1 : [) and.r,,: 2. If you usethc Newton-Raphson or the modified secantmcthod,use an initial guessof .r; - 0.7. If you usc thc secantmethod, Lrseinitial guesscsof ,r; 1 :0 and
Opti mizqtion
CHAPTER OBJECTIVES The primaryobjectiveof the presentchapteris to introduceyou to how optinrization can be usedtcldetermineminirrraand rnirximaof bothone-dimensional and multidimensional functions.Specificobjectivesand topicscoveredare ' t ' ' ' ' ' ' ' t
Understanding why and whereoptirnizafionoccursin engineering and scientific problemsolving. Recognizingthe differencebetwecnone-dimensional and multidimensional optirnization. D i s t i n g u i s h i nbge t w c e ng l o b a la n dl o c a lo p t i m u . problemso thatit can bc solvedwith a Knowing how to recasta nraximization r n i n i r n i z i nagl g o r i t h n r . Bcing ableto definethe goldenrati
7.1
YOU'VEGOT A PROBLEM jurrper can be projectedupwarrdat a specifiedvelocity. n object like a bLrngee If is subjectto Iineardrag,its altitudeas a functionof time can be computed as
t66
t--
-ry, * T) tr - e I(tn'I)t) Z: zr* 1 (,,.,
I ( f I
(,
Z,I INTRODUCTION AND BACKGROUND
Maximum elevation
F'GURE7.1 E l e v a l t a no t q f u l l a l i ; t r ' o t f t i m e f o r o n o b f t r c l i r r r i l o / / i p r o f e c l e t l u S t , . " , ' o ,rcvli l h i r n i r t t r o l r t , - l r tr
w h e r e : : a l l i t u c l (er n )a h i l v ct h cc a r t h ' ss u d : r c e( d c t ] n c da s : : 0 ) . l o : t h c i n i t i a lt l t i t u c l e ( m ) ,l r : r n i l s s( k g ) ,( : a l i n c a r d r a gc o e l ' l ' i c i c(nktg / s ) ,q r : i n i t i ' r lv c l o c i t y( r n / s )a, n dI : tinrc(s). No(c that (irr tlris liurnulation"positivcvckrcityis corrsidcrcd to bc in thc rrpwlrci d i r c c t i o n . G i v c n t h e l b l l o w i n g p a r a r n c l c r v,t\l:u9c. s1 :{ l r r . r / s r . - ':, , l O 0 m , 1 1: r- 5 . 5 r n / r , r i : t l Ok g , a n d r ' : l 5 k g / s ,E c 1(.7 .I ) c a n b c u s c dt o c u l c u l a t ct h c j r r r n pre' s a l t i t u c l cA. s d i s p l a y c d i n F i g . T . l , t h c j u r n p c r r i s c s t o a p c a k c l c v a t i o n1o9l '0i rnbroaut al h o u t l : - 1s . S u p p o s tch a ty o u a r cg i v c nt h c - j o ho l ' d c t c r r r i n i ntgh ce x l c t l i r n eo l ' t h cp c a kc l c v a l i o n . T h c d e t c r m i n a t i oonl ' s u c hc x l r c m cv a l u e si s r c l c r r c dt o l s o p l i r n i z a t i o nT.h i s c h a p t c rw i l l introduccy()uto how the cornputcris uscdto nrakcsuchdetcrn'riniltions.
7 . 1 INTRODUCTION AND BACKGROUND In thc nrostgcneralscnsc,optinri:/.ati()rl is thc ploccsso1'crcatirrgsonrcthingthat is as c f f b c t i v ca s p o s s i b l eA. s c r r g i u c e r w t o n t i n u < l u sdl cy s i g nd c v i c c sa n c pl r o d u c t tsh a ( s .c n r u s e p e r l i r r mt a s k si n a n c f f i c i c n lI a s h i o nl i l r t h c l c a s tc o s t .T h L r sc, n g i n c e r sa r c a l w a y sc o n frontingoptirnizittionpnrblerls that uttcnrptto balarrccpellirrnranccand linritatiorrs.In adclition. scientistsIravcintcreslirr optirnalphenorncna firlnr thc pcukclcvationo1' rurnging projectilesto the minirnunrli'eccncrgy. perspective. Frour a nrathcnraticirl optinriz.ation dcirlswith l'indingthe nraxinraand rninimaof a l'unctiorr that dcpendson one or m()revariablcs.The goal is to detcrnlincthe valuesof tlre variablesthat yield rnaximaor minirnalbr the I'unction.Thcsecan then be substituted back into thc functionto conrputcits optinralvalucs. A l t h o u g ht h e s c s o l u t i o n sc a n s o r n c t i r n cbsc o b t a i n c da n a l y t i c a l l yn, r o s tp r a c t i c a l optimizationproblenrsrequirc nurnerical,cornputcrsolutions.From a numcricalstandpoint, optirniz.atit'rn is sinrilar in spirit to lhe root locationrncthodswc just covercdin Chaps.5 and6. That is. both involveguessinganclsearching tilr a pointon a function.The fundarne'ntal dif'lbrcnccbetweenthc two typesol'problernsis illustratedin Fig.7.2. Root Iocationinvolvessearchingfor thc locationwherethe I'unctioncqualszcro. ln contrast, involvessearchingfor the function'scxtrenrep
168
OPTIMIZATION
T m
l'('t) : o l"(r) < o l(x) : o
/'("t) = o > "f"(,r) 0
TT
tht gri
F I G U R E7 . 2 A f u n c t i oonf o s i n g l ev o r i o b l iel l u s t r c f itnhqed i f f e r e n cbee l w e e n r o o f so n do p l r r n o
As can be seenin Fig. 7.2,theoptirnunrs arethepointswhcrethccurveis flat.In mathenrirticalterms,this correspondsto the .r value whcre thc derivative / '(.r) is equalt0 zelo. Additionally,thc secondderivative,.1"(,r), indicateswhethertheoptimurnis a rninimumor a maximum:il' .t'"(x) < 0, the point is a maximLlm;if .l'"(.r) > 0, the pointis a minimun Now, understanding thc relationship betwecnrootsandoptimawouldsuggest a posible strategyfor finding the latter.That is, you can diff-crentiate thc functionandlocateth root (thatis, the zero)of the ncw tunction.ln fhct.someoptirnizationrnethods dojustthis by solvingthe root problenr:.f'(.r1: 11. E X A M P L EZ . l
sat AS t__
ofr on( prc sea me
D e t e r m i n i n gt h e O p t i m u m A n o l y t i c o l l yb y R o o tL o c o t i o n Problem Stotement. Deterrninethe tirnc and magnitudeil1'thepeakelevation basedon E q . 0 . 1 ) . U s e t h e l b l l o w i n g p a r a n r e t evr a l u e sf o r y o u r c a l c u l a l i o n.:g : 9 . 8 1 m / s t , ;o : 100m, u0 - -55m/s, nt : u0 kg, andr: : l-5kg/c.
* {oJ( PqU
S o l u t i o n . E q u a t i o n( 7 . 1 )c a nb e d i l f ' e r e n t i a t e t odg i v e
rh;
ltou
dz
n
:
\ ( / t t t) t -
D'e
tn8
(;
(l -
e-
i'l"'t';
Note that because1)- Ll-.ldt, this is actLrally thc ccluation fbr the vclocity.Themaximun elevationoccursat the value ol I that drives this gquationto zero.Thus,the problen anrountsto deterrnining the root.Forthis case.this can be accomplished by settingthedcrivativeto zeroand solvingEq. (E7.L l.) analyticallylbr
r :'!rn (r + !a ) tng/ r' \
r ivcs S u b s t i t u t i n gt h e p i l r i i r n e l e r g
,:
l5ts5t 8o r n/ r r \ :3.8.r166s 80r9.81)/ l.s \
min
INTRODUCTION AND BACKGROUND This value along with the parameterscan then be substitutedinto Eq. (7.1) to computethe maximum elevation as z:
80 / 8 0 r QR l r \ 8 0 r, 9= ."8, ,1r r- r . 8 3 t 6 6 ) : 1 0 0 + : ; | 5 0 + " " . ; ; " , , ) ( t _ . _ , 1 58 0 , 3 8 3 1 6 "6"1, _ 1 9 2 . g 6 0m9
15\
rs /'
ls
We can verify that the resultis a maximum by differentiatingEq. (E7.1.1)to obtainthe secondderivative d2z -;;: (ll-
c - - u t ) €- . t ,/ , , t t - , n - ( t-/(' .' t/t, n \ t "ntlt ln
- m - 9^. 8-1 ;
The fhct that the secondderivativeis negativetells us that we have a maximum. Further, the result makesphysicalsensesincethe accelerationshouldbe solely equalto the force of gravity at the maximum when the vertical velocity (and hencedrag) is zero. Althoughan analyticalsolutionwaspossiblefbr this case,we couldhaveobtainedthe sameresultusingthe root locationmethodsdescribedin Chaps.5 and 6. This will be left as a homeworkexercise.
Althoughit is certainlypossibleto approachoptimizationas a rootsproblem,a variety of directnumericaloptimizationmethodsareavailable. Thesemethodsareavailablefbr both problems.As the name implies,one-dimensional one-dimensional and multidimensiontrl problemsinvolvefunctionsthertdependon a singledependent variable.As in Fig. L3a, the searchthenconsistsof climbingor desccnding peaksandvalleys.Multidione-dimensional mensionalproblemsinvolve lunctionsthat dependon two or more dependentvariables.
FIGURE 7.3 ( o )C n e ' d i m e n s i o n o lp t i m i z o t i o nT h i sf l g u r eo l s o i l l u s t r o t ehso w m i n i m i z c i i oonf 1 1 l r 1i s i ot t h e m o r i m i z o l i o n equivcrlen o f - . / ( , r ) ( b )T w o - d i m e n s i o noopl l i m i z o t i o nN. o t e t h o tt h i s l r g u r ec o n b e t o l e n t o r e p r e s e nei i t h e ro m o x i m i z o i i o(nc o n t o u risn c r e o s ei n e l e v o l i o nu p t o lhe moximumiike o mountcrin) or c minimizotionicontoursdecreosein eJevotion down to the m i n i m u ml i k eo v o l l e v )
Optimum/ir'..u'r
Minimum.l("r)
r70
OPTIMIZATION
7.
ln the samespirit,a two-dimensional optimizalioncan againbe visualizedassearchingout peirksand valleys(Fig.7 .3b).However,just as in realhiking, we arenot constrained towalk a singledirection;insteadthe topographyis examinedto efTicientlyreachthegoal. Finally,the processof findrng a maximum versusfinding a minimumis essentially identicalbecausethe samevalue x* both minimizes/(x) and maximizes-/(,r). This equivalenceis illustratedgraphicallyfor a one-dirnensional ftrnctionin Fi_c. 7.3d. In the next section,we will describesome of the mole common approaches for onedimensionaloptimization.Then we will providea brief descriptionof how MATLABcan be employedto determineoptimafor multidinrensional functions.
7.2 ONE-DIMENSIONALOPTIMIZATION This sectionwill describetechniques to find the mrnrmnmor maxrmumof a function ofa s i n g l e v a r i a b l e . l ( . . ) . A u s e f u l i m a g e i n t h i s r e g a r di s t h e o n e - d i n r e n s i o n" raol l l e r coaster"-like lirnction dcpicteclin Fig. 7.4. Recall liorn Clraps.-5ancl6 that root location was complicatedby the fact that severalrootscan occur tbr a singlefunction.Similarly, h o t hl o c a lu n dg l o h u lo p t i m i rc u r o c c u ri n t r p t i n r i z a t i o n . A gkhul olttittrtrmreprescntsthe very best solution.A local optinrunr,thoughnotth very best.is betterthunits inrmecliate neighbols.Casesthat includelocaloptimaarecalld multimodol.lnsuchcases,we will almostalwaysbe intercstcd in findingtheglobaloptimunl In addition,wc mustbe concerned aboutmistakinga localrcsulttor theglobaloptimum. Justas in rool location.optimizrtionin one dimensioncan be dividc'dintobracketing andoperrmethods.As dcscribedin thenextsection,thc golden-section search is anexamplc of a bracketingnrethodthatis very sirnilarin spiritto thc hisectionnlethodlbrrootlocation This is firllowcdby a somcwhatmoresophisticated inter. brackctingapproach-parabolic polation.We will then show how thcsctwo nrethodsarecornbinedand irnplemented wit M A T L A B ' s I r ni n b n d f u n c t i o n .
FIGURE 7.4 A {unctionthirl osymptoticolly opproocheszero ot p us ond minLrs oc ond hcrstrvo moximum ond two rninimum p o n l s i n t h - -v i c i n i t yo f l h e o r i q i n .T h e l w o p o i r l s1 ot h - -r i g h tr r r e o c o o p t i m o , w h e r e c sf h et w o t o t h e l e f to r e q o b o l .
7. In aft
the pr( ofi
the lett rolr Euc whe
Mul
wher
q
'l
In ad provir termir T roots i guess tween The rc xr
Global maxtmum
FIGUR Euclid s rotioof smoile
Global
F-
1-
2.2 ONE.DIMENSIONAL OPTIMIZATION
l7l
7.2.1 Golden-SectionSeorch In manycultures,certainnumbersareascribedrnagicaiqualities.For example,we in theWest quantities. areall familiarwith "lucky 7" and "Fridaythe 13th."Beyondsuchsuperstitious thereareseveralwell-known numbersthat havesuchinterestingandpowerful mathematical propertiesthat they could truly be called"rnagical."The mostcomnlonof thesearethe ratio of a circle'scircumferenceto its diameterz and the baseof the natLlrallogarithme. Although not as widely known, the golden ratio shouldsurelybe includedin the pantheonof remarkablenumbers.This quantity,which is typicallyreprescnted by the Greek leuer Q (pronounced;fee), was originally definedby ELrclid(ca. 300 BCE) becauseof its role in the constructionof the pentagramor five-pointedstar.As depictedin Fig. 7.-5. Euclid'sdefinitionreads:"A straightIine is saidto havebeencut in extremeandmeanratio when,as the whole line is to the greatersegment,so is the greaterto the lesser." The actr"ral valueof the golclenratiocan be derivedby expressing Euclicl'sdefinitionas t1IL2 Ir
lr (':
(7.2)
Multiplying by t 1/(.2and collectingtermsyields (7.3)
o'-Q-I:0 where@ : L r / ( z. The positiveroot of this equationis the goldenlatit'r
4,:
l+\,6
:
(7.1)
1 . 6t 8 0 . r - r 9 8 8 7 4 9 8. 9 .
pleasingin Westerncultures. The goldenratio has long beenconsideredaesthctically In addition,it arisesin a varietyof ilthcr contextsincludingbiology.For our pulposes,it providesthe basisfbr thc golden-sectiorl rncthoclfbl dcscarch,a sirnple,general-purpose function. terminingthe optimumof a single-variablc searchis sirnilarin spirit to the'bisectionapproachlbl locating The golden-section by a lorver rootsin Chap.-5.Recallthat bisectionhingedon definingan interval,specil'ied guess(-r,)and an upperguess({,) that bracketeda singleroot.The prcsenceof a root bethat.l'(.r,)and.l(.r,,)had difl'ercntsigns. tweenthesebounclswas veriflcclby deterrninimg The root was thcn estimatedas the midoointof this intcrval:
'.,
'tt I x,,
(7.51
FIGURE7.5 so lhot fhe Euclidsdefinitionof the golden roiio is bo-sedon dividing o iine i.rfolwo segmc-nls roiio of ihe whole line io the lorgersegmentis eguol lo the roiroof the lcrrgersegrnenlio ihe smo//ersegmeni Ihis roiio is co//edthe go/den rofio (t
,
t-+l-,
_
(2-
172
OPTIMIZATION The final stepin a bisectioniteratjoninvolveddetennininga new smallerbracket.Thiswas done by replacingwhicheverof the bounds-r1or-r,/had a functiorrvalue with the samesign as-l(r,.).A kev advantageof this approachwas that the new valner, replacedoneof theold bounds. Now supposethat insteadof a root. we were interestedin detenniningthe minimurn of function.As with bisection,we can startby dcfining an intervalthat a one-dimensional andhence containsa singleanswer.That is, the intervalshouldcontaina singlenrinirnurrr. as for bisection.where.l,and4 is calleduninrodal.We can adoptthe samenomenclature definedthe lower and upperbounds,respcctively. of suchan interval.However,in conlrasl to bisection,we needa new strategyfor findinga rninimumwithin the interval.Ratherthan value(which is sufflcientto detecta sign change,andhencea usinga singleintermediate zero), we would need two intermediatcfunction valuesto detcct whcthera minimum occun'ed. points, The key to makingthis approachetficicntis thc rvisechoiceof the intermediate As in bisection,the goal is to nrininrizelirrrctionevaluations by rcplacingold valueswith nerv values.For bisection.this was accornplished by choosingthe midpoint.Forthe golclen-section sc'arch,the two intcrnediirtepoints are chosenaccordingto the golden ratio: .Yt:.\/ +d
(76)
.r):xu-d
(1.1)
where d : ( ( h - l ) ( : r , -, , r r )
(7.81
Thc functionis evaluatcdat thcsetwo inleriolnoints.Two resultscan occur: l.
)
I t ' , a s i n F i g . 7 . 6 a , . l ( x.rll( - r : ) , t h e n . l ( . r , ) i s t h c r . n i n i n r u n r , a n d t h e d o m a i n o f . r t o t h e For leli cll'x,,liorn -r,to r), clurbe clintinatcdbecauscit doesnot containthc rninirnum. the new .r,lirr thc next round. this casc,.r, bscclmes If I ( v:) . ./ (,rr), then/ (.r,)is the rrinimum andthc donrainol' r to thc rightof .r,,fiom .rr to ,r,,would bc eliminatcd.Fclrthis case,.\r bccotnesthc new .r,,lirr thc ncxtround.
Now. hereis thc neirlhcnelit {l'clmthc uscol'thc goldenratio.Becallscthe oliginal.r, and r, u,crechosenusingthe goldenlatio, we do not havc to recalculatcall thc function valuesfor thc next iteration.For example.tbr thc caseillustratedin Fig.7.6,theold"r,becomesthe new rr. This meirnsthat we alrcadyhavethe valuelbr the new.l(r.), sinceitis the sameas the l'unctionvalueat the old.r,. To cornpletethc algorithrn,we needonly detcrminethe new.r,. This is donewith Eq. (7.6) with r/cornputedwith Eq. (7.8) basedon the new valueso1',r,rud.r,,.Asimila approachwould lre useclfbr thc alternatecascwherethe optirlLrnrf-cllin the leli subinterval. For this case.the new.rrwould be computedwith Eq. (7.7). rapidly. As the iterationsarerepeated, the intervalcontainingthe extremumis reduced In fact,eachroundthe intervalis reducedby a 1'actor of d - I (about61.87c).Thatnleans that after l0 rounds,the intervalis shrunkto about0.6l8r{)or 0.008or 0.8% of itsinitiai length.Afier 20 rounds,it is about0.0066%. This is not quite as good as the reduction achicvedwith bisection(50Vo\.birt this is a harclerDroblem.
EXAMPLE 7.2
Gol<
Prob
50lu
2.2 ONE-DIMENSIONAL OPTIMIZATION
r73
S
n d )f rt e f,
rl-d+xl
St .n
(al
n S.
rh le :n
x2
-t1
t I I
New (b)
ne or
FIGURE 7.6 (o) The initiclstep of the golden-section seorcholgorithminvolveschoosingiwo inleriorpornts occordingto the golden rotio (b) The secondstep involvesdeflnlngo new intervolthot e n c o m o o s s el n s eoollmum.
)m rd. .rl
on )eis
EXAMPLE 2.2 G o l d e n - S e c t i o Sn e o r c h ProblemStotement. Use the golden-section searchto find the minimum of 1
irh lar al. Llv. tns :ial ron
"f ( x t : - - 2l 0s i n r within the interval from -r,: 0 to jr,,: 4. Solution. First,the goldenratio is usedto createthe two interiorpoints: d : 0.61803(4 - 0) : 2.4'.721 rr :0 +2.4121:2.4121 xz:4-2.4121 :1.5279
174
OPTIMIZATION The function can be evaluatedat the interior ooints: l(x:):
1.52192 - 2 s i n ( l. 5 2 1 9 ) : - 1 . 1 6 4 7 l0
I (rr) :
2.412t2 - 2 s i n ( 2 . 4 1 2 1:) - 0 . 6 3 0 0 t0
Because.[(xt1 ..1 (,r1),our best estimateof the minimum at this point is thatit n focatedat r : | .5219with a valueof ,f (r) : -l .7641.ln addition,we alsoknow thatthe minimum is in the intervaldefinedby -r,,.rr,and.r,. Thus,for the next iteration.thelower b o u n dr e n r a i r l s . :t / 0 , a n dx , b e c o n r etsh e u p p e rb o u n d ,t h a ti s ,. r , , : 2 . 1 7 2 1. I n a d d i t i o n , the fbrmer-r. valuebecomesthe new .rr, thatis,x, - | .5279.ln adclition,we do nothave to recalculate/(x1), it was determinedon the previousiterationas l (.5219) : -1.1641, All that remainsis to use Eqs.(7.8) and(7.7)to computethe new valueof r/andr,: d : 0 . 6 1 8 0 3 ( 2 . 1 1 2 1 0- ) :
or es
1.5219
or the
r z : 2 . 4 J 2 1 - 1 . 5 2 7 9: 0 . 9 4 4 3 T h e f u n c t i o n e v a l u a t i o n a t ./r(.0i s. 9 9 4 3 ): l.,53l0.Sincethisvalueislcssthanthe f u n c t i o nv a l u ea t . r , . t h e m i n i m u mi s l ( 1 . 5 2 1 9 ) : - 1 . 7 6 1 1, a n d i t i s i n t h e i n t c r v apl r e scribedby rr, -r,. and,r,,.The processcan be repcatcd,with the resultstatrulated hcre:
Th xt
I 2 3 4 5 6 7 B
0 0 o 9443 4 s443 3050 30,sc) 3050 3 9 0I
J'$,) 0 0 _53r0 5 3t 0 7595 7595 7595 7742
x2
I 5279 o 944'3 5279 3050 5279 4427 3 9 0I 4427
f (xr) 7647 -5310 7647 7595 7647 7755 7742 7755
xt
24721 I 5279 I.BB85 ) 5279 1 66-s6 I 5279 I 4427 I 4752
l&) --0 6300 *1 7647 , t 5432 --17647 t 7136 t 7647 -1 7755 -t 7732
xu
J'(x,,)
40000 :tIt36 -o 6.100 2 4721 (l c,,lOO 2 4721 B8B5 t.5432 BBB-5 ) 5432 6656 I 7136 5279 t 7647 -t 7647 5279
a
2 A/21 ) 5279 0 9443 0 5836 0 360/ a 2229 0 I3l8 0 0B5r
Note that the current minimum is highlightedtbr cvery iteration.Alier the eighth i t e r a t i o nt,h e m i n i m u r no c c u r sa t x : 1 . 4 4 2 1 w i t h a l i r n c t i o nv a l u c c t l ' - ] t . 1 1 5 5T. h u s . t h e r e s u l ti s c c r n v e r g i nogn t h e t r u ev a l u eo f - 1 . 1 1 5 7 a t . r : 1 . 1 2 1 6 .
Recallthat for bisection(Sec.5.4), an exactupperboundfor the crror can be calculatedat eachiteration.Using similarreasoning. an upperboundfbr golden-section search canbe derivedaslbllows:Oncean iterationis complete,theoptimumwill eitherfall inone of two intervals.lf the optimumfunctionvalueis at,rr, it will be in the lower interval (.r,, .r,..r,).If optimumfirnctionvalueis at -r,,it will be in the upperinterval(.r...r,..r,,).Because the interiorpointsale symmetrical,eithercasecan be usedto definetheerror.
Fit ap
No Co
gol wo nuI l.
Z.2 ONE-DIMENSIONAL OPTIMIZATION
Lookingat the upperinterval(xr,r,, :r,,).if the true valuewereat the fiu'left,the maximum distancefrom the estimatewould be Ar.:-TI -,r2 : . r t I ( Q - D Q , - r r ) - x , ,+ ( Q - l ) ( x , ,- x i ) : (rr - .t,,)+ 2(0 - l)(.r,, - ,r1) : (20 - 3)(.r,,- .rr) or 0.2361(-r,,-.r',). If the true vaiuewereat the far right, the maximumilistancefiorn the estimatewould be A,r1, : :
-{,, -'f ,{rr
I .t't -
(Q -
l)(x,, - -ri)
: (r,, - rr) - (Q l)(.r,,- .r1) : ( 2 - d ) ( . r u- . r r ) Therefbre,thiscasewould represent themaximumerror.This resultcan or 0.3820(,r,,- ,r1). thenbe normalizecl to thc optimalvaluefcrrthat iteraticln .r,,n, to yicld le
t._..1
t , , : ( 2 - O ) l ' : ! - - - - : J lx t 0 0 %
e.sl
I r,'nr I
; 79
This cstimateprovidesa basisfbr tcrminatingthe iterations. searchfor minimizationis presentcdin An M-file f'unctionfbr the golden-section returnsthe locationof the rninimum,the valueof the function,the Fig.7.7.The l'uncticln approximate crror,artclthe numberof iterations. The M-file can bc uscdto solvethe problenrfrorn Example7.1.
1? > ) ' q - - - ' .i 8 1 ; v O - 5 5 ; n t = 8 0 ; r . : = 1 \ ; z r ) - I 0 l ) ; (zt)+rnlc*(vO+m*!.1.,'c)*(I erp( :=Q(L) r: > | x n i n , f rn i ri , .:ra , i L c r I = g o 1 ctn i n ( z , O , i', )
l6
>-
)7 29 7B il
xmin
c/m*1,) )
nr*q,i.*L)
;
=
3.8317 lnii n 192.86r) 9
th ne
u;h ne xF SC
6.9356c
(l0'-r
N o t i c eh o w b e c a u s et h i s i s a m u x i m i z a t i o nw, e h a v ee n t e r e dt h c n e g a t i v eo f E q . ( 7 . 1 ) . C o n s e q u e n t l [ym , i n c o r r e s p o n ct losa m a x i m u mh e i g h to l ' 1 9 2 . 8 6 0 9 . You nraybe wonderingwhy we havestressed the reduccdfunctiorrevaluations of thc goldcn-sectionsL-ilrch. Of course,lor solvirrga single optirnization,thc speedsavings would bc negligible.However,there are two importantcontextswhere mininrizingthe numberof functioncvaluationscan be important.Theseare l.
searchalgorithmmay be a Many evaluations.Therearc caseswherethe golden-section part of a much largercalculation.In suchcases.it may be calledmany times.Therefbre, kecpingfunctiortevaluations to a minimumcoulclpay greatdividendsfor suchcases.
176
OPTIMIZATION
r -n-r io-r Ix, fx,e-, irer l.goJdrin lf ,x',xu,es,maxic,varargin) % goldrrLin: mj-nimizaLior-r golclen section search % : fx,fx,ea,iterj=goldmin(f,xl,xu,es,maxit,pI,p2,...) % uses grolden sectjon search to find t.he minimum of f % inpr'rg; t = name of functior1 Z oo r'l , xl lower and Jppe' oLesses (dlefault - C.0001?) % es = de:;ired relative error (default = 561 % maxit = naximum allowable iterations p1 ,p2, , .. = additionaL parameters used by f % 9:
? ? ? ?
^rrrrirrts.
x = locaLj.on of mrnrmum fx = minimum function value ea - approxirnaLe relative error = number of iterations iter
(?)
FI
if rr.rrgin.3,error('at least 3 input argumenLs required'),end if nargin<4 rsempty(es), es=0.0001;end if nargin<5 isempt-y(maxit), maxit=50;end p h i . ( 1 + s q r r -( 5 ) ) / 2 ; iter-0; while(1) xI); d = (phi.-1)* (xu xI = xl r cl; x2 = xu - d; if t(x7,\tararqirr{:}) < f (x2,vararcJin{:}) xopt = x1; x1 = 7); xopt
-
ul
7.
Pa pr,
co ca
SO
x2,.
tic
vl.
end iter=iter+1; if xopt---0, ea = (2 - phi) * abs( (xu i-f ea <= es I iter >= maxit,br:eak, end end x-xol';[x.i x 'pL,'/a'or:in :] ;
xl)
/
xopt)
*
100;end
F I G U R E7 . 7 A " A / i ; l e 1 3 j p r 5 . r m i n p t h e m r n i m u m o f o l u n c i i o n w i t h t h e o o l d e n - . s e c l i o ns e o r c f r
7
Time-consunringevaluation.For pedagogicalreasons,we use simple functionsil most of our examples.You shouldunderstandthat a functioncan be very compler and tinre-consunling to evaluate.For exarnple,optimizationcan be usedto estimatr the parameters of a modelconsistingof a systemo1'diff'erentialequations.Forsucl cases,fhe "function" involvestime-consumingmodel integration.Any method thal m i n i r n i z e s u c he v a l u a t i o nw s o u l db e a d v a n t a s e o u s .
wl op
7.3 $(AAAPLE
Po
Pr
Sr
2.2 ONE-DIMENSIONAL OPTIMIZATION
177
Parabolic approximation of maximum
Truemaximum ./(.r) True function
Parabolic function
F I G U R E7 . 8 G r o p h i c odl- - p i c t i oonf p o r o b oi c i n t e r p o l o t i o n
7.2.2
Porobolic
Interpolotion
Parabolicinterpolationtltkesadvantagcol'the lirct that a sccor.rd-ordcr polynornialolien providcsa good approxirnation to the shapcof l(.r) ncaran optimum(Fig. 7.g). .fustits lhcrc is tlnll'ttnc sl.raight linc corrnccting two points,thereis only 9nc parabola c o n n e c t i ntgh r e ep o i n t sT . h u s .i 1 ' w ch a v ct h r c cp o i n t st h a t . y o i n t h l yr a c k eat n o p t i m u m w . ccltnl.ita parabolato the points.Thcn wc candil'lcrcntiatcit, sctthc rcsultcqualto zcro,ancl solvelbr atr estitnaleo1'lhcoptimal-r. It can bc shownlhloughsontcalgcbraicrlanipulat i o n st h a tt h c r c s u l ti s rl:.rl-
(7.l0l
where-r', -r., ancl,r, iirc thc initial gucsses, and.rr is the valueo1'.rthat corresporrds tq the o p t i r r r u nvta l u cu l ' t h cp a r . a h o l fi ci t t o t h eg u e s s e s . E X A M P2L. E 3
P o r o b o l i cI n t e r p o l o t i o n ProblemStotement. Use parabolicinterpolationto approxirnatc thc rnipirnurnol' /(r) :
in Lex ate rch lat
rl lo
2sin'r
r v i t hi n i t i a lg u e s s eosf , r , - 0 . . r , :
l , a n d _ r . . :- 1 .
S o l u t i o n . T h c l u n c t i o nv a l u e sa t t h c t h r e eg u e s s ecsa n b e e v a l u a t e d : r
-
r'-l .
-A
f)
.l(r,):0 l(r,): -1.-s829 /(r',):3.1136
r78
OPTIMIZATION
7.3
into Eq. (7.10)to give and substituted r'. -
* 3.11361 - 0 - 4 ) 2[ - r . 5 8 2 9 - 01 l (l - 0)2L-1.5829 2 ( 1 - 0 ) [ - r . 5 8 2 9- 3 . i l 3 6 1- ( l - - + [)- 1 . s 8 2 9- 0 ]
I
whe func
which frasa functionvalueof /(1.50-55): -1.1691. Next, a strategysirrrilarto the golden-sectionsearchcan be employedto determine which point should be discarded.Becausethe function value for the new point is lower than for the interrnediatepoint (.r:) and the new x value is to the right of the intermediate point.the lower guess(x,) is discarded. Therefore,lbr the next iteration: rr :l r: : I.-50-5-5 xl: 4
Exar
.li.r'): -1.5829 .l'Q): -1.76e1 ./k.):3'1136
r v h i c hc a nb e s u b s t i t u t ei dn t o E q . ( 7 . l 0 ) t o g i v e ra = 1.50.55 :
I ( r . 5 0 - 5-5 I ) 2[ - I . 7 6 9 t- . ] . 1 l 3 6 l 2 ( 1 . - s 0 5 -r5) t- - 1 . 7 6 9-t 3 . l l 3 6 l - ( t . 5 0 5 s 1 ) l - t . 7 6 9 1
1.4903
which lrasa functionvalueofl( I .4903): -l .1114.The processcan be repeated, withthc r-e.suIts tabulatedher-e: .rl
I 2 3 4 5
J@,)
0 000cr 0 0000 I0000 r 5829 I0000 ,t 5B2a r o o o o - t 5829 1 4 2 5 6 - t 7 75 7
x2
0000 5055 4903 4256 4266
f (x2)
r.t
f k,l
- r 5 8 2 9 4 0000 3 r 36 I 7 6 9 1 4 0000 J I I 36 t 7714 5055 - t 7691 -1 7757 4qa:J - t /714 -) 7757 49A3 I 7714
X,
1 .5055 t 4903 I 4256 I 4266 14 2 7 5
"f(rJ -1 /691 -1 7711 -) 7757 -t 7757 -)7757
Thus, within five iterations,the resultis convergingrapidlyon the true valueof -1.7757 a t . r: 1 . 4 2 J 6 .
7.2.3
Thu itera
MATTAB Function: rminbnd
Recallthat in Sec.6.4 we describedthe built-in MATLAB functionf zero. Thisfunction combined severalroot-finding methodsinto a single algorithm that balancedreliability with efliciency. The fminbn.f function provides a sin.rilarapproachfor one-dimensionalminimiza. tion. It combinesthe slow. dependablegolden-sectionsearchwith the faster,butpossibly unreliable.parabolicinterpolation.It first attemptsparabolicinterpolationandkeeps ap plying it as long as acceptable resultsareobtained.If not,it usesthe golden-section search to set mattersin hand.
7.3
MU
Asid func a rol valle a har
2.3 MULTIDIMENSIONAL OPTIMIZATION A simple expressionof its syntaxis = fminbnd(functian,xl,x2)
fxmrn, fvafl
wherex and f va1 arethe locationand valueof the minimum, function is the nameof the function being evaluated,and xl and x2 arethe boundsof the interval being searched. Here is a simple MATLAB sessionthat uses fminbnd to solve the problem from Example7.l. > > g = 9 . B1 ; v 0 - 5 5 r m= BO; c = 1 5 ; z A- L 0 a ; >>
z=@(t)
(20+n/c*(v0+m"g/c)x
\1-exp(
c/m"t))-m*g/c*t)
;
X=
3.8317 f-192.8609 As with f zero, optional parameterscan be specified using optimser. we can display calculation details: options
= ofitimset
( 'display'
,
'iter')
For example,
;
fnr i
Func-count 1 2 -l
4 5 5 1 B 9 OptimizaLion the
current
f (x)
X
3.05573 4.94427 r.88854 3 .81544 3.85836 3.83332 3 .83r52 3.83166 3.83159
rB9.159 187.19 I/I,87I 792.851 -r92 .851 192 . B6r 792 . B6r 192. B(r1 792.861
Procedure inirial go 1den qolden parabo 1i cpirrabolic perrabolic parabolic P.,rrabolic parabo 1i c
ter.mrnated: x
OPTIC)NS.ToLX of
satisfies 1.000000e
the
termrnation
criteria
usinq
004
3 .83,r1 Thus, after three iterations.the nrethodswitclresf'romgolden to parabolic,and alier eight iterations.the minimum is determinedto a toleranceof 0.0001.
7.3 MUTTIDIMENSIONAT OPTIMIZATION Aside from one-dimensionalfunctions, optimization also deals with multidimensional functions.Recallfrom Fig.7 .3a that our visual imageof a one-dimensionalsearchwas like a roller coaster.For two-dimensionalcases,the image becomesthat of mountainsand valleys(Fie.1.3b).As in the lbllowing example,MAILAB's graphiccapabilitiesprovide a handy meansto visualizesuchfunctions.
r80 E X A M P L E7 . 4
7.4
OPTIMIZATION V i s u o l i z i n gq T w o - D i m e n s i o n qFlu n c t i o n Problem Stqtemeni. Use MATLAB's graphicalcapabilitiesto display the following function and visually estimateits minimum in the range-2 < xr < 0 and 0 < .r2 < 3: . f ( x t ' x ) : 2 i x t - x z - ' l -2 x l + 2 x r r z + x i Solution. The fbllowing scriptgenerates contourand meshplotsof the function: x - l i n s p a c e ( 2 , C), 4 A ) ; i, = l i n s p a c e ( 0 , 3 , 4 0 ) ; t ] : ' Y l = m e s h q t r i d( x ' Y ) ; z = . 2 + x - '+{ 2 * x . ^ 2 + 2* x . * Y + Y .' ' 2 ; subploL i'7,2,7); c ! ; = c o n t o l l r ( X , Y , Z ' ) ; ( -l :. r b e l ( c s ) ; xlabel ('r-1' ) ;\'l;rbel (':<-2' ) ; (a) Contour ploL' ) ;,Jrid; trtle(' subplol \I,2,2 t; cs=surf c lX,'{,Z); zmin=f l,ror(min (Z)); z m a x - r - - e iI ( m . r x( Z ) ) ; x l a b c l { ' y _ 1 ' ) ; : r I a b e l ( ' : ' , . _ ?), '; z l a b e l title('(b) M es l - r t r ' l o t ' ) ;
nLtn on {fAt
uat funr
7.i
Sta mu dire han
whr the (' f (x_I,:<._2)' ) ;
the
As displayed in lrig. 7.9, both plots indicatc that function has a minimurn value of about -l and.r, : l.-5. . f t . r ' , " r 2 ) : 0 t o I l o c a t e da t a b o u t r l :
FIGURE7.9 (o) Confourond (b) rneshplc,lsof o Mo-dimensionolfunctron
Bo wh
forc fere
* lo) | the
(a)
{b)
7.4 CASESTUDY
t8l
Techniques for multidimensionalunconstrained optimizationcan be classifiedin a numberof ways.For purposesof the presentdiscussion,we will divide them depending on whetherthey require derivative evaluation.Those that require derivativesare called gradient,or descent(or ascent),methods.The approaches that do not requirederivativeevaluationare callednongradient, or direct, methods.As describednext. the built-in MATLAB functionfniinsearch is a directmethod. 7.3.1 IVIATLAB Functiohs fminsearch StandardMMLAB has a function fminsearch that can be usedto determinethe minilnum of a rnultidimensionalfunction. It is basedon the Nelder-Meadmethod,which is a direct-searchmethod that uses
fvall
= fminsearch( function,xl,x2)
wherexmrn and frzal arethe locationandvalueof theminimum,fttnction is thenameof the tunctionbeing evaluated,and xl and x2 are the boundsof the intervalbeing searched. Hereis a simpleMATLAB sessionthat usesf rninsearch to determinerninimumfor thefunctionwe just graphedin Example7.4: : . > f = @( x )
2+x\1 ) -x(2) +2*xlI)
-1. OO0O =
^ 2 + 2 * x ( l * x ( 2 ) + x ( t : , )^ 2 ; )
1.5001)
f v e ri
0.7500
ENERGY EAUILIBRIUM AND MINIMUM POTENTIAL Bockground. As in Fig. 7.70a,an unloadedspring can be attachedto a wall mount. When a horizontalforce is applied,the spring stretches.The displacementis relatedto the force by Hookes law, F : kx. The potential energyof the deformed stateconsistsof the differencebetweenthe strainenergyofthe springand the work doneby the force: PE(i :0.5kx2 * Fx
(7.11)
7.IO FIGURE forcesiretches spring oitoched to o wollmount. of o horizoniol {b}Applicotion {o)An unlooded isdescribed thespring forceonddisplocement by Hookes low wheretherelotionship beiween
OPTIMIZATION it,I+:l.S-,1,,i l ,t 'i.,,|.
continued
Equation(7.11)definesa parabola.Sincethe potentialenergywill beat a equilibrium, the solution for displacementcan be viewed as a one-dimensional tion problem. Becausethis equation is so easy to differentiate,we can solvefor p l a c e m e nat s . r : F / k . F o r e x a m p l ei,f k : 2 N / c m a n d F : 5 N , r = 5 N / ( 2 2.5 cm. A more interestingtwo-dimensionalcaseis shown in Fig. 7.11.In this two are degreesof freedom in that the system can move both horizontallyand In the same way that we approachedthe one-dimensionalsystem,the equilibrium mationsare the valuesof x, and x, that minimize the potentialenergy:
5€
The
12
^_l2
+ (L,,- x ) 2 * L")
+ o.skb(
Thu loca
If the parametersare ko: 9 N/cm, k, : 2 N/cm, Lo: 70 cnt Lo: l0 cm,F, = 2 Fz = 4 N, use MATLAB to solve for the displacementsand the potentialenergy.
7.I I FIGURE (o)unlooded A 1"wo-spring system: ond(bllocded.
PROBTEM
three iterations the root of Eq. t
Example7.1 along the formula =-x2*8x-12
the maximum this function analytir
thatEq. (7.10)yi guesses ofr, - 0, the following f
=3+6.r+5x2+3 6e minimum by findi Use bisection r
-1.5x6+2r4+1 fiurction. methods to
fu dl values of r.
PROBLEMS
continued Solution.
An M-file can be developedto hold the potentialenergyfunction:
p = P H ( x , k a , l . : b ,L a , L b , F ' I , l -2 ) function ^ . 1) - L a ' 2 ; P E a =0 . 5 * k a * ( s q r t ( x ( 1 ) ^ 2 + ( L a - r ( 2 ) ) ) ^ 2 ' 2 + ( s q r t ( x ( l , b + x ( . 1) PEb=0.5*kb* \'2)) ) Lb)'l; W - F l * x ( 1 ) + F 2* x ( 2 ) ; p - P F l a +P E b- W ; The solution can be obtained with the fminsearch
function:
k a = 9 ; k b - 2 ; L a = 1 0 ; t , l ; =i 0 ; F 1 , ,: , ;. E " t , - , ] I x , f ] - f r n i n s e a r . c h ( B P E , i - " 0 .! r , 0 . 5 l
4 .9\2 i
I .2'.0 l 9
f-
- 9 . (,42,: Thus, at equilibrium,the potentialenergyis -9.6422 N.cm. The connectingpoint is located4.9523 cm to the right and 1.2'759cm aboveits original position.
PROBtEMS (c) Dil-l-erentiatc' the lirnction and then Lrsea root-location methodIo solve lor the maxirlurn.l(.r)and the correspondingvalueul'r. 7.5 Solve for the value of -xthat rnuximizes.f(,r) in Prob.7.4 search.Employ initial guessesof using the golden-section 12 J ( r=) - r r + 8 r ,ti : 0 and-r,,: 2, and perfbrrnthree iterations. themaxinrurnand the colrespondingvalueol' 7.(r RepeatProb. 7.5, except use parabolicintcrpolation. Delermine ly ( i.e..usingdill-erentiation). Enrplo-v rforlhisfunctirxanalvtical initialguessesof ,r) : 0.,r, : I. andxr : 2. rnd perVedfy thatEq. (7.10)yields the sanreresultsbasedon lbrm threeiterations. of,t, : fl, .v. - 1. xp1l.v..: $. initialguesses 7.7 Enploy the l'o|l<'tringnlethodsto t'ind the nrarirnum olthelbllowinglunction: Consider I ( r ) : 4 x - l . u x 2* 1 . 2 - 1 3 0 . 3 - r a '(r) r = 3 { 6-r* -5-r + 3xr +,l-ra (a) Golden-section search(.rr: -2, -r, : 4, 8, : l(n.r. by findingthe root ol'the derivativeol' thenrinimunr (b) Parabolicinterpolation(,tr : 1.7-5,r'z : 2. ir : 2.-5, 2 Usebisectionwith initial [uessesof -r,: function. iterations: -5). r r =l ' 7.ll Considerthe fbllowing function: Given r 1 y=)- 1 . 5 - t n .f (xl : ro + 2-tr + 8x2 + 5r +2x" +12l threeiterationsof the Newton-Riiphsonmethod Perform valfherootof Eq. (E7.| .l). Usethe parameter E x a m p7l e. 1a l o n gw i t h a n i n i t i a lg u e s so f r : 3 s thefonnula Given
thefunction. Plot methodsto prove that the function is conUseanalytical forall valuesof -r. cave
Use analyticaland graphicalmethodsto show the lirnction has a minirnum fbr some value ol' x in the range *2 < x < l.
r84
OPTIMIZATION
7,9 Employ the fbllowing mcthodsto fincl the minitnurn of thc function fiorn Prob. 7.8: (a) Golden-section search(,r,: -2, -v,,: I, r. -- lc/c). (b) Parabolicinterpolation(,rr : -2, -y:: -1. .r., : l, iterations:5).
thcfbllowingtr.rnction: 7.1{)Considcr 3 .l (.r) : 2.r * -
P constant: 9.8 I rn/s:.attd,t'n- the initial height(rn).Usethe golclen-sectionsearch to determine the nraximum height givcn -t'u: I nt, u' : 25 rn/s,and0,,: 50". Iterateuntilthe approxirnate ertor falls below €,: lc/ousinginitialguesses of ,r,: g and-r,,: $Qp. 7.16 The dcflection of a unifbrm bearnsubjectto a linearly distributedloadcan be computedas increasing
0.4 I
( d - 1 )o . z
0
J
,= _5(-ys+2L)r'-Lrrl Perform IO itcrationsof parabolicinterpolationto locate I)()EIL the minimum.Commenton tlreconvclgcnceof yttur rcsults GiverrthatL: 600 crn,C = 50,000kN/cnr:,1: 30,000cm{, (.rr: 0.1. ,r: : 0.-5,-rr : 5) 7.ll Devclopan M-filc that is expresslydcsignedto locate ahd ur,,: 2.5 kN/cm, detcrtnincthe point of maximumde(b) usingthe golden-scction sealch scarch.[n othcr words, {lection(a) graphically', a rnaxirnumwith thc golden-scction sct if up so that it dircctly finds the traxitnum ratherthan until tlie approximatcerror falls below e, : l% with initial finding thc rninimunrof -/'(.r). Thc subroutineshouldhavc guesscsof -r,: 0 ancl.r,, L. 7.17 A ob.icctwith a ntassof 100 kg is prolectedupward thc tblloia,ingf'eaturcs: liom tlre surluccol'thc carth at a vclocity of 50 rn/s.[f the . Iteratcuntil thc rclativcclror talls bclow a sk)ppingcri- ob.icctis subjcctto lincaldrag(r' : l5 kg/s),usethegoldentcrionor cxcecdsa ntaxinruntnunrbclol'itcration.. scctionscarchto detcrrnincthc rnaxirnumheighttheobjeo . Rcturnboth thc optinral"t and./(.r). attains. curvedefinedby is a bcll-shapcd Test y()ur prograrnwith thc sarncproblctnas Exanrplc7.1. 7.Itl The nornralclistribution
FIGUREP7.22
lne,specificgrow,h .sls . onlibioiic versusthe fooo t
7.23 A compounclA will I tank reactor. The product B separatlon unit. Unreacted processengineer has fbund is a function of the convers will result in the lowest cos constant.
7.12 Dcvclop arr M-f ilc to krcittc l ntinimum with thc goldcn-scction search.Ralhcrthanusingthc Inaxitrtuuitcrations und L,q. (7.9) as thc stoppingcritcria,dctcnninc(hc
of scarchto clc(crttrirrc the location Usc thc goldcn-scction numbcrof itcralionsnccdcdt()attaina dcsircdtolcrance.'fcst the inflcctionpoint of this curvc firr positivc.r. y o u r f u n c t i o nb y s o l v i n gE x r n t p l c7 . 2 u s i n gL ' , , ,: 0 . ( X X ) 1 . 7 . 1 9 l J s c t h c f r l i n r ; r r , r I r : l rl u n c t i o n t o d c t c r m i n teh e 7.1-1t)cvelop an M-filc (o irnplcntcntparabolicintcrpoltl- rlinimrnl ot' tion (o krcatca nrininturn.Thc subroutincshouldhavc thc / ( - t .. r , ): 2 . v ) , 2 . 2 5 . r . l '- 1 . 7 - 5+, r .1 . 5 - r 2 lirllowingl-caturcs: the r t'lr ltlnction to detcrrnine 7.20 Usc tllc f nriri,i{r.r . andhavcthc progllut gcuBascit on tu,oinitialgucsscs, ofl.ttitxirtrunr cratcthc third initiirlvalucat thc rnidpoint()f thc intcrlal. . Check whethclthc gtrcsscsbrackctit tlaxitrturn.Il ntlt, / 1 , r . . r ' ): 4 - r * 2 . r ' - 1 - - 1 . 12 . r r + 2 - r r . ' - 3 r 2 but thc subroutincshouldnot inrplcntcntthc algorithrn, 7 . 2 1 C i v e nt l r cl i r l l o r v i n gl u n c t i o n : shouldrcturnirncrr()rnrcssagc. . Itcratcuntil thc rclativccrror lllls bckrrva stoppinScri./ (,r. .r'): -tl-r * -rl + l2r' + 4r'l 2,rr' tcriorior cxcccdsl rnaxintuntttutnhcltll'itcrationr. . Rcturnboth thc optinral,r and./(,r). Dctcnnine thc rninirnuttr(a) glaphically.(b) nurnetically theresult with (hc lrLj rr;r,.Li,:li l'unctitln.ancl(c) suhstittrte Tcst 1,pprpfogramrvith thc samcptoblctrtas Exanrplc7.3. (b) back into thc l'unutionio clcternrinctlic rninimum ol' arc takcn at ccrtain poiltts 7.14 PrcssurcmcasurL'nlcnts (.r, i'). bchind an airlbil ovcr tiuic. Thc data bcst l'its thc curvc I 7.22 Thc spccilic gt'owthratc of a vcast that producesan _r: 6 cos-r I.5 sin.t l'rom.r: 0 to 6 s. Llsclilur iterations (.: is a lirnctionol'thc lixld conccntr.ttiot'l antibiotic rncthodto l'indthc nrininrunrprcssurc. o1'thcgoldcn-search Sctr,:2and,r,,:'1. 2t with 7.15 Thc trajcctoryo1'aball can bc cot.nputcd 4+0.8r+(l+0.2(] r''*rir r : ( t i u l H { ) ) . -t - = j , 2t ,j crrs'11"
As dcpictedin Fig. P7.22,growth gocs to zc() at veryl0$ conccntrations clucto lcxrdlirnitation.It alsogoesto zelo8l due to toxicitycitects.Findthevalueof whcre ,r': the height(m). d,, : thc initial angle(radiansl, high conccntri.itions u. : the initial vclocity (rn/s), .g : thc gravitational r u ( r l r i eh g r o ut h i s u r t r : r x i n r u m .
0
cosr:. f(,,___
7.24 A finite-element mode loading and monrenrsrF.ig f(x,y)--5.r2-5xy*2
where-r : end displacernen v a l u e so f r . a n d 1 . t h a tn t i n u n 7.25 The Streeter-phelp.t mo, dissolvedoxygen .on."ntrrn, chargeof sewage (Fig. p7.25
o: o.r- ;*!L p k a * k . - k , ,. S/, -;(t
- e - ^ , ,, ,
I
r---r__'
I FIGUREP7.24 A contilever beom
r85
PROBLEMS
rt )S
0,4 8 ld_1)0.2
(mg/L)
ly
or'
0
0510 c (mg/L)
lc.
14,
ch ial rrd .he :n-
by
rof the
IIGURE P7.22
FIGURE P7.25
Thespecific growthrcle of o yeosf thot producescn onlibiol c vefsus the food concentrotion.
A dissolvedcxygen "sog" below o point dischorgeof sewoge Inroo rver.
whcre o : dissolvedoxygen conccnlration(mg/L). o, : (mg/L),r : lraveltirne(d), oxygensaturatiorl conccntrrti()n L,, : biochemicaloxygendenrand(BOD) concentralion irt thc rnixing point (rng/L), kz : rate ol'decompositionol' d I ) , f r ,:, r e a e r a t i o n B O D ( d r ) .t , : r a t e o l ' s e t t l i n g o l ' B O( D r). ratL'(d arrd5,,: sediurent oxygendemand(mg/L/d). As indicatcdin Fig. P7.2-5,F.r1.(P7.25)producesln levela,, srrme oxygen"sag" thal reachesa criticalr.nininrunr l, I r"" r |r""l traveltirnet, bclow thc point discharge. This point is called ' 6 ( ' l Cost=clt-l I \.\1,r "cri(icirl"hecauseit rcpresents the lrrcationwherebiotuthat L\{l-.rl)-/ I (likc I'ish)would be thc most stressed. depend on oxygcn rnodelof a cantileverbearnsubjectto ?J4A finite-element r:ivcn (Fig. P7.24)is civen by optinrizing Deternrinethc critical tlavcl tirrre'and concentriltiun, hoding andmonrents t h c l b l l o w i n gv a l u e s : rl = .r.f - ).r.\'+ i..).\'- .\' - | ..\.\' J{.r. ( , ,: 0 . 6 t l I r , ,:, l ( ) u r g / L k,r:0. t d |
7J3A compound A will be converted into B in a stirred hnkreactor. TheproductB and unreactedA ure-purified in a loparation unit. UnreactedA is recycled to thc reactor.A pocess engineer hasfoundthat the initial costol'thc syslcm ilafunction ol'theconversionr.,. Find the conversionthat rillresult in thelowestc()stsystenr.C is a proportionality on$ant.
r = enddisplacenrent and .y : end mol.nent.Find the of.randl thatrninimizc11.r,r'). TheStreeter-Phclps mtxlel can bc usedto cornputethe oxygen concentration in a river helowa pnin(clis(Fig.P7.25), ofsewage :ally )sult num
o = 0' ', - - J t t ! ( c t ' , t- d - ( i , i + t ' ) / ) li,r*k,-ft,,' Sr -p(l-e-^"t
t.:0.05d
I
L,,:-50rng/L
5r,: I rng/L/d
distributionol' ptrllutantconcen7.26 The two-dirnensional trationin u channclcan be describedby r ' ( r . r ) : 7 . 9 t O . l 3 . r* 0 . 2 1 . t ' -0 . 0 5 , r : - 0 . 0 1 6 r ' r- 0 . ( X ) 7 - r I
ej.25\
Determinetheexacrt Iocationol'thepeakconcentration given t h e l u n c t i o na n dt h e k r r o w l e c l g c ' l hl haet p e u kl i e sr v i t h i nt h e -
,(,\.i
{1(,.,r./..
ttt,tl'rltt
-
I
---rr.
7.27 A total chiirgeQ is unitbrnrlydistributedarounda t'ing-
:s an
sh',rned cr-rnductrq rr\th rrli\rl. r \ thrr.:c r1 i- \r':q11-{ .-t 1 o\:\mc€ -\ lr()m tnt center ()1Lr\er\ng tFrg. P1 .21 t.The \orce exerted on thc charge by the ring is given by
)' low ro at le of
RGURE P7.24 Aconli everbeom
|
'I :
-
-
', rt xO' r
4ten 1t2 1 47131t rrCr/(Nmr;,r7: l0 iC' w h e r e a , , : 8 . t 1x5 l 0 Q:2x and d : 0.9 m. Deterrninethe distancex whet'ethe force is a mlxir.t'turtr.
r86
PR
OPTIMIZATION
ro
r_
D 20,000
(r
10,000 0
FIGURE P7.27 motoris a flnc7.2t1The torquctransrlittedto an indr"rction tion of the slip betwccnthe rotationof the statorfield and thc rotrrrspced.r,whele slip is detincdas tl
FIGUREP7.31 Twofrictionless mosses cor lineorelostic springs.
FIGUREP7.29 Plotof drcg versus velocity for cn oirfoil
tlR
n
The combination of the two f'actorsleads to a minimum clrag ( a ) lf o :0.6 and W : 16.000,dcterrninethe minimun drag anclthc velocity at which it occurs. ( b ) In adclition,devclopa sensitivityanllysisto determim d how this optirrum varics in responseto a range l5s(l- s) ,' w: 12,000to 20,(xx)with o : 0.6. ( I - . r ) ( 4 , s r- 3 s * z l ) 7.-10 Rollcr bcaringsarc sub.iectto fatiguetailure FigureP7.28showstlris function.Usc u numcricalmcthod lirrgc contrct loacls/i tFig. P7.-10).Thc problenrof to dctcrminethc slip at which tlrc rlaximum torqueoccurs. thc location ol' the maxirnum stressakrngthc -r axiscan 7.29 Thc total drag on an airtiril can bc cstirnatcdby shownto bc cquivulcnt(o maxirnizingthefunction:
whcre rr : rcvolutionspcr seconclof rotatingstatorspccd and ri,,: rotor spced.KirchhotT'slaws can bc usedto show fbrm) artd slip that the torquc (cxpressedin dirncnsionlcss arc rclatcdby
D : 1 1 . 1 1 1 o Y ' +( # ) ' :T F-r'iction
Liit
0..1
.',4+'F
Torclueironsmifted to on inductoros o functionof slip
__/
rr'Ir
Find thc .r that rraxirnizcs/(.r').
whcrc D : drag,o : ratio of air densitybctweenthc flight As scen altitudcandscalevcl, lV: wcight.and V: r,clocity. in Fig. P7.29,tlrc two l'actorscontributingto drag alc i'rflected difl'ercntlyas velocity increases.Whcreasfrietion drag increascswith velocity,thc drag due to lifi dccreascs.
FIGURE P7.28
-/t
FIGURE P7.3O Roller beorings.
\
04
\
---_]+.r l+r-l
7.31 In a similar fashionr Sec. 7.4, develop the poten tem depicted in Fig. P7.3L plots in MATLAB. Minimiz to determine the equilibrium ltc forcing function F: l0( Jd t6 - l5 N/m. 732 As an agricultural engi open channel to carr) ine the optimal dime perimeterfor a cross-section dimensionsuniversal'l 733 Use the funcrion rmins ofthe shortest ladder that rea ftnce to the building's wall (l where ?34 The length of rhe long 6e corner depicted in Fig.
P7.32
r87
PROBLEMS
P7.3| hictionless mosses fo o woll by o poirof connected e - -n"s,f 'i -. r < . . i n,^Jq-
FIGUREP7.33 um ine of
by ing be
ln a similarlashionto the case study describedin ?.4,derelop the poterrtialenergyf'unctionfol the sysin Fig. P7..il. Developcontour and surflce depicted tnMATLAB.Minimizethc potcntialcnergyl'unction letermine theequilibriurndisplacenrents .r, and -r. given F : 100N und the paral)rcters forcing tunction k,,:20 tr,= l5 N/rn. Asanagricultural engineer.you rnustdesigna trapeopen channel to carry irrigationwater (Fig. P7.32) netheoptinrilldinrensionsto rninimizethe wettecl fbrr cross-sectit'rnal areaof 50 rn'. Are the relativc l universal Usethelunctiontm i ri:;e:a r ch to deternrine the length shortest ladderthat reachesliorrr the ground over the tolhebuilding's wall (Fig. P7.33).Testit lirr the case L _ .t -
A l o d d e rl e o n i n go g o i n s o l f e n c eo n d i u s tt o u c h i n go w o l l
computing the value of e that minimizes the fbllowing function:
L(o): Ll- + -. sfnfi
"''
stn(ft-a-H)
For the casewhere ur1 : r.lr2: 2 m, usea numericalrnethod describedin this chapter (including MATLAB's built-in capabilities) to developa plot ol'L versusa rangeola's from 4 - 5t o 1 3 5 " .
,t ^-
Thelengthof the longestladder that can negotiate comer depictedin Fig. P7.,1,1 can be determinedby
P7.32
_-."', '..
FIGURE P7.34 A ioddernegolioting formed o corner bytrrohollwoys
:l!:li;
fi
,i
Lineor Syrtems ii
3. I
ovERVtEW Whof Are Lineqr Algebroic Equotions? In Part Two, we O"rrrrn::;.1|,Te value xrhar satisfied a single equation,f r) : 0. Now, we deal with the caseof determining the values* ri;*r, . . . , x, thatsimultaneously satisfy a set of equations: f,lx,.xr-....x,):0
fik,y,
,i:o
: xr,. fJ.xp
:
.,,
,, ',,''.'
,xr) : 0 Such systemsare either linear or nonlinear.In part ,*:..*. deal with lincar algebraic equarions that are of the generalform a | x t * a n x z + . . . + a l o x n: [ , aztxt * azzxz* .. . * a2,,x,,: fi,
(PT3.t)
-: a 1 , 1 x 1* a n Z X ZI
...*
ar,,x, :
12,
wherethe rz'sareconstantcoefficients. the b.s arecon_ stanrs.. rhe * unknowns.and n is ,f,. nrri., of 1.: equatrons. All otheralgebraicequationsarenonlinear.
linear Algebroic Equotionsin Engineeringond Science fundamenlat equarions of engineering
::iy^,:l T are based ano. scrence on conservationlaws. Sometamllar quantitiesthat conformto such laws are mass, energy..anO momenturn. In mathematical terms,these teadto balanceor continuiryequarions rhar llT'p,": .relare systembehavioras represented by ifre levels or
r89
t90
PART3 LINEARSYSTEMS
Feed
@t
FIGURE PT3.I Two lpes of syslemsthot ccn be modeledusinglineorolgebrclcequotions.(ol lumpedvorioble systemthot involvescoupled linitecomponentsond (b) distributedvcrioble systemlhot involves o confinuum.
responseof the quantitybeing modeledto the propertiesor characteristics of the system alrdthe externalstimuli or fbrcingfunctionsactingon the system. As an example,the principleof massconservation can be usedto formulatea model for a seriesof chemicalreactors(Fig.PT3.l a). For this case,the quantitybeingrnodeled is the massof the chemicalin eachreactor.The systempropertiesare the reactioncharacteristics of the chernicaland the reactors'sizesand f-lowrates.The forcing lunctions arethe feed ratesof the chemicalinto the system. When we studieclrootsof equations,you saw how single-component systemsresultin a single equationthat can be solved using root-locationtechniques.Multicolnponentsystems resultin a coupledset of mathematicalequationsthat r.nustbe solveclsinrultaneously. The equationsarecoupledbecausethe individualpartsofthe systerrale influencedby other parts.For example,in Fig. PT3.lc, reactor4 receiveschemicalinputsfiom reactors2 and3, Consequently,its responseis dependenton the quantityofchernical in theseotherreactors. When thesedependenciesare expressedrnathematically,the resLrltingequationsare often of the linear algebraicfolm of Eq. (PT3.l ). The,t's areusuallyrneasures of the magnitudesof the responses of the individualcomponents. Using Fig. PT3.ln as an example, "r1 might quantifythe amountof chernicalrnassin thefirst reactor,.r2 rnightquantifytheamount in the second,and so lbrth. The a's typically representthe propertiesandcharacteristics that bearon the interactionsbetweencomponents.For instance,thea's for Fig. PT3.la mightbe reflectiveof the flow ratesof massbetweenthereactors.Finally.theb's usuallyrepresent the forcing functionsactingon the system,suchasthe feedrate. Multicomponentproblemsof thesetypes arisefiom both lurnped(macro-)or distributed (nricro-) variable mathematicalmodels.Lrrmpedvoriable probLemsinvolve coupled
3.2 PARTORGANIZATION
t9r
finite components.Examples include trusses,reactors,and electric circuits. The three bungeejumpersat the beginningof Chap.8 area lumpedsystem. Conversely,distributed variable problems attempt to describethe spatial detail of systernson a continuousor semicontinuous basis.The distributionof chemicalsalongthe lengthof an elonqated,rectangularreactor(Fig. PT3.lb) is an exantpleof a continuous variablemodel. Diffbrential equationsderivedfiom conservationlaws specify the distribution of the dependentvariable fbr such systems.These differentiai equationscan be solved numericallyby convertingthem to an equivalentsystemof simultaneousalgebraic equations. The solutionof suchsetsof equationsleplesentsa major applicationareatbr the methcldsin the fbllowing chapters.TheseeqLrations arecoupledbecausethe variablesat one location are dependenton the variablesin acljoiningregions.For example,the concentrationat the middle of the reactor in Fig. PT3.11,is a function of the concentrationin adjoining regions.Similar examplescould be developedfor the spatialdistributionof temperature, m o n r e n l u mo.r e l e c t r i c i t l . Aside fiom phvsicalsystems.simultiineous linearal-uebraic equationsalso arisein a varietyof mathernatical problemcontexts.Theseresultwhen mathematical functionsare requiredto satisfyseveralconditionssirnultaneously. Eachconditionresultsin an equation that containsknown coefflcientsand uuknownvariables.The techniques discussed in this part can be used to solve fbr the unknownswhen the equationsare linear and algebraic. Somervidely usednunrericaltechniquesthat employ simultaneous equationsare legression analysisand splineintel'polation.
3.2
PARTORGANIZATION Due to its importancein fbrmulating and solving linear alsebraicequations.Chap. 8 provides a brief overviewof tnotrir algebru.Aside from coveling the rudimentsof matrix representationanclrnanipulation,the cl.rapteralso describeshow man'icesare handledin MATLAB. Chttpter 9 is devotedto the most fundamentaltechniquefor solving linear algebraic systems:GausseLimittolion.Before launchinginto a detaileddiscussionof this technique, a pleliminary sectiondealswith simple methodsfor solving small systerns.These approachesarepresented oneoftlre rrrethodsto provideyou witlr visualinsightandbecause the eliminationof unknowns-represents the basislbr Gausselimination. After this prelirr.rinary material,"naive" Gausseliminationis discussed.We starl with this "stripped-down"versionbecauseit allows the fundarnentaltechniqueto be elaborated on without complicatingdetails.Then,in subsequent sections,we discusspotentialproblems of the naive appr-oirch and presenta number of mocliflcationsto mjnirlize and circunlventtheseproblenrs.The fbcus of this discussionwill be the processof switching rows,orpdrlial pit'oting.The chapterendswith a brief descriptionof eflicientmethodsfbr solving tridiagonal ntatrrces. Chapter10 illustrateshow Gausseliminationcanbe lbrmulatedas an LI) factoriZation. Such solutiontechniquesare valuable1br caseswhere many right-hand-sidevectorsneed to be evaluated.The chaoterends witlr a brief outline of how MATLAB solveslinear s\/stet)-ls.
r92
PART3 LINEARSYSTEMS Chapter -1-1 startswith a descriptionof how LU factorizationcan be employedto efficiently calculatethe mati.r inverse,which has tremendousutility in analyzingstimulusresponserelationshipsof physicalsystems.The remainderof the chapteris devotedto the importantconceptof matrix condition.The conditionnumberis introducedas a measureof the roundoff errorsthat can result when solving ill-conditionedmatrices. Chapter I2 dealswith iterative solutiontechniques.which are similar in spirit to the approximatemethodsfor rootsof equationsdiscussedin Chap.6. That is. they involve guessing a solutionand then iteratingto obtaina refinedestimate.The emphasisis on the GaussSeidel method, although a description is provided of an alternative approach, the Jacobi method.The chapterendswith a brief descriptionof how nonlinearsimultaneousequations canbe solved.
nrl effiuluso the [e of o the UESS-
,.I. t l"
LineorAlgebroicEqrotions ond Motrices
AUSS-
Tcobi tions
CHAPTEROBJECTIVES The primary objectiveof this chapteris to acquaintyou with linear algebraicequations and their relationshipto matricesand matrix algebra.Specificobjectivesand topics coveredare e . ' . .
U n d e r s t a n d i nmga t r i xn o t a t i o n . Being able to identify the following typesof matrices:identity,diagonal, symmetric,triangular,and tridiagonal. Knowing how to perlbrm matrix multiplication and being able to assesswhen it is feasible. Knowing how to representa systemof linear algebraicequationsin matrix form. Knowing how to solve linear algebraicequationswith left division and matrix inversionin MNLAII.
\OU'\E
GO\ A.PROBLEN\
upposerhat three jumpers are connectedby bungeecords.Figure 8.ln shows them It U.ing held in placeverticerllyso that eachcord is fully extendedbut unstretched.We \ crn define threedistances..t'1.,t'2.and .\. as measureddovnv'ard from eachof their 9/ grar it1 takeshold and thejumpers s ill evenunstretchedpo:itions. ,.l.tierrhel are reJeased. positionsshownin Fig. 8.1b. tually cometo the equilibriun-r Supposethat you are askedto computethe displacernentof eachof thejumpers.If we assumethat eachcord behavesas a linear spring and follows Hooke's law, free-bodydiagramscan be developedfor eachjumper as depictedin Fig. 8.2. forcebalancecanbe writtenforeachjumper: Using Newton'ssecondlaw, a steady-state m 1 gI k z ( x z- 1 1 )- f t 1 r 1: Q t 1 t l g* k r ( , r : - x ) - k z ( x z- . r r ) : 0 tntS - k: (.r: - irz) : 0
r94
LINEAR ATGEBRAIC EQUATIONS AND MATRICES
ffi
",:Ol?r
- nu 1
ffi
Iri
.:
n
ffi
z'^r
- ni l )
li:
?r
ffi
il
t:tl
ffi
lr) _H
Arxr
I'l
I
n
tr
?r
t2(r, - 11)
I
i
ffi
tl
lr) 11
tl
tr (a) Unstretched
tt
( b ) Stretched
nrrg k.(x2- x)
FIGURE 8.I Threeindividuos conneciedbv bunqeecords
ill
m tg t.(.r1 - "r2)
m$
FIGURE 8.2 Free-body diogroms.
wherenr; : the massof jumper r (kg), A; : the springconstantfor cordT(N/m),-ri : the displacernentof jumper I measureddownwardliom the equiiibrium position (m), andg = gravitationalacceleration 19.81nr/sr).Collectingtermsgives ( k 1l k ) x 1
-kzxz
:mtg
-ftz.tr * (kz I kz)xz- k:-rr : mzg
(8.1)
-ktxzlk3xl-7n19 Thus. the problem reducesto solving a systelnof three simultaneousequationsfor the threeunknown displacements.Becausewe have useda linear law for the cords,these equationsare linear algebraicequations.Chapters8 through l2 will introduceyou to how MATLAB is used to solve such systemsof equations.
8.I
MATRIXALGEBRAOVERVIEW Knowledgeof matricesis essentialfor understandingthe solutionof linear algebraicequations. The tbllowing sectionsoutline how matricesprovide a conciseway to representand maninulale l i n e r ra l q e h r a i c
II
MATRIX ALGEBRA OVERVIEW
ll
iiil
C o l u m n3
I
I
I at3
a:t
at2
azt
azz .t..:&it:'',
azu
onl
o.)
a,r,'
Qt,
I
-Row2
-
t^t
a,n3
FIGURE 8.3 A moirix.
8.1.1 Mqfrix
Nototion
A matrix consistsof a rectangulararray of elementsrepresentedby a single symbol. As depictedin Fig. 8.3, [A] is the shorthandnotationfor the matrix andai 1 designatesan individual element of the matrix. A horizontalset of elementsis called a row and a vertical set is called a column.The first subscriptI alwaysdesignatesthe numberof the row in which the elementlies.The second subscriptTdesignatesthe column.For example,elementa23is in row 2 and column 3. The matrix in Fig. 8.3 has m rows and n columns and is said to have a dimensionof m by n (or m x n). It is referredto as an m by n matrix. Matrices with row dimensionm : l. suchas [b]:
[bl
b2
brl
are called row yectors. Note that for simplicity, the first subscriptof each element is dropped.Also, it shouldbe mentionedthat thereare times when it is desirableto employ a specialshorthandnotation to distinguisha row matrix from other types of matrices.One way to accomplishthis is to employ specialopen-toppedbrackets,as in fbl .l Matrices with column dimensionn : l. suchas
lnl j tL
-
L:
(8.2)
are ref'erredto as column vectors. For simplicity, the secondsubscriptis dropped.As with the row vector,thereare occasionswhen it is desirableto employ a specialshorthandnotation to distinguisha column matrix from other typesof matrices.One way to accomplishthis is to employ specialbrackets,as in {c}. I In addition to specialbrackets,we will use caseto distinguishbetweenvectors(lowercase)and matrices (uppercase).
r96
LINEARALGEBRAIC EAUATIONSAND MATRICES Matrices where rn : tttl
l-r,,
n ate called .square matrices. For example. a 3 x 3 matrix is atrl
I A l : I a r , a ) - ) . r r ,I
L,;,";: ,;,1
The diagonalconsistingof the elementsas, a22,and a.j3is termed theprincipal or maitt diagonal of the matrix. Squarematricesare particularly important when solving setsof simultaneouslinear equations.For such systems,the number of equations(correspondingto rows) and the numberof unknowns(correspondingto columns)must be equalfor a uniquesolutionto be possible'Consequently,squarematricesof coefTicients are encounteredwhen dealinewith suchsystems There are a number of specialforms of squarematricesthat are important and should be noted: A symmetricmatrix is one wherethe rows equalthe columns-that is, a; : aii for all i i's and.l's.For example,
[st2f tAl:ll3 7l L2 7 8_l i s a 3 x 3 s y n r m e t r inc r a t r i x . A diagonal nmtrix is a square matrix where all elements off the main diagonal are equal to zero, as in
fo,, I lAl=l a-) ) | orr) L Note that wherelarge blocks of elementsate zero.they are left blank. An identi4' matrix is a diagonalmatrix where all elementson the main diagonalare equalto 1, as in
[rl
tAl:l
I
I
LlJ
The symbol Ul is usedto denotethe identity matrix. The identity matrix haspropertiessimilar to unity. That is,
[A]Ul: [/][A]: tAl An upper triangular malrx is one whereall the elementsbelow the main diagonalare zero. as in
[,4]: ['''
rl12
o,rl
422
ott
I
ctstJ
8.I MATRIXALGEBRA OVERVIEW
197
A lower triangular malrrx is one whereall elementsabovethe main diagonalarezero, as ln I
f o,, tAl
:
|
i
i
o11
1,,;,,; ,,,] "',
I
i
A bandedntotix has all elernentsequalto zero,with the exceptionof a band centered on the main diagonal:
ln l:
lI
f al
at)
I
a:z
dt.r
,,,
Or, I aat aqc)
t .a Z t I |
L
(/tl
Now that we have specifiedwhat we meanby a matrix, we can definesomeoperatingrules that govern its use.Two nr by n natrices are equalif, and only if, every elementin the first is equal to every elementin the second-that is, [A] : [B] if ai1 : bi1 for all i andj. Addition of two matrices,say, [Al and [B|, is accomplishedby addingcorresponding terms in eachmatrix. The elementsof the resultingmatrix [C] are computedas c'i1: ct,i1b,i f o r i : 1 , 2 , . . . , m a n dj : 1 , 2 , . . . , n . S i m i l a r l yt,h e s u b t r a c t i oonf t w o m a t r i c e ss.a y . [E] ninus [F], is obtainedby subtractingcorrespondingterms,as in -fij
forl: l,2,...,mand j:l,2....,n.ltfollowsdirectlyfromtheprecedingdefinitions that addition and subtractioncan be performedonly betweenmatriceshaving the same dimensions. Both addition and subtractionare commutative;
t A l + t B l: [ B ] +[ A ] and associative:
( t A l +t B l )+ [ c ] : [ A l + ( t B ]+ t c l ) The multiplication of a matrix [A] by a scalarg is obtainedby multiplying every element of [Alby S. For example,for a 3 x 3 matrix: fPo,,
I on.,
Rart Pat,,f
i o , , i o r ,I efltt
I
I I j
II
8.1.2 Motrix Operoting Rules
I D I: s l A l: l i o l
I i
I
The precedingmatrix has a bandwidthof 3 and is given a specialname-the tridiagonal matri.r.
dij:eij
I
,n,,
I
I
I i I
i I
r98
LINEARALGEBRAIC EAUATIONSAND MATRICES
q-l
[A'l^"n
F,'I
t,
rl ->
[::]Ll
[Blnt
=
[C],"t
;) Interiordimensions are equal. multiplication is possible
[sxs-1x7-22
II Exteriordimensionsdefine t h e d i m e n s i o n so f t h e r e s u l t
FIGURE 8.4 Visuol depiction of howtherowsondcolumns lineupin
FIGURE 8.5 Motrixmultlplicotion conbe per{ormed onlyif
m o t r i xm u l t i p l i c o i i o n .
l h e i n n e rd i m e n s i o nosr e e q u o l .
The productof two matricesis represented as [C] : tAltBl, wherethe elementsof [C] are dellned as { 1i :
+
) .oislt11 t:l
{8.3)
where n : the column dimensionof [A] and the row dimensionof [B]. That is, the c;, elenent is obtainedby addingthe productof individual elenents from the lth row of thefir$ matrix, in this case[A], by theTthcolumn of the secondmatrix [B]. Figure 8.4 depictshow the rows and columnsline up in matrix multiplication. According to this definition, matrix multiplication can be perfbrmedonly if the first matrix has as many columnsas the nunber of rows in the secondmatrix. Thus, if [A] is an m by n matrix, [B] could be an n by / matrix. For this case,the resulting[C] matrix would have the dimensionof m by /. However, if [B] were an m by i matrix, the multiplication could not be performed.Figure 8.5 provides an easy way to check whethertwo matrices can be multiplied. If the dimensionsof the matricesare suitable.matrix rnultiolicationis associative:
(tA JtB[c] l ) : tA l (tB l tcl ) 'aLnd distributive:
t A l ( t B l +[ c ] ) : [ A l [ B ] +t A l t c l or ( [ A ] + t B l ) l c l : [ A ] [ c ]+ l B l l c l However,multiplicationis not generallycommutative:
tAttBl+ lBltAl Thot ic rh,' order nf mrrltinlic:rtion is
B.'I MATRIXALGEBRA OVERVIEW Althoughmultiplicationis possible,matrix divisionis not a definedoperation.However.if a matrix [A] is squareand nonsingular.thereis anothermatrix [Al r, cailed the inver.seof [Aj, fbr which tAltAl
' : 1 J e ]: 1 1 1 [A]
Thus,the multiplicationof a matrixby theinverseis analogous to division,in the sensethat a numberdividedby itself is equalto l. That is, nultiplicationof a matrix by its inverse l e a d st o t h e i d e n t i t ym a t r i x . The inverseof a 2 x 2 matrix can be representedsimply by
lAl
': ail422 -
anq2l
I
o" -n'tl
L-at
itrrJ
Similarfbrmulastbr higher-dimensional matricesaremuchmoreinvolved.ChapterI I will deal rvith techniqueslbr using numericalmethodsand the computerto calculatethe inverse lbr suchsystems. Two othermatrix manipulationsthirtwill have Lrtilityin our discussionare the transposeand the augmentationof a matrix. The trartslto,se of a matrix involvestransformingits rows into columnsand its columnsinto rows. For examole.for the 3 x 3 ntatrix: 0t)
f u,,
a,,l
l A l : l , I . ; ( t ) - , , i ,I Lr;; 't2 r,.,J t h e t n r n s p o s ed .e s i g n a t eld, e 1 ri.s d e l i n e da s (t,t a.,l
l - , , ,'
t A l r: I , , ; ' u . , u , .I
L,. ,;. ",;l In otherwords, element of
the rr;; the transpose is equalto the zu 7; elementof the original matl'lx. The transposehas a variety of functionsin matrix algebra.One simple advantageis that it allows a column vector to be written as a row and vice versa.For example.if J , -r l -r.
['' l
, -. t l.r I
[., J
tnen {c}r :
lc1 c2 c:l
In addition,the transposehas numerousmathematicalapplications. The final matrixmanipulationthat will haveutility in our discussionis uuqmenratiott. A tnartrixis augrnentedby the additionof a column (or colunurs)to the original matrix. For example,supposewe have a 3 x 3 matrix of coefficients.We might wish to augmentthis matrix [A] with a 3 x 3 identity matrix to yield a 3-by-6-dimensionalmatrix: [,t,,
ott
I LaSt
a1t
d1
l ool
a2t a3t
021 {41
0l0l 0 0 r_j
200
LINEARALGEBRAIC EAUATIONSAND MATRICES onthe Such an expressionhas utility when we must perform a set of identicaloperations matrix rows of two matrices.Thus. we can perlbrm the operationson the singleaugmented ratherthan on the two individual matrices.
EXAMPTE 8.1
M A T L A BM o t r i x M o n i o u l o t i o n s Problem Stotement. The following exampleillustrateshow a variety of matrixmanipulationsareimplementedwith MATLAB. It is bestapproachedas a hands-onexercise onthe computer. S o l u t i o n . C r e a t e a 3x 3 m a t r i x : >>
=
A
[1
5
b;7
4
2;-_t
61]
6 2
,-4 -3
6
The transposeof [A] can be obtainedusing the ' operator: >>
A',
-i
1 5 6
-3
.i6 27
Next we will createanother3 x 3 matrix on a row basis.First createthreerow vectors: >> x _ 1B 6 gJ; >> Y = t 5 8 1l; >> z -- t4 B 2l;
Then we can combinetheseto tbrn.rthe matrix >>
B =
lx;
y;
z)
p
-5
B
4B
9 1 2
We can add [A] and [Bl together: >>C=A+B
c= 9 212 1
1l 14
l5 3 9
MATRIXALGEBRA OVERVIEW Further,we can subtract[B] from [C] to anive back at [A]: >>C=C-B
1 'l
5 4
6 2
-3
6
7
Becausetheir inner dimensionsare equal,[A] and [B] can be multiplied >> A*B
7 44 -26
94 9A 86
26 7r 1
Note that [A] and [B] can also be multiplied on an element-by-element basis by including a period with the multiplication operator as in >> A. *B
B -35
30 32 48
12
54 2 14
A2 x 3 matrix can be set up >> D =
t1 4 3;5
B 1l;
If [A] is multiplied times [D], an error message will occur >> A*D .'?! LTta' usrnq Inner matrix dimensions
must agree.
However, if we reverse the order of multiplication so that the inner dimensions match, matrix multiplication works >> D*A
ro
20 58
53
35 53
The matrix inverse can be computed with the inv function: >>
l\1
lT
-
=
fnv(Al
4.2462 -a.8462 0.8308
0.01s4 0.3846 4.3231
-0.2154 0.6L54 -0.4769
202
LINEARALGEBRAIC EAUATIONSAND MATRICES To testthat this is the conect result,the inversecan be multiplied by the originalmatrixto give the identityrratrix: >>
AXAI -
a rl:;
-0.0(r00 1.0000 -0.0000
1.0000 0.0i100 0.0000
0.0000 0.rr000 1.0000
The eye functioncan be usedto generate an identitymatrix: 22
f
= e)re(3)
I=
0 1
0 0t)
Finally,matricescan be augmentedsirnplyas in >> Aug
=
tA
Il
Lrrn
t4:
010
361
Note that the climensionsof a matrix can be determined bv tlre size firnction:
3
8.I .3 RepresenlingLineor Algebroic Equotionsin Motrix Form It shoulclbe clear that r.natricespror,'idea concisenotation tbr representingsimultaneous linearequations. For example,a 3 x 3 setof linearequations, d r t r r * a 1 2 . t 2- f 0 t 1 . V :
bl
a 2 2 \ 2 1 o 2 1 . r 3:
b.
* a 3 2 . Y 2I { 1 3 1 1 3:
fi,
t t 2 1 x 1* 4:rrr
(8.4)
can be expressecl as [A]{r} : {b}
(8.5)
EQUATIONS WITHMATLAB ALGEBRAIC 8.2 SOLVINGLINEAR
where [A] is the matrix of coefficients:
otl ,,,-l l oo.t , t a:2 .,:: I I a): arrJ
--
tdl
ttrl
Lrl-rr
{D} is the columnvectorof constants: lblr :
lb1 bz
bt)
and {,,r}is the columnvectorof unknowns: {r}r :
[.r1 x2 r:]
Recall the definition of matrix rnultiplicationtEq. (8.3)l to convinceyourselfthat Also, realizethat Eq. (8.5)is a valid matrixmultiplicaEqs.(8.4)and (8.5)areequivalent. tion becausethe numberof columnsn of the first matrix [A] is equalto the numberof rows n of the secondmatrix {-r}. This part of the book is devotedto solvingEq. (8.5)lbr {x}. A formal way to obtaina solution using matrix algebrais to multiply eachside of the equationby the inverseof [A] to yield
tAl
r t A l { r }: t A l 1 i . b }
Because[A] '[A] equalsthe identitymatrix,the equationbecomes
tr) : tAl
r{b}
(8.6)
Therefbre,theequationhasbeensolvedfbr {r}. This is anotherexampleof how theinverse playsa role in matrix algebrathatis similarto division.lt shouldbe notedthatthis is not a very efficient way to solve a systemof equations.Thus, other approachesare employedin numericalalgorithms.However,as discussedin Section11.1.2,the matrix inverseitself hasgreatvaluein the engineeringanalyses of suchsystems. It shouldbe notedthatsystemswith moreequations(rows)thanunknowns(columns), regressionwhere A typical exampleis least-squares m > n, are said to be overdetermined. systemswith less an equationwith n coefficientsis fitto m datapoints(x,.y).Conversely, A typical example of equationsthan unknowns,,??< n, are said to be nndertletermined. underdetermined svstemsis numericaloDtimization.
EQUATIONSWITH MATTAB 8.2 SOTVINGTINEARATGEBRAIC MATLAB providestwo directwaysto solvesystemsof linearalgebraicequations. The mosl efficientway is to employthe backslash, or "left-division,"operatorasin >>r=A\b
T h e s e c o n di s t o u s em a t r i xi n r e r s i o n : >>
x
=
inv(,q)*b
204
LINEARALGEBRAIC EQUATIONS AND MATRICES As statedat the end of Section8.1.3,the matrix inversesolutionis lessefficient thanusing the backslash.Both optionsare illustratedin the following example.
E X A M P L E8 . 2
S o l v i n gt h e B u n g e eJ u m p e rP r o b l e mw i t h M A T L A B ProblemStotement. UseMATLAB to solvethebungeejumperproblemdescribedatthe beginningof this chapter.The parametersfor the problem are
Moss (kg)
Jumper
Spring Constont (N/m)
6A 7A BO
Top(1) M i d d i e( 2 ) Bottom(3)
Unstretched Cord Length (m)
50 100 50
2A 2A 2A
Solution. Substitutingtheseparametervaluesinto Eq. (8.1)gives
-i:B-,8] : [-l;: {;t}{::::;l I L o -so so-l[', J Izs+.s Start up MATLAB and enterthe coefficientmatrix and the righrhand-sidevector: >> K = t150
100 0; 100 150
50;0
50 501
v-
150 100 rJ >> mq -
-100 150 -50 [588.6;
0 50 50 586.7;
184.81
588.6000 586.7000 784.8000 Employing left division yields >> x -
K\m9
4r.2020 55.9170 71.5130 Alternatively, multiplying the inverse of the coeflicient matrix by the right-hand-sidevec. t o r g i v e s t h e s a m er e s u l t :
8.3 CASESTUDY =
>> x
inv(K)*mg
4r.2A2A 55.9170 7I.613A Becausethe jumpers were connectedby 20-m cords,their initial positionsrelative to the platform is >> xl
= t20;40;501;
Thus, their final positionscan be calculatedas >> xf
= x+xi
6I.2A2a 95.9rtA 131.6130 The results, which are displayed in Fig. 8.6, make sense. The first cord is extendedthe longest becauseit has a lower spring constant and is subject to the most weight (all three jumpers). Notice that the secondand third cords are extended aboutthe sameamount.Becauseit is subjectto the weight of two jumpers,one might expectthe secondcord to be extendedIonger than the third. However,becauseit is stiffer (i.e., it has a higher spring constant),it stretchesless than expectedbased on the weight it carries.
(a) (bl
FIGURE 8.6 Positions of three i n d i v i d u ocl o snnecied by bungeecords. (o)Unsiretched ond (b)stretched
CURRENTS AND VOLTAGES IN CIRCUITS Bockground. Recallthat in Chap. 1 (Table 1.1),we summarizedsomemodelsand associatedconservationlaws that figure prominently in engineering.As in Fig. 8.7, each model representsa systemof interactingelements.Consequently,steady-state balancesderived liom the conservationlaws yield systemsof simultaneousequations.In many cases, such systemsare linear and hencecan be expressedin matrix form. The presentcasestudy focuseson one such application:circuit analysis. A common problem in electrical engineering involves determining the currents and voltagesat various locationsin resistorcircuits. Theseproblemsare solved using Kirclr(orpoint) rulestalesthatthealgebraicsumof hoff'scurrentandvoltagerules.Thecurrent (Fig. all currentsenteringa node must be zero 8.812), or
fi:o
(8.7)
where all current enteringthe node is consideredpositive in sign. The current rule is an applicationof the principle of conservationof charge(recallTable 1.1).
EQUATIONS AND MATRICES LINEARALGEBRAIC
206
*_.r--*_ il
( a ) C h e m i c ael n g i n e e r i n g
f,,
( b ) C i v i le n g i n e e r i n g
(a)
ffi,ryl M Circuit (c) Electricalengineering
R,
--+AA---]+
WI
i,j
(b)
(d) Mechanicalengineering
FIGURE8.8 FIGURE8.7
Srhamntir
Engineeringsyslemswhich, ot steodyslote,con be modeledwilh lineorolgebroic equolrons.
ronro
rule of (o)Kirchhoff's current o n d ( b ) C h m ' lso w .
The voltage (or loop) rule specifiesthat the algebraic sum of the potential differences (that is, voltage changes)in any loop must equal zero. For a resistor circuit, thisis expressedas
fc-I^:o
(8.s)
of where f is the emf (electromotive force) of the voltage sources,and R is the resistance any resistors on the loop. Note that the secondterm derives from Ohm's law (Fig. 8.8b), which statesthat the voltage drop across an ideal resistor is equal to the product of the cunent and the resistance.Kirchhoff's voltage rule is an expressionof the consentationol enerSy. Application of theserules results in systemsof simultaneouslinear algebraic Solution. equationsbecausethe various loops within a circuit are interconnected.For example,consider the circuit shown in Fig. 8.9. The currents associatedwith this circuit are unknown
onesimply no greatdifficultybecause anddirection.This presents bothin magnitude assumesa direction for each current. If the resultant solution from Kirchhoff's lawsis negative,then the assumeddirection was incorrect.For example,Fig. 8.10 showssone assumedcurrents.
8.3 CASESTUDY
ili
continued
,il
'll I
!l
R:5O Vt:200Y
7o:0V
a n =i s o
R:20f)
A resistor circuitlo be solvedusingsimultoneous
FIGURE 8.IO
i;n"^' ^l^.h.^;.
AssumedcLruenr direction5
an, rnf innc
Given these assumptions, Kirchhoff's current rule is applied at each node to yield in*isz*jrr:0 ios* isz- ls+ :0 i a 3- 1 2 :
Q
i5a-iaj:Q Application of the voltage rule to each of the two loops gives ios-isz-ls+:0 -is,rRs+- i+-tR+:* i12Ri2* l52R52: Q -iosRos - iszR-srI ieRe - 200 : 0 or, substituting the resistances from Fig. 8.9 and bringing constants to the right-hand side, *151s+- 5ia3- 10i32* 10i52: Q -20ios-10is:f5ip:200 Therefore, the problem amounts to solving six equations with six unknown currents. These
equationscan be expressedin matrix form as
[r l0
l0 lo t0
l
r
-l
0
0
I
0 o l0
Ls -ro
0
-l
-1 o
0 o
o
_20 o
-t0
0
0 I
-15
0l
l12
0 |
r52
r -1 I | --sl ol
t3z .oJ r)4
iqz
:{,L
208
EQUATIONS AND MATRICES LINEARALGEBRAIC
continued Although impractical to solve by hand, this system is easily handled by The solution is >>A=t111000 0 -1 0 1 *1 0 0 0 -1 0 0 1 0 0 0 0 1 -1 0 10 -10 0 -15 *5 5 *10 A -20 0 ol; >> b=t0 0 0 0 a 2001,; >> current=A\b = current 6.1538 -4.6L54 -1.5385 -6.1538 *1.5385 -1.5385
8.1 ci
comm€ the oril E.2 At
tAl
rcI LE tFl
Thus, with proper interpretation of the signs of the result, the circuit curents and ages are as shown in Fig. 8. 1L The advantages of using MATLAB for problems of this should be evident.
Answe (a) WI (b) Ide (c) Wl
J t2'
(d) Pe (l
FIGURE8.I I obtoined using MATLAB Thesolution forcurrents ondvoltoges V:
169.23
(4
PROBLEMS
PROBTEMS Givena squarematrix [A], writc a single line MATLAB thatwill createa new matrix [Ang] that consistsof original matrix[AJ augnrentedby an identity matrix [11. A numberof mafices are definedas
[137f lr 2 7l
L2 0 4J
il pt:l:--,i ,'l
l r t, -_ (\
[r s 8l
r r l = 1 7t r l
L4 0 6_l
1-l r r r = f ro 7 .11 Ll
LG):17
6 4J
thefbllowingquestionsregardingthcse matrices: Whatarethe dimensionsof the matrices,l Identifythe square,coluntn, and row matrices. Whatarethe values of the clcments..ar,. b.1, dzt, ezz, /r:,8r:? Perform thefollowing operations:
(r)trt + tBl (1)7 x [B]
(2)IAI+ lFl 1-s1 [A] x tBl
FIGUREP8.5
(3)tBl _ tEl (6)(C],
(7) [Bl x [Al ( 1 0 )[ 1 ] x t 8 l
(8) [D]7 r n 1 [ E l rx [ E l
(e) tAl x { C} ( t 2 ){ C } rx { C }
8.3 Writethefollowingsetof equations in rnatrixforrn: 50:5,r:-7.r: 4rz+7.\*30:o rr - 7.r.r: ^10, 3xzf 5;:r UseMATLAB to solvelbr theunknorvns. In addition,usert to computethetranspose andthe inverseof thecoefficient matrix. 8.4 Threematrices aredefinedas
-,'l , ,",:ffi l'],",= l],.,:; [,i, (a) Perfbrm all possible nrultiplicationslhat can be com_ puted betweenpairs of thesematrices. (b) Justify why the remainirigpairs cannot be rnultiplied. (c) Use the resultsof (a) to illustratewhy the order of multiplication is important. 8.5 Five reactorslinked by pipes are shown in Fie. pg.5. The rate of massflow througheachpipe is computed as the prodr"rctof flow (Q) and concentration{r.). At steady state, the mass llow irrto and out of each reactor must be equal.
2ro
EAUATIONSAND MATRICES TINEAR ALGEBRAIC
For example, for the first reactor, a mass brilance can be written as 0 o r r : ofr Q t t ' t :
/-'
cos]0 -Fjcos60 *F1.71 -Fr sin -10
F.rsin 60 * Fr ,
\
nr :0:
F : I F 1c o s 3 0 I
F z nI H z
F rs i n 3 01 F 2 . ,I V :
for node 3: /r'n
algebraicequationsin matrix form and then use MATLAB to solve fbr the unknowns. 8.7 Considerthe threemass-fourspringsystemin Fig.P8.?, Determining the equationsof motion from XF' : ma,for eachmassusing its free-bodydiagramresultsin thefollow' ing differentialequations: /kr+ k"\
:0: 5 / --' n 'u
\-
P8.7 TIGURE
rr*lrlrr-l-lr::0
for node 2: )/--
I
Q r s c tI Q n c t
Write massbalancesfor the remainingreactorsin Fig. P8.5 and expressthe equationsin matrix fbrm. Then use MATLAB to solve for the concentrationsin eachreactor. 8.6 An importantproblem in sffucturalengineeringis that of finding the forcesin a staticallydeterminatetruss(Fig. P8.6). This type of structurecan be describedasa systemof coupled linear algcbraicequationsderived from force balances.The sum of the forces in both horizontal and vertical directions must be zero at each node, becausethe system is at rest. Therefore,for node 1: Lf,:0:-Fr \- n'' : 0 :
+"x?
r - :- 0 :"
\m:/
-F, - F.rcos60-l Fr.r,
: D a u o : F : s i n 6 o ' *F : , ,r v : where F,.,,is the externalhorizontal force applied to node I (wherea positiveforce is from left to right) and {.,. is the external vertical force appliedto node i (where a positiveforce is upward). Thus, in this problem, the 1000-lb downward fbrce on node I correspondsto F,.,: - 1000.For this casc, all other F,.,'s and {,,,'s are zero. Expressthis set of linear
/kr\ \mt/ /l'\
/,t'\ /k,*t,\ tr (31'' -(
;,-
rzr /
\
,t:
\m2 l
/
/k.\ /k,'" \ ( a ) * r +-{ ^ r r ^ 4 } x , : o \.n.r/
\
m3 /
andmr=mt= w h e r e f t -, k t : 1 0 N / m , k . : f t r : 3 0 N / m , mz: I kg. The three equationscan be written in matrixform: Q : {Accelerationvector} + lk I m matrixl{displacementvectorx } At a specifictime where:r, : 0.05 m, r: : 0.04 m, andl,= 0.03 m, this forms a tridiagonal matrix. Use MATLABto solve for the accelerationof eachmass. 8.8 Solve
[ 3 + 2 i + ' lJ : r [ _- l I 2 + , I i I I i r . l[ z : f 8.9 Perform the same computationas in Example8.2,but use five parachutistswith the following characteristics
F I G U R EP 8 . 6
Moss Jumper
(ks)
Spring Conslont (N/m)
I 2 3 4 5
55 75 60 75 90
BO 50 70 r00 20
urltr"*ffi Cord tength (m) t0 10 t0
r0 t0
8.10 Tl identica the bott 2.5kg,t displac 8.11 Pe the circ 8.12 Pi the circ 8.13 D two mal lZl is n multipli the prol 8.14 D the tran plemen
PROBLEMS
2ll
Threemassesare suspendedvertically by a seriesof springswheremass I is at the top and mass3 is at bottom. lf g : 9.81 m/s', ffir : 2 kg, m2: 3 kg, 2, : kg,andtheft's: l0 kg/s2,useMATLAB to solve for the ntsr Perform the samecomputationas in Sec. 8.3, but for circuit in Fig. P8.I l. Perform the samecomputationas in Sec. 8.3, but for circuit in Fig.P8.12. Develop, debug,and test your own M-file to multiply matrices-thatis, [X] : lYl[Zl, where lY] is m by n and isn by p. Employ f or. . . end loops to implement the ion andinclude error trapsto flag bad cases.Test program usingthe matrices from Prob. 8.4 Develop, debug,and test your own M-file to generate tmnspose of a matrix. Employ f or. . . end loops to imthetranspose. Testit on the matricesfiom Prob. 8.4.
R=30O
,
R:20c)
R:10O Vr : 150volts
R:2O R=5O Vo: 0 volts
F I G U R EP 8 . I I
R=35O
1
Vr : 10 volts
R:8f) Vo= 150volts
F I G U RP E8 . I 2
ti
. ...
GoussEliminotion
CHAPTEROBJECTIVES The primary objectiveof this chapteris to describethe Gausseliminationalgorithm fbr solving linear algebraicequations.Specificobjectivesand topics coveredare ' . ' . . '
Knowing how to solve small setsof linear equationswith the graphicalmethod and Cramer'srule. Understandinghow to implementfbrward eliminationand back substitutionasin Gausselimination. Understandinghow to count flops to evaluatethe efficiency of an algorithm. Understanding the conceptsof singularityand ill-condition. Understandinghow partial pivoting is implementedand how it differs from completepivoting. Recognizinghow the bandedstructureof a tridiagonalsystemcan be exploited to obtain extremelvef1icientsolutions.
t the end of Chap. 8, we statedthat MATLAB provides two simple and direct left-division, methodsfbr solvingsystemsof linearalgebraicequations:
>>
x
=
A\D
and matrix inversion, >> x
=
inv (A) *b
Chapters9 and l0 provide backgroundon how such solutionsare obtained.Thismaterial is includedto provide insight into how MATLAB operates.In addition,it is intended to show how you can build your own solution algorithmsin computationalenvironments that do not haveMATLAB's built-incapabilities.
9.I SOLVINGSMALLNUMBERS OF EQUATIONS
2r3
The techniquedescribedin thischapteris cllleclGausseliminationbecause it involves combiningequafionsto climinateunknowns.Althoughit is clneof the eirrliestrnethodsfbr solving simultaneous equctions,it remainsarnongthe most importantalgorithnrsin use today and is the basisfur linear equationsolving orl many popular sotiwarepackagesincluclingMA|LAB.
9 . 1 SOIVING StylAtt NUMBERSOF EQUATIONS Betbreproceediugto Gausselinrination,we will describeseveralmethodsthat are appropriate fbr solving small (n < 3) setsof simultaneousequationsand that do not recluirea computer.Theseare the graphicalmethocl,Cramer'srule. anclthe eliminationof unknowns.
9.1.1 fhe GrophicolMethod A graphicalsolutionis obtainablefor two linearequationsby plottingthen on Cartesian coordinateswith one axis colresponding to -r1and the othel to -r2.Becausethe equations arelinear.eachequationwill plot as a straightline. For example,supposethat we havethe tbllowing equations: 3 - r r* 2 . r r : -,-.
--L l,^
-
l8 )
I f v e e . \ u n c r h e t . ir>, t h e z b : c i l : t a , { e a a n . > o l r c c a c h t t l t h t s e t e u i l U o i l s r o r - r t ! 3
-rl:--.tl+9 -t-t:
I -f I t-rl
The equationsare now in the forrn of straightlines-that is..t2 : (slope)-rr * intercept. When theseequationsare graphed,the valuesof r1 and x2 at the intersectionof the lines representthe solution (Fig. 9. l ). For this case,the solutionis ,r1 : 4 andxz : 3 . by a planein a For threesimultaneous equations. eachequationwould be represented three-dimensional coordinatesystem.The pointwherethe threeplanesintersectwould representthe solution. Beyond three equations,graphical methodsbreak down and. consequently,have little practical value fbr solving simultaneousequations.However,they are usefulin visualizingpropertiesof the solutions. For example.Fig. 9.2 depictsthreecasesthatcan poseproblemswhen solvingsetsof parallellines. linearequations. Fig. 9.2a showsthe casewherethe two equationsrepresent For suchsituations,thereis no solutionbecausethe lines nevercross.Figure9.2b depicts the casewhele the two linesarecoincident.For suchsituationsthereis an inllnite number of sollrtions.Both typesof systemsaresaidto be singular. In addition,systernsthat are very closeto being singular(Fig. 9.2c) can also cause problems.Thesesysternsare said tobe ill-conditirnecl.Graphically,this conespondsto the fact thatit is diiTicultto identifythe exactpoint at which the linesintelsect.Ill-conditioned systemswill alsoposeproblenrswhen they areencountered duringthe numericalsolution This is becausethey will be extremelysensitiveto roundofTerror. of linearequations.
il
2r4
GAUSSELIMINATION
6rt
9.I FIGURE ofthe Theintersection lineorolgebroicequotions. of o setof Mo simultoneous solutlon Grcphicol the solution. linesrepresenls
EX
9.2 FIGURE (o)nosoluiion, ond solutions, systems: ondilfconditioned of sinqulor depiction {b)lnflnlte Grophicol visuolly to detecl isdifficuh oresoclosetholihepointoI intersection (c)ilicondiiioned theslopes system wtrere
9.1.2
Determinqnls qnd Cromer's Rule
Cramer's rule is anothersolution techniquethat is best suitedto small numbersof equa' tions. Before describing this method, we will briefly review the conceptof the determinant, which is usedto implementCramer'srule. In addition,the determinanthasrelevancet0the evaluationof the ill-conditionine of a matrix.
SOLVINGSMALLNUMBERS OF EAUATIONS
2r5
Determinonts. The determinantcan be illustratedfor a set of threeequations: [A]{,t} : {b} where [A] is the coefficientmatrix [Al:
ort an anf f I a21 a22 azz I tl LA3t
432
AT J
The determinantof this systemis formed from the coefficientsof [Aj and is represented as D :
latt I u.,
i" ;;
at2 Ltt
at3 o)1
o .,
" .;
Although the determinantD and the coefficientmatrix [Al are composedof the same elements,they are completelydifferent mathematicalconcepts.That is why they are distinguishedvisually by using bracketsto enclosethe matrix and straightlines to enclosethe determinant.In contrastto a matrix, the determinantis a single number.For example,the value of the determinantfor two simultaneousecuations
ott D : latr I I A)t A)) | - - l
is calculatedby D:atla22-an(7zt For the third-ordercase,the determinantcan be computedas
ott o"l ot'l D : a ,' ,l al as !z | o ,-rl al o: rt t a . r : l* o ,-.l ,al:or " a J 2 .?.r.rl l
(e.1)
where the 2 by 2 determinantsare called minors. Determinonts Problem Stotement. Computevaluesfor the determinantsof the systemsrepresentedin Figs.9.l and9.2. S o l u t i o n . F o r F i g . 9 . 1:
D-
JL
-1
2
:3(:2)-2(-1):8
ForFig.9 2a: L _ !
D-
|
2--l 1
: - ; ( ll ) - t l ; l :/0- 1 \ L
\L,/
ForFig.9.2b: D:l
l-1 2
-r : - t I( : 2 )1(-l):0
l-r
z
216
GAUSSELIMINATION For Fig. 9.2c: tr
D_
I
)' ,', _:l :-
I
/-?
2
\s/
?\
___1)_ll_l:_0.04
In the foregoing example,the singular systemshad zero determinants.Additionally, the resultssuggestthat the systemthartis ahnostsingular'(Fig. 9.2c) has a detenninant that is close to zero. These ideas will be pursuedfurther in our subsequentdiscussionof illconditioningin Chap.I l. equaCromer's Rule. This nrle statesthat eachunknownin a systemof linearalgebraic tions may be expressedas a fraction of two determinantswith denorninatorD andwiththe numeratorobtained from D by replacing the colurnn of coefficientsof the unknownin q u e s t i o nb y t h e c o n s t a n tbs r . b 2 , . . . . b , , .F o r e x a m p l ef,o r t h r e ee q u a t i o n sx,1 w o u l d b e computedas
.,,:
lbr oe 4r:i I bz azz azt I I tr^ a1. 4.. l ,
EXAMPLE 9.2 Cromer's Rule ProblemStotement. Use Cramer'srule to solve 0 . 3 ; r r+ 0 . 5 2 x 21 0 . 5 . r r+
r::
-0.01
x2f l.9rj:
0.67 0 . 1 r r + 0 . 3 x 2 * 0 . 5 1 3: - 0 . 4 4
Solution. The determinantDcanbe evaluatedas [Eq. (9.1)]:
a:o:lol: l?l-o"l3i A:31.'l3i o:l:-ooo22 The solutioncan be calculatedas
.r I -
-0.0r 0.52 I 0.67 r 1.9 -0.44 0.3 0.5 -4.0022 10.3 -0.01
-0.0022
I I
i o.r 0.67 r.eI | 0.I
0.03278 - 1 4 . 9 :
-0.44 0.5| -0.0022
0.0649 _ _)a < -0.0022
-0.01 10.3 0.52
I o.s r 0.6l I o.r 0.3 *0.44
-0.043s6
,0.0022
-0.0022
: 19.8
SOLVINGSMALTNUMBERS OF EAUATIONS
217
The ae-' Function. The determinantcan be computeddirectly in MATLAB with the det function. For example,using the systemfrom the previousexample: >> A-10.3 0.52 I;0.5 >>
1 1.9;0.1
0.3 0.51;
D=det (A)
D= 0 .0422
Cramer'srule can be appliedto compute.{i as in >> A(:,1)= t-0.0I;0.61 ; -0.441 -0.0100 0.6700 -0.4400 >> x1=det X1
0.5200 1.0000 0.3000
1.0000 1.9000 0.5000
(A) /D
=
-14.9000 For more than three equations,Cramer's rule becomesimpractical because,as the number of equationsincreases,the determinantsare time consumingto evaluateby hand (or by computer).Consequently,more efficient alternativesare used.Someof thesealternativesarebasedon the last noncomputersolutiontechniquecoveredin Section9.1.3-the elimination of unknowns.
9.1.3 Eliminotionof Unknowns The elimination of unknownsby combiningequationsis an algebraicapproachthat can be illustratedfor a setof two equations:
(e.2) (e.3)
a 1 1 x 1* a 1 2 X 2 : 1 1 1 aZtxtla22x2:fi2
The basic strategyis to multiply the equationsby constantsso that one of the unknowns will be eliminatedwhen the two equationsare combined.The result is a single equation that can be solvedfor the remainingunknown.This valuecan thenbe substitutedinto either of the original equationsto computethe other variable. For example,Eq. (9.2)might be multipliedby a21andEq. (9.3)by all to give aztTltxt I
a 2 1 A 1 2 X :2 A Z l b t
a 1 1 a 2 1 X y{ a 1 1 a 2 2 x 2:
attbz
(e.4) (9.5)
SubtractingEq. (9.4) from Eq. (9.5) will, therefore,eliminatethe x1 term from the equations to yield a t t a 2 2 x-2 a y l a l 2 x z: a t t b Z- a y b t which can be solvedfor a 1 1 b 2-
a21b1
ii ti
GAUSSELIMINATION
218
Equation(9.6)can thenbe substituted into Eq. (9.2),which can be solvedfbr a 2 2 b 1- u p b 2 d 1 1 4 2 2-
02yAp
Noticethat Eqs.(9.6)and (9.7)follow directlyh'om Cramer'srule:
th' anl lb,
,._t-_-t_
aybl - a12lt2
attl
Il l, , , a n l 1O t t
'l lo" u bt lrr.'
1,.3
_.1____J___i_I
lQtr tl I c.lrr
0|tt)2 -
A2jal?
d 1 1 b 2-
a21b1
A))
I
I
1
an J tltt
I
The eliminationof unknownscan be extendedto systemswith rnorethan two or three eqLrations. However, the numerouscalculationsthat are requiredfor larger systemsmake the methodextremelytediousto irnplementby hand.However,as describedin Section9,2, the techniquecan be formalizedand readily programrnedlbr the conlputer.
9.2
NAIVE GAUSSELIMINATION In Section9.1.3,the eliminationof unknownswas usedto solve a pair of simultaneous equations. The procedureconsistedof two steps(Fig.9.3): 1. 2.
one of the unknownstiom the equations, The equationswere manipulatedto elin.rinate had one equationwith one unknown. of this elimination step was that we The result into could be solved directly andthe l'esultback-substituted Consequently, this equation solve for the remaining unknown. one of the original equationsto
This basicapproachcan be extendedto largesetsofequationsby developinga systemand to back-substitute. Gausselimination atic schemeor algorithrnto eliminateunkr.rowns is the most basicof theseschemes. This sectionincludesthe systematictechniquestor forward eliminationandbacksubfor areideallysuited stitutionthatcompliseGausselimination.Althoughthesetechniques implementationon computers,some modifications will be required to obtain a reliable algorithrn.In particular,the computerprogram must avoid division by zero. The tbllowing method is called "naive" Gausselimination beciiuseit does not avoid this problem. Section 9.3 will deal with the additional featuresrequired for an et'fectivecomputer progranl. The approachis designedto solve a generalsetof n equations: d1111 * q s z x z l 4 r - t x :* ' " ayll
* ( r 2 X 2I d z r j r : * '
I ay,,x,,: $t
(9.84)
. ' I a 2 , r x , :, S .
(9.8r)
1a,.-r- :
(9.8c)
::
2r9
9.2 NAIVEGAUSSELIMINATION
drz an [r,'
dz2 azt
lt"
att
a32
Lo!
I
o' ['"
,rrl
o','
,' ""
o',,'
| {
,,,4 (a) Forward
elimination
4rr
I
i h: xt:
b"z/a'\z
(b) Back substitution
(b'r- a|34lftt'22
-t, : (b, - dr:.r:- aptzlfa,
FIGURE 9.3 Theh.azo phoses (olforword of Gouss eliminotion: eliminotion ond(b)bocksubstitution
As was the casewith the solutionof two equations,the techniquefor n equationsconsists of two phases:eliminationof unknownsand solutionthroughback substiturion. Forword Eliminotion of Unknowns. The first phaseis designedto reducethe set of equationsto an uppertriangularsystem(Fig. 9.3a).The initial stepwill be to elininate the first unknownr1 from the secondthroughthe nth equations.To do this, multiply Eq. (9.8a) by a21f all to give a 2 1 x y* \ o p * z alt
*Lanxz+ atl
..'+
9or,*, att
-
eht br
(e.e)
Qtt
Now this equationcan be subtractedfrom Eq. (9.8b) to give
(,,, - 9 u ,r).,,+ .. + (,,,,- #,,,,) r , - b,- !) !6, or tt'rrxz+ . .. I a'"r-r,,: 6, where thc prime indicatesthat the elementshavebeenchangedfrom their original values. The procedureis then repeatedfor the remainingequations.For instance,Eq. (9.8a) can be multiplied by a3llayl and the result subtractedfrom the third equation.Repeating the procedurefor the remainingequationsresultsin the lbllowing modified system: ( t 1 1 x 1* c t p l 2 l
n 1 - 3 - t*3 . . . *
a1ux,,: $,
a \ t x t * a \ . x 3 a " ' * r t l , , x ,:, 6 \
(9.10a) rq InA\
220
GAUSSELIMINATION
c t r , x z | a 1 ] , r *3. . . * c t . . , , x , r : 6 \
a',rrx,,:6',,
a ' r ' r . rl ox u 2rr,r*...r
(9.10c)
(9.10d)
For the foregoingsteps,Eq. (9.8a)is calledthe pivot equationandall is calledthe pivot elentenr.Note that the processof multiplying the first row by u21f tr11is equivalent to to dividing itby all and multiplying it by a21. Sometimesthe division operationis refened as nonnalization.We makethis distinctionbecausea zeropivot elementcan interferewith normalizationby causinga division by zero.We will returnto this importantissueafterwe completeour descriptionof naiveGausselimination. The next step is to eliminatex2 from Eq. (9.10c) through (9.10d). To do this,multiply Eq. (9.10r) by aj2la\, and subtractthe result from Eq. (9. lOc). Perform a simila elimination for the remainingequationsto yield { 7 1 1 . \l '-1a 7 l t , f , tI d l r r . r r
"'LA1nX,,:lt,
tr\ar2+ aztu * .. . I a'r,,x,: 11', ai{tr...la'i,,x,:['!
:: ai,l4 1 ... t al,,x,: b',' where the double prime indicatesthat the elementshave beenmodified twice. The procedurecan be continuedusing the remaining pivot equations.The finalmanipulationin the sequenceis to usethe (n - I )th equationto eliminatefhe x,,-1 termfrom the rth equation.At this point, the systemwill have beentransformedto an uppertriangular system: a t t x t I a n x z I r z 1 3 J *3 . . . I a 1 r r , r : f i 1 a'zzrz* c/ar4 + . . . I a'r,rr, : 6, - 6\ a'1.,4* ... * a'1,,r,,
(9.1 ld)
(9.11r) (9.llc)
: l o ) , i it;'* , , : h " t t Bock Substitution. Equation(9.11d)can now be solvedfor x,,: -ht \1 n
t)
(r-l) ann
into the (n - l )th equationto solve for ir,,- 1. Thepn This result can be back-substituted by thefollowcedure,which is repeatedto evaluatethe remaining:r's, can be represented ins fbrmula: il
6ri tt _ 'r
\/-
i:i+l
, r 1 1 - -l-)J" , --t.l
fori:n-l,n-2,
E}
22r
9.2 NAIVEGAUSS ETIMINATION E X A M P L9E. 3
Noive GoussEliminotion Problem Stqtement. Use Gausseliminationto solve 3xr- 0.lrz -0.2x2: 7x2-0.34: 0.lxr* 0.3x1 0.2x2+ l0;t: -
(Ee.3.l)
7.85 -19.3
rFq ? ?r
(E9.3.3)
71.1
Solution. The first part of the procedureis fbrward elimination.Multiply Eq. (E9.3.1) by 0.1l3 and subtractthe resultfrom Eq. (89.3.2)to give -19.5617
7.00333x:- 0.293333n :
Then multiply Eq. (89.3.1)by 0.313and subtractit from Eq. (89.3.3).After theseoperat i o n s .t h e s e to f e q u a t i o niss 0 . 2 x 3:
7.85
- - 19.5617 - 0.293333x: 7.00333x: - 0.190000x2 * 10.0200x:: 70.6150
rFq 1 4) (89.3.5) (E9.3.6)
To completethe lbrward elimination,rr rrIUStbe removedfiom Eq. (E9.3.6).To accomplish this, multiply Eq. (E9.3.5)by -0.19000011.00333and subtractthe result from Eq. (E9.3.6).This eliminatesr? from the third equationand reducesthe systemto an upper triangularform, as in (E9,3.7) 7.85 3xr0 .l x z 0 . 2 x 31.00333x2- 0.293333xt: -19.561'7
(E9.3.8)
10.0120-tj: 70.0843
(E9.3.9)
We can now solve these equationsby back substitution.First, Eq. (89.3.9)can be solvedfor
70'0843 : 7.oooo3 10.0120 into Eq. (E9.3.8).which can then be solvedfor This result can be back-substituted
- t9.5617+ 0.293333(7.00003) : -2.50000 7.00333 Finally,"r:: 7.00003andx2 : -2.50000 canbe substituted backinto Eq. (E9.3.7),which can be solvedfor
7.85+ 0.1(-2.50000) + 0.2(7.00003) : 3.00000 Although thereis a slight round-off eror, the resultsare very closeto the exact solutionof rr : 3, ^z: -2.5, &I1d;rj : 7. This can be verified by substitutingthe resultsinto the original equationset: -r(-r)- o. I (- 2.-5) 0.2(7.00003): 7.81999= 7.85 0 . 1 ( 3 )+ 7 ( - 2 . 5 ) - 0 . 3 ( 7 . 0 0 0 0 3: ) - 1 9 . 3 0 0 0 0: - 1 9 . 3 0 . 3 ( 3 )- 0 . 2 ( - 2 . 5 ) + l 0 ( 7 . 0 0 0 0 3 :) 1 1 . 4 0 0 3= 1 1 . 4
222
GAUSSELIMINATION
function x = caussNaive(A,b) % GaussNaive: naive Gauss eliminatiorr % x = GaussNaive(A,b): Gauss elimination ? input: % A=coefficientmaLrix % b = rigirr hand side vector % outpuL; ? x=solutionvector lm,nl = size (A) ; if m-=n, error('Matrix nb - n+1; Aug - [A b];
A must be square');
withouL
pivoting.
end
? forward elimination for k = 1:n-1 for i = k+1:n - Aug(i,k) /Auq(k,k) ; factor = Aug(i,k:nb) -factor*Aug(k,k:nb) Aug(i,k:nb) ; end end ? back subsr'rue ion x = zeros{n,l); x(n) - Aug(n,nb)./Auq(n,n) ; ior i = n-1:-i:1 - (Au.J(i,nb)-Aug(i, i+1:n) *x(i+1:n) ) iAug(i,i) x(i)
;
end
F I G U R E9 . 4 A n M J i l ei o r m p i e m ennot i v eG o u s se l i m i n a t i o n .
9.2.I
MATTAB M-file! GaussNaive
An M-file that implementsnaive Gausselimination is listed in Fig. 9.4. Noticethatthe coefficient matrix A and the right-hand-sidevector b are combinedin the augmented matrix Aug. Thus, the operationsare performedon Auq ratherthan separatelyon A andb. Two nestedloops provide a conciserepresentation of the tbrward eliminationstep. Al oLtterloop movesdown the matrix from one pivot row to the next. The inner loopmoves below the pivot row to each of the subsequentrows where elimination is to takeplam. Finally, the actual elimination is representedby a single line that takes advantage of MATLAB's ability to perform matrix operations. The back-substitutionstep follows directly from Eqs. (9.12) and (9.13).Again, MATLAB's ability to perform matrix operationsallows Eq. (9.13) to be programmed asa sinsleline.
9.2.2 Operotion Counting The executiontine of Gausseliminationdependson the amountof .floating-point operatilv
9.2 NAIVE GAUSS ELIMINATION
223
Therefore,totalingup theseoperationsprovidesinsightinto which partsofthe algorithmare most tirneconsumingand how computationtime increasesas the systemgetslarger. Before analyzingnaive Causselirnination,we will first define somequantitiesthat facilitateoperationcounting: il1
il1
111
\-, r,;' -, \- r'ir /--' /-'"' ;
I
i
I
tn
;-t
i-l
\L - r
: I * I + | +...11:nt
\-;
:
L'
In
\ - r , ; r + o r ir -- \ - r , ; r r \ -, 'o, r i r //_.t
| +2 +3 +... +t1t
nr_ k+l
It: t n ( n r*
l)
(9.| 1a,b)
/_6", i-l
:
nr) I
(9.14cd)
(9.14e)
O(nt)
\-;r : l.+ 2,+3-+...+ ,rz-m(ml1)-(2mll) :!*o@2) lJ
(:e.t1.f )
6-i
where O(rl") means"terms of order tn" and lower." Now let us examinethe naiveGausseliminationalgorithm(Fig.9.4) in detail.We will first count the flops in the eliminationstage.On the first passthroughthe outer loop, k : I . Therefbre,the limits on the inner loop are from i: 2ro n. Accordingto Eq. (9.14d),this meansthat the numberof iterationsof the innerloop will be f/ - - t : t t _
2+l:n-
(9.I s)
I
i:2
For every one ofthese iterations,thereis one division to calculatethe factor.The next line then performs a multiplication and a subtractiontbr each column element from 2 to nb. Becausenb : n f 1, going fiom 2 to izb resultsin r multiplicationsand n subtractions. Togetherwith the singledivision,this amountsto fl + I multiplications/divisions and n pass iteration inner loop. The for the first addition/subtractionstor every of the total (n l) multiplication/divisions l)(n) throughthe outerloop is thereforeQt l)(n 1 and addition/subtractions. iterationsof the Similarreasoningcan be usedto estimatethe flopsfor the subsequent outer loop.Thesecan be summarizedas
'"r,t,tu."n";;;;;,.ion Outer Loop k
tr,r,",tolp
aatiri""Tsritr".tt"r
i
Flops
I 2
2,n 3, rt
{rr- I j(r) ln-2)ln-t)
Flops
{,r-llfu+1) l,'- 2)l,l
: k
rr- l
kll
n
ln
k ) l n+ I - k )
( r) ( 2 )
n, n
(ir
k)(n+2-k)
( r) { 3 )
flops for eliminationcan be computedas Therefbre,the total addition/subtraction ir
rr I
I
k ' t ( n +t - f r ) : f
k ( 2 n *t ) + / . 2 1
(9.16)
tl li
224
GAUSSELIMINATION
n ( nt t , i t - Q t *t t )f r + f t ' A:t k:t t-l Applying someof the relationshipsfrom Eq. (9.14) yields
l n 3+ o ( , ) ) - [ n 3+ o r n z t ] *[ ] , , t + o r n z ; l : | * o r r , J L.''
J
A similar analysislbr the multiplication/divisionflops yields
[ n 3+ o l f ) ] -
[ n 3+ o ( n ) ]+ [ ] , , t * o r r ' ; ] : | * o r r t , t)tl L_-
)
(9.r9)
'
Summingtheseresultsgives ln' ^ J
(9.201
lO(tt')
Thus, the total number of flops is equal to 2nt 13 plus an irdditionalcomponent proportional to terms of orclernr and lower.The result is written in this way becauseasn gets large,the O (n?) andlower termsbeconrenegligible.We are thereforejustifieclin concluding that for largen, the effort involved in forward eliminationconvergeson 2n3f 3. Becauseonly a singleloop is used,back substitutionis much simplerto evaluate. The numberof addition/subtraction flops is equalto n(n - l) 12. Becauseof the extradivision prior to the loop, the numberof multiplication/division flops is n(n -l l)12. Thesecanbe added to arrive at a total of n2+ o1ny
(9.2\
EXA
Thus, the total efTortin naive Gausseliminationcan be representedas
oUr',+ n2+o(i,) -ll-I1r\ T * Forward elimination
2!
* oCl
(9.221
Back substitution
Two usefulgeneralconclusionscanbe drawn ftom this analysis: l.
As the systemgets larger,the computationtime increasesgreatly.As in Table 9.1,the amountof flops increasesnearly three ordersof magnitudefor every order of magnitudeincreasein the numberof eouations.
T A B T E9 . 1
N u m b e ro f f l o p sf o r n o i v eG o u s se l i m i n o t i o n .
n
Eliminqfion
t0 r00 r000
705 6 7 15 5 0 667xlaE
Bock Substituiion
r00 r0000 I x lOo
Tolol Flops
805 6 8I 5 5 0 668 x IOB
2tt313
667 666667 6 . 6 7x I A B
Percenl Due to Eliminction
87.58% 9853% 99 85%
9.3 PIVOTING 2.
9.3
225
Most of the effort is incurredin the eliminationstep.Thus, effortsto make the method n.roreefficient shouldprobably focus on this step.
PTVOT|NG The primary reasonthat the foregoing techniqueis called "naive" is that during both the elimination and the back-substitutionphases,it is possiblethat a division by zero can occur.For examole.if we usenaive Gausseliminationto solve 2 - r 21 1 . 1 1 8 4rr * 6-ru*7.rt - -3 2x1 - 3x2 -f 613 5 the normalizationof the first row would involve division by ay1:0. Problemsmay also arisewhen the pivot elementis close,ratherthan exactlyequal,to zerobecauseif the magnitude of the pivot elementis small comparedto the other elements,then round-off errors can be introduced. to determinethe coefficient Therefore,beforeeachrow is nomralized,it is advantageous with the largestabsolutevalue in the column below the pivot element.The rows can thenbe switchedso that the largestelementis the pivot element.This is calledpartial pivoting. If columnsas well as rows are searchedfor the largestelementand then switched,the procedureis called contpletepivoting. Cornpletepivoting is rarely usedbecauseswitching columns changesthe order of the .r-'sand, consequently,adds signilicant and usually unjustified complexity to the computerprogram. The following example illustrates the advantagesof partial pivoting. Aside from avoiding division by zero,pivoting also minimizesround-off eror. As such,it also serves as a partial remedy fbr ill-conditioning.
9.4 EXAMPLE
Poriiol Pivoting ,
ProblemStotement. Use Gausseliminationto solve 0.0003xr+ 3.0000x2:2.0001 : 1.0000 1.0000xr* 1.0000.12
, ; r
Note that in this fom.rthe first pivot element,arr : 0.0003,is very closeto zero.Then repeat the computation,but partial pivot by reversingtl'reorder of the equations.The exact solutionis 'r1 : I /3 andxz : 213. Solution. Multiplyingthe first equationby 1/(0.0003)yields - r - fr 1 0 , 0 0 0 x 2 : 6 6 6 1
'
which can be usedto eliminater1 from the secondequation: -9999x2:
-6666
which can be solved fbr:r2 : 213.This result can be substitutedback into the first equation to evaluate"rr| : ^t-
2.0001- 3(2/3) 0.0003
(E9.4. r)
226
GAUSSELIMINATION Due to subtractivecancellation,the result is very sensitiveto the number of significant f i g u r e sc a n i e di n t h ec o m p u t a t i o n :
Significonr Figures 3 4 5 6 7
.rt
x2
0 667 0.6667 4 66667 0 666667 0.6666667
I
11
0 0000 0 30000 0 330000 0 3330000
AbsoluteVolue of PercenlRelotive Error for x,
r099 r00 t0 I 0t
Note how the solutionfor.r, is highly dependenton the numberof significantfigures.Thn is becausein Eq. (E9.4.1), we are subtractingtwo almost-equalnumbers. On the other hand,if the equationsare solvedin reverseorder,the row with thelarger pivot elementis norrnalized.The equationsare 1.0000xr* 1.0000x2: 1.0000 0.0003xr* 3.0000-tr:2.0001 Elimination and substitutionagain yields x3 :213. For different numbersof significant figures,rr can be computedfrorn the first equation,as in
| - (213) I This caseis much lesssensitiveto the numberof significantfiguresin the computation:
Significont Figures
3 4 5 6 7
xl
x2
4 667 4 6667 0.66667 4 666667 0.6666667
0 333 0 33-?-? 0 33333 0 333333 0 3333333
AbsoluteVolue of PercenlRelotive Error for x,
0l 00r 0 00t 0 000r 0.0000
Thus, a pivot strategyis rnuch rnore satisfactory.
9.3.1 MATLAB M'file3 GaussPivot An M-file that implementsGausselimination with partial pivoting is listed in Fig.9.5.It is identicalto the M-file for naive Gausselirninationpresentedprel'iouslyin Section9.2.1 with the exceptionof the bold porlion that implementspartial pivoting. Notice how the built-in MATLAB function max is usedto determinethe largestavailable coefficient in the column below the pivot element.The rnax function hasthe syntax [y,r]
= max(x)
9.4 TRIDIAGONAL SYSTEMS
227
function x = caussPivot(A,b) pivotingr % GaussPivot: causs eLimination % x = GaussPivot (A,b) : Gauss eliminarion % input: ? A=coefficienLmatrix ? b - right. hand side vector Z outpuc: % x=solutionvector
vrith pivor,ing
Im,n]=size (A) ; A must be square'); if m.-=n, error('Matr.ix end nb=n+1 ; Aug= [A b] ; % forward elimination for k - 1:n-l eo partiaL pivoting t b i g , i l = m a x ( a b E( A u g ( k : n , k ) ) ) ; ipr=i+k-1; if ipr-=k AUS([k,ipr],: )=Aug( [ipr,k],: ); end for i = k+1:n factor=Augr (i, k) /Auq (k, k) ; Aug ( i, k : nb) =Aug( i, k : nb) - factor*Aug (k, k : nb) ; end ? back substitution x=zeroS(n,1); x ( n ) - 4 L 1 g( n , n b ) / A u g ( n , n ) ; for i = n-1:-1:1 x ( i ) = ( A u g ( i , n b ) - A u g r( i , i + 1 : n ) * x ( i + 1 : n ) ) / A u g ( i , i ) ; end
FIGURE 9.5 A n M J i l e t o i m p l e m e ntfh eG o u s se l i m i n o t r own i i h p o r i i o lp i v o t i n g
9.4 TRIDIAGONATSYSTEMS Certain matriceshave a particularstructurethat can be exploitedto developefficient solution schemes.For example,a bandedmatrix is a squarematrix that has all elementsequal to zero, with the exceptionof a bandcenteredon the main diagonal. A tridiagonal systemhas a bandwidthof 3 and can be expressedgenerallyas
fi8r e2 .fz 8z e3 ft
r1
rl
.{1
r.
-
41
(9.23)
€tt I
.f,,-l
8n-t
Xn-l
' n- |
GAUSSELIMINATION
Notice that we have changedour notationfor the coefficientsfrom a's and b's to e's,/s, ma,g's,and r's. This was doneto avoid storinglargenumbersof useiesszerosin thesquare modificationis advantageous becausethe resultingalgonthn trix of a's. This space-saving requireslesscomputerrnemory. An algorithmto solvesuchsystemscanbe directly patternedafterGausseliminationthat is, using forward elimination and back substitution.Howeveq becausemostof thc matrix elementsare already zero, much less effort is expendedthan for a full matrix.This efficiency is illustratedin the following exarnple. E X A M P L E9 . 5
S o l u t i o no f o T r i d i o g o n o lS y s t e m
Stotement.Solve the following tridiagonalsystem: Problem
-, ll;tt_lTfl ':i -:l t:tI t,3;1-l
-l f2.04 2 .04 l-r
Solution. As with Gausselimination,the tirst step involves transformingthematrixh upper triangular form. This is done by multiplying the first equationby thef actore2lf ynl subtractingthe resultfrom the secondequation.This createsa zero in placeof e2andtransforms the other coefficientsto new values. f z : . f zt'j:
?a
T+t
:2.04 -
t ' )- ? , ' , : 0 . 8 f,
-l
1) : I '550 2JJ4e I t 4 0 . 8 t: 2 0 , 8 1.04
Notice that g, is unmodifiedbecausethe elementaboveit in the first row is zero. istranr After performinga similal calculationlbr the third andfourth rows,the system formed to the upper triangularform
,.lot-:os lI;i I:I - i33I [204 ,.i,Jl;;I I ift',';f I Now back substitutioncan be appliedto generatethe linal solution:
,o: 'io: '+#: t'3 - 83r-l
,r:
fa ,'z - 82x3
,fz rl ^l
-
8lJ2
.ft
r5e48o
14.22r- (-l)159.480 :124.538 1.395 - (- l) 124.538 20.800 : 93.718 1.550 40.800- (- l)93.778 :65.910 2.040
9.5 CASESTUDY
function x = Tridiaq(e,f,g,r) % Tridiag: Tridiaqonal equation solver banded system t x = Tridiag(e, f,g,r): Tridiagonal system solver. % input: ? e=subdiagonalvector Z f= diagonalvector g = superdiagonal 2 vector 2 r = right hand side vector % output: ? x=solutionvector n=lengrth (f ) ; ? forward f ar
V
-
elimination ).r
= e(k) /f (k-1-); factor - factor*g{k-1); = f{k) f(k) - factor*r(k-1); = r(k) r(k) end % back substirurion (n) ; x(n) = r(n)/f for k = n-1:-1:1 x(k) = (r(k)-s(k)*x(k+1) )/f
(k) ;
end
FIGURE 9.6 AnMJileto soveo tridiogonol sysiern.
9.4.1
MATLAB M-file! rridiag
An M-file that solvesa tridiagonalsystemof equationsis listed in Fig. 9.6. Note that the algorithmdoesnot includepartialpivoting. Although pivoting is sometimesrequired,most tridiagonalsystemsroutinely solvedin engineeringand sciencedo not requirepivoting. Recall that the computationaleffort for Gauss elimination was proportional to n3. Becauseof its sparseness, the effort involved in solving tridiagonalsystemsis proportional to n. Consequently,the algorithm in Fig. 9.6 executesmuch, much fasterthan Gausselimination,particularlyfor large systems.
MODELOF A HEATED ROD Bockground. Linear algebraicequationscan arise when modeling distributed systems. For example,Fig.9.7 shows a long, thin rod positionedbetweentwo walls that are held at constant temperatures.Heat flows through the rod as well as between the rod and the surrounding air. For the steady-statecase,a differential equation basedon heat conservation can be written for such a system as d2T
_ +h'(7.- z):0
ax-
(9.24)
230
GAUSSELIMINATION
continued 7,,: 20
I
fo: 40
ti{it
-- Lx;'7,,:20
Is = 200
{ ,
,r=0
FIGURE9.7 twowollsof constont temperoture rodpositioned between butdifferent A noninsuloted uniform The {inite-d'flerence representolion employsf6ur inls1ie,66is5.
where I: temperature("C), x : distancealong the rod (m), h' : aheat transfercoefficient betweenthe rod and the surroundingair 1m-2), ztrdTo: the air temperature("C). Given values for the parameters,forcing functions, and boundary conditions,calculus can be usedto developan analyticalsolution.For example,if h' :0.01, 4, : 20, T(0)= 40, and (10) : 200, the solutionis t^ 20 T :13.4523n0'tx - 53.4523e-0 +
(9.2'
In Although it provided a solutionhere,calculusdoesnot work for all suchproblems. such instances,numerical methods provide a valuable alternative. In this case study,we will use finite differencesto transformthis differentialequationinto a tridiagonalsystem of linear algebraic equationswhich can be readily solved using the numerical methodsdescribed in this chapter. by Solution. Equation(9.24) canbe transformedinto a setof linear algebraicequations conceptualizingthe rod as consistingof a seriesof nodes.For example,the rod in Fig.9.7 is divided into six equispacednodes.Sincethe rod has a length of 10, the spacingbetween nodesis A,x :2. Calculus was necessaryto solve Eq. (9.20 becauseit includes a secondderivative. As we learnedin Sec.4.3.4,finite-differenceapproximationsprovide a meansto transfom derivatives into algebraic form. For example, the second derivative at each node canh approximated as dzT
Ti+t -ZTi *Ti-r
d-t
Lx2
where I designatesthe temperatureat node i. This approximation can be substitutedinto Eq. (9.2q to give Ti+r-2Tr * Ti*t n, -2
+h'(7"-4):0
9.5 CASESTUDY
continued Collecting terms and substitutingthe parametersgives -Ti*r*2.04Ti-
(e.26)
4+r:0.8
Thus, Eq. (9.24)has beentransformedfrom a differentialequationinto an algebraicequation. Equation(9.26) can now be appliedto eachof the interior nodes:
-To*2.04T1-4:0.8 - Tt * 2.0472- Z: : 0.8 - Tz 12.04\ - Z+: 0.8 - Tt -t 2.0474- Is : 0.8
(9.27)
The values of the fixed end temperatures,Io : 40 and Ts:200, can be substitutedand moved to the right-handside.The resultsarefour equationswith four unknownsexpressed in matrix form as
a; ffi' 'it;]{i^l:{}i}
(9.28)
So our original differentialequationhas been convertedinto an equivalentsystemof lineal algebraicequations.Consequently,we can usethe techniquesdescribedin this chapter to solve for the temperatures.For example, using MATLAB >> A=12.A4 -1 0 0 -r 2.04 -1 0 a -r 2 .0,1 -L t) 0 -1,2.A4); >> b=i40.8 0.8 0.8 200.81'; >> T= (A\ b) ' T= 93.',i18a
65.9698
'r24.5382
r59.41 9a
A plot can also be developed comparing these results with the analytical solution obtained with Eq. (9.25), .
L
j'
i
-
t;
>> x=[0:2:101; >> xanal=f0:101; >> TT-0 (r) 13.452 -j*exp (0.1*xanal) (-0.1*xanal)+20; t-,r.rj=1:(Ydl--); >> plot it ,'l , 'o' , ranal,
-53.4523*exp
. . .
Tartal )
As in Fig. 9.8, the numericalresultsare quite closeto thoseobtainedwith calculus.
232
GAUSSELIMINATION
confinued
9.1 | the r (Fig.
Analytical (linel and numerical (pointsl solutions
220
9.2 | 200
4
180
Chec equa
160
9.3 ( 140 r-. 120
(a) S
100
rl (b) c
80
fi
(c) c 9.4 (
60
10
(a) C
FIGURE9.8 versus rod.Boihonolyticol ondnumericol A plotof temperoture distonce olongo heoted {line) lpoints)solutionsore disployed.
(b) u
(c) U tt
(d) s In addition to being a linear system,notice that Eq. (9.28) is also tridiagonal.Wecan use an efficient solutionschemelike the M-file in Fis. 9.6 to obtain the solution: >> e=10 -1 -1 -11; >> f=12.04 2.04 2.04 2.04); >> g=i-1 -1 -1 01. >> r= [40.8 0.8 0.8 200.8] ; >> TridiaS (e, f, q, r )
65.9598
93.7185
r2,4.5382
tc 9.5 C
c
l.(
r59 .4195
The system is tridiagonal because each node depends only on its adjacent Becausewe numberedthe nodes sequentially,the resulting equationsare tridiagonal. casesoften occur when solving differentialequationsbasedon conservationlaws.
(a) So (b) Co (c) On reg (d) Sol (e) Sol Intr
9.6 ci' )
(a) Sol con
PROBLEMS
PROBTEMS Determine the number of total flops as a function of number of equationsn for the tridiagonal algorithrn
9.6). Use thegraphical methodto solve 8 . r t : 2 4 4.rt - { *r 6 , r : :
(b) Substituteyour results into the original equations to check your answL.t's. 9.7 Given the cquations 2r1-6x2-
-r-r:-38
-3rr-;r:*7.r.t:-34
31
your resultsby substitutingrhem back into the
*8rr +
- r 2 - 2 x 1: - 2 0
(a) Solvebv Gaussc.limination with partialpivoting.Show all stepsol the computation. - l . l r 1* (b) Sr"rbstitutc your rcsults into the original equationsto l0-r2: l16 checkyour answers. - 2xt+ ll .4.b : 174 9,8 Pcrforn.rthc samccalculationsasin Example9.5, but lbr graphicallyand check your resultsby substituting the tridiagonal Solve system: them buclinto thc equati(rn\. -0.4 0nthebasisof thegraphicalsolution,what do yor.rexpect Ios l tll . r I -c).4 0.8 - 0.a regarding theconditionof thc system) | | | L1 Compute thedeterminant. {).4 o8j|.r. L Given thesystemof equations 9.9 Figure P9.9 shows three reactorslinked by pipes. As - 3 t : + 1 x ' ,: 2 indicated, the rate of transf'erof chenticalsthrough each pipe is equal to a f'low rate (Q, with units of cubic meters x112.12- .rr : 3 per second)rnultiplied by the concentrationof the reactor :2 5 . t -12 - r 2 fronr which thc flow originates(c, with unirsof milligrarns Compute thedeterninant. per cubic metcr).If the systcmis at a steadystate,the transUseCramer'srule to solve fbr thc r's. f'cr into each reactorwill balancethe transferout. Develop UseGauss eliminationwi(h partialpil,otingto solvetor r.niiss-balarrce equations for lhe reactors and solve the the-r's. threc simultaneous linear algebraic equations for their Substitute your resultsback into the original equations conccntratlons. tocheck yoursolut.ion. Given theequations Given thesystemof equations
:{f,
0 . 5 ; -t 1 . r 2 : 1 . 0 2 -t 2 1x2:
9.-5 -18.8
Qnct
Solve graphically. the deterrninant. {b)Compute (c)0n the basis of (a) and (b), what would you expect regarding the system'scondition? by the elinrinationof unknowr.rs. {d)Solve (t)Solveagain.but with a,, modified slightly to 0.52. your results. Interpret 9,6Giventheequatiolrs 1 0 - *t 12 - r 2 - , r 3 :
200 mg/s Qz(z
Qr.: 120 Qrr: 40 Q,z= 90 Qz::60 Qyt:3o
27
- 3.t1- 6-t2* 2.r3: -61.5 . r 1* . r 2 * 5 . r : : - 2 1 . 5 (l) Solve bv naiveGuusselirnination.Shorr,' all stepsof the computatton.
FIGUREP9.9 Threereociors linkedby pipes Theroteof mosstrcnsfer through eochpipeis equo to theproductof f ow Q cnd concenirotion c of thereccforfromwhichthef ow orgrnoles.
234
GAUSS ELIMINATION
9.10 A civil engincer involved in constructionrcquires the lbllowin-edifl'crentialequationrcsultsfiom a steady-$ate .1800,5800, and 5700 rnr of sand, line gravel, and coarse massbalancefbr a chemicalin a onc-dimensional clnal: gravel,respectivcly,lbr a building plojcct. Thcre arc thrcc d-t llt: 0:D --U ,--kc pits from which thesematerialscan be obtaincd.The cornd.\' d.r pits of thcse is 1-rosition r.'u'hcre c : conccntri.ition, / : tinlc, r : distance,D = diffusion coclllcicnt. U - f-luiclvelocity. and li = a first-order dccay late. Convert this ciitilrcntial ccluationto an equiva. Fine Grovel Sond Coorse Grovel lent systcmof simultaneous algcbraicequations. GivenD = : 311and c(10) : 20. strivethese 2, U : l, k -- 0.2, r'(0.1 P iIr 30 t5 55 equationsfi'om.r : 0 to l0 and dcvelopaplot of concentfa' ?it2 45 25 30 tion velsusdistance. Pir3 2a 25 55 9.13 A stagecxtractionprocessis depictedin Fig.P9.13. In sr-rchsystems,a streamcontainirtga weight tiactionr',,,ofa Hou, nran-vcubic melels rxlrst be hauleclfiorn u'achpit in chemical enters fronr the lcft at a mass f'low late of F,. Siniultaneously, a solvcntcarryinga rvcighttiactionx,nof orderto meetthe cnginecr'snceds? the same entersflom the right at a flow rate0fF,. chemical 9.ll Arr clectricalengineersupervise's theploductionof three Thus, lbr nrassbalancccan be reprcsentcd stagc i, a as typcs of clectrical componcnts.Three kinds of rr-raterial' ) t
rnctal.plastic,anclrubber-are rctpriledtirl production.The amountsneededto prodLrcceachcomponentare
Componenf
Metol (g/ component)
Plosric (9/ component)
t5 l7 t9
03c o4a
I
2 J
Rubber (9/ component)
o 2
0 5-5
(P9.l 3n) At cach stage.an equilibriumis assumcdto be established bctween,r', andr, as in ''i K : {P9.ll}) wheleK is calleda distribution Equation(P9.13b) coefficient. can be solvedfbr -r,and substitutcdinto Eq. (P9.13a)to yield F, \ /F, \ rPg.ljir | (;=K Ir, ':g r- t' r a /) r ' , \ r t / lf Fr : 500 kg/h, .\'in: 0. 1, F.: 1000kg/h, rin = 0, and K : 4, delerminethe valuesof _r',,u, anclx.,,,,if a tlve-stage rcactoris uscd.Note that Eq. (P9.l3c) must be modified to accountfbr thc inflou, wei-shtliactions when appliedto the filst and laststages. r' ,-
If totalsof 3.89,0.09-5,and 0.282 kg of rnetal,plastic,and lubber',respectivcly,arc availablecachday, how nranyconrponentscan be producedper day'? 9.12 As cle.scribed in Scc. 9.3, lineal algebraicequationscan arise in the solution of difl'erentialcquations.For example.
/ ('\
FIGUR
9.14 A viscous pipe se( mechan flows ir to obtai Q, I Qt) 3Qt
o,= Qt=
o.=
9 . 1 5A followi BC,AL
74k
P9.I3 FIGURE A stoqe extroclionprocess
<_FIOW:
A_
-Flow:
F,+
FIGUI
PROBLEMS
235 -lJ - CD -AD r DE
A,+AD:o A,*AB:0 74+ BC+(3/s)BD:0 AB - G/s)BD:0
A / 5 t CE : 0 (3l5tBD:0
CD+ (415)BD:O - D E - ( 3 / s ) C E: 0
-BC+(315)CE:0
E \ . 1( 4 l s ) C E - 0
9.1(t A pentodiagonal systemwitlr a bandwidth of flve can generally beexprcssed as .ft ,9r hr e2 .fz 8z h2
P9.r4
dt
e1
h
ht
Br
pump deliversa unit llow (Q, ) of a highly 9.11A peristaltic viscous tluid.The network is depictedin Fig. P9.14. Every section hasthe samelength and dianreter.Thc massand I energybalancecan be simplitied to obtain thc in everypipc. Solve the following systemof equations in theflow in every stream.
d,,
t
€tt I d,
Q z + 2 Q t -2 Q : : O Q s t 2 Q o- 2 Q r : 0
€rt
8n
I
.f,,
rl r2 r3
,{l
3 Q .- 2 Q ( , : 0
Jr-t
.rl
0,=0,+O, Qt= Qti
Qs
Qs=Qo* Q,
fttl
15A trussis loaded as shown in Fig P9.15. Using the in-esetof equations,solve for the l0 unknowns.AB. ,AD.BD.CD. DE, CE, A,. A,, and E,
24 kN
xrt
Developan M-file to etficientlvsolvesuchsystemswithor,rt pivotingin a similar fashionto the algorithmusedfbr tridiagonal matrices in Sec.9.4.1.Tcst it fbr the following case:
-2
lo Lo
P9.r5
-l 9 -4 3 al 1 ,1 0 -7
t_" t_;
0 I lf
-3
:ltiitil
..
LU Foctorizotion
OBJECTIVES CHAPTER The primary objective of this chapter is to acquaint you with LIJ factorizationt. Specific objectivesand topics coveredare .
' . .
.
Understandingthat LIJ factortzationinvolves decomposingthe coetlcient matrix into two triangularmatricesthat can then be usedto efficiently evaluatedifferent right-hand-sidevectors. Knowing how to expressGausseliminationas an LU factortzation. Given an LIJ factorization.knowing how to evaluatemultiple right-hand-side vectors. a Recognizingthat Cholesky'smethodprovidesan efficient way to decompose symmetricmatrix and that the resultingtriangularmatrix and its transposecanbe usedto evaluateright-hand-sidevectorsefficiently. Understandingin generalterms what happenswhen MATLAB's backslash operatoris usedto solvelinearsystems.
s describedin Chap. 9, Gausselimination is designedto solve systemsof linear algebraicequations:
[A]{r}: {b}
(i0.1)
Although it certainly representsa soundway to solve such systems,it becomesinefficienl when solving equationswith the sarnecoefficients[A], but with different right-hand'side constants{b}. | [n the parlanceof numericalmethods,the tenns "factorization"and "decotttposition"are synonyrnous. Tobe consistentwith the MATLAB doculnentation,we havechosento employ the terminology LU factori:ationlor the subiectof this chapter.Note that LLl tlecompositionis very comntonly usedto describethe sameapproach
OVERVIEW OF [U FACTORIZAIION Recallthat Gausseliminationinvolvestwo steps:fbrwardeliminationand back substitution(Fig. 9.3).As we learnedin Section9.2.2,the forwald-elimination stepcompriscs the bulk of the conrputational etlbrt.This is particularlytruefor largesystemsof equations. LU factoriz.ationmethodsseparatethe time-consumingelimination of the matrix [A] from the rnanipulationsof the right-handside {b}. Thus, once [A] has been "factored" or "decornposed," multipleright-hand-side vectorscan be evaluatedin an efTicientmanncr. Inter-estingly. Gausseliminationitselfcanbe cxpressed as an LU facrorizttlion. Bcfole shclwinghow this can be done.let us lirst providea mathernatical oven,iewof the factorizationstrategy.
I O.I
OVERVIEWOF LU FACTORIZATION Just as was the casewith Gausselimination.LU factorizationrequirespivotingto avoid divisionby zero.However,to simplifythe followingdescription, we will omit pivoting.In addition,the fbllowing explanationis limited to a setof threesimultaneous equations. The resultscan be directlyextencled to ir-dimensional systems. Equation( 10.I I can be rearranged to give
l A l { r } - { b }: 0
( 1 0 . 2)
Supposethat Eq. ( 10.2)could be expressed as au uppertriangr"rlar system.For example,fbr a3x3svstenr:
['l
Ll12 ,r': Lltt
0
l/ rr
I
:1t_
I
,r,.-l
..,f-
d1 dt dz
t 0.3)
Recognizethat this is similar to the manipLrlation that occursin the first step of Gauss elimination. That is, elimination is used to reducethe system to upper triangular form. Equation(10.3)can alsobe expressed in rnatrixnotationancirearranged give 11'r
t u l { . r -} { d }: 0
(r 0 . r )
Now assumethat thereis a lower diagonalnatrix witl.rl's on the diagonal, t/l-
tLt
Ir
-
0 0l
li.,
I
0l
( r0 . 5 )
Ll, /r: l l that has the propertythat when Eq. (10.a)is premultipliedby it, Eq. (10.2)is the resulr. That is.
t t l I t u ] { , r-} { c t l l : t A l { r } - [ b ]
(10.6)
If this equationholds,it follows fiom the rulesfor matrixnultiplicationthat
lLllul: lAl
(r 0 . 7 )
ILlldj : lbl
(10.8)
and
238
IU FACTORIZATION
tAl {.r}: {ll}
tut f..torirution
A
{ lul
lLl
I
tLl
._-
{d}: {b}
I
V {d}
I
.lul
Substitution
V ir) : trl)
,".0
I
t {ri
|,o
F I G U R EI O . I ihe stepsin 1Uloctorizotion.
A two-stepstrategy(seeFig. I 0. I ) for obtainingsolutionscan be basedon Eqs.( 10.3), ( 1 0 . 7 )a, n d( 1 0 . 8 ) : 1. 2.
LU factorizationstep. [A] is factoredor "decomposed"into lower [L] and upper[U] triangularmatrices. Substitutionstep.[t] and [U] areusedto determinea solution{x } for a right-hand side aninter{b}. This stepitselfconsistsof two steps.First,Eq. (10.8)is usedto gerrerate mediatevector{z/}by forward substitution.Then,the resultis substitutedinto Eq.(10.3) which can be solvedby back substitutionfbr {x }.
Now let us shorvhow Gansselinrinationcanbe imnlementedin this wav.
I o.2 GAUSSETIMINATIONAS [U FACTORIZATION Although it might appearat face value to be unrelatedto LU factorization,Gausselimination can be usedto decompose[Al into [Ll and IU]. This can be easilyseenfor [U], which is a direct product of the fbrward elimination.Recall that the forward-eliminationstepis intendedto reducethe original coefficientmatrix [A] to the form
ot' ac 4rr -l f ,I a i , ar., lUl:10 I t--
L0
;;l 0 ass J
whiclr is in the desireduppertriangularformaf.
ri AS IU FACTORIZATION I O.2 GAUSSELIMINATION Though it might not be as apparent,the matrix [L] is also producedduring the step. This can be readily illustratedfor a three-equationsystem, f a r r o t z . 7 1I3 -rl I o., a22 ott I r r 1o.; aj2 ar )
_ti.: l;:
The first stepin Gausselimination is to multiply row I by the f'actor[recall Eq. (9.9)] 0il
Jzt:-
0)l
and subtractthe result from the secondrow to eliminatea.,. Similarly,row I is multiplied by A2t | 1t Ltll
and the result subtractedfiom the third row to eliminatea,,. The final stepis to multiply the modified secondrow by ..
t..-| 1: -
a\, azz
and subtractthe result from the third row to eliminatea-,,. Now supposethat we merely perfbrm all these manipulationson the matrix [A]. Clearly,if we do not want to changethe equations,we alsohaveto do the sameto the righthand side {b}. But thereis absolutelyno reasonthat we have to perform the manipulations Thus,we could savethe/'s and manipulate{b} later. simultaneously. Recall that the whole idea behind the Where do we store the factors/,,,J.r, andf...r'? elirninationwas to createzerosin at1,a,,t,and a3,.Thus,we can storerr in a.,,f., in a.r, andl,, in a.,,.After elimination,the [A] matrix can therefbrebe written as
l - . , ',
If,
{/lt apl
ur or,I
(r0.10)
L,t , .fp ai:,) This matrix, in fact, representsan efficient storageof the LU factorizationof [Al, lAl -
(10.1l)
tLltul
where f lUl :10
on
a12 n', T a ' r , a \ , ,1
(r0.r2)
0 0l 1 0l
0 0 .r 3 )
L o t o;1.)
and
[r 7rl:lft
L .fsr .f:z 1I The following exampleconfirms that [A] : tLl[n.
I.UFACTORIZATION E X A M P L El 0 . l
l U F o c t o r i z o t i o nw i t h G o u s s E l i m i n o t i o n Problem Stqtement. Derive an LU factorizationbasedon the Gauss eliminationoerformed previously in Example9.3. Solution. ln Example 9.3, we used Gausselimination to solve a set of linear algebraic equationsthat had the following coefficientmatrix:
-0.1 -0.21 [ 3 -0.3 7 lAl:10.1 1 0 . 2 r0 I 10.3 After forward elimination,the fbllowing uppertriangularmatrix was obtained:
-0.r -0.2 [3 I 0 : 1 0 7 . 0 0 3 3 3 . 2 9 3 3 313 tul 1 0 . 0 1 2I0 0 L0 The f'actorsemployedto obtain the uppertriangularmatrix can be assembledinto a lower triangularmatrix. The elementsd21and d11were eliminatedby using the factors
. f u: ,
0.1
:0.0333333 /,, :,
0.3
:0.1000000
and the elementd12woSeliminatedby using the factor -0.19 .f., :
7J0333
: -0.0271.100
Thus. the lower triangularmatrix is
1 0l 0 [ r 0l tll :10.0333333 L 0.100000-0.0271300r J Consequentlv. the LU factorizationis tI -
I
0
otT3 tt
-0.1
-0.2 I
I r 0 | l 0 7 . 0 0 3 3 3- 0 . 2 9 3 3 3| 3 [A):[LilU):10.033333] 1 0 . 0 1 2 I0 0 1 0 . 1 0 0 0 0 0 - 0 . 0 2 7 1 3 0 0l l L 0
Thisresultcanbe verifiedby performingthemultiplication of [L][L/]to give [
3
L
0.3
-0.1
lLltul:l}.jeeeeee 7
-0.2
-0.3 1 I
-0.2 9.99996 )
wheretheminordiscreoancies aredueto roundoff. Afier the matrix is decomposed. a solutioncan be generatedfor a panicularright-handside vector {b}. This is done in two steps.First, a forward-substitutionstepis executedby solving Eq. (10.8)for {d}. It is importantto recognizethat this merely amountsto performside ing the eliminationmanipulationson {bl. Thus, at the end of this step,the right-hand
EXAA
IO.2 GAUSSELIMINATION AS TUFACTORIZATION will be in the samestatethat it would have beenhad we performedforward manipulation on [,4]and {}} sirnultaneously. The fbrward-substitutionstepcan be representedconciselyas i-l
s-, , tl,:fi, Ll,ibi
fori :
1.2.....n
j:l
The second step then merely amounts to irnplementingback substitutionto solve Eq. ( 10.3).Again, it is importantto recognizethat this is identicalto the back-substitution phaseof conventionalGausseliminarionlcomparewith Eqs.(9.l2) and (9.l3)l; t,
:
drf Q,,, il
d; -
u11x1 | j:i+1
'fl:-uii
,l0.2 EXAMPLE
T h e S u b s t i t u t i oS n teps Problem Stotement' Completethe probleminitiateciin Example10.1 by generatingthe final solution with forward and back substitution. Soluiion. As just stated,the intent of forward substitutionis to imposethe elimination manipulationsthat we had formerly appliedto [A] on the right-hand-sidevector {b}. Recall that the systembeingsolvedis
-0.r -92lJ,yr | - 3|. 7 - 0 . 3 . , I [ 7 . 8 s I l0 -0.2 l l l : l - l e . . rI Lo.: r o I [ * r | | t r . +J andthattheforward-elimination phaseof conventional Gausseliminationresultedin T3
-0.1
-0.2
(
7.85
tfxr I -02e3333 : | -'i?i'tI 7oo,i3 I. | Ilj l 0 t0.0lt0 lrr L0 zo.os+:
-J
l
I
J
The forward-subsriturion phaseis implementedby applyingEq. (10.g):
o,J..,.., ? 3l{:l fo L 0 . 1 0 0 0 0 0 - 0 . 0 2 7 1 3 0 0I J [ , 1 r
7.85 -19.3 1t.4
or multiplying out the left-handside: : d1 7.85 0.0333333dr+ :-19.3 d2 0.100000dr-- 0.0271300d2+ d3 11.4 We cair solve the tirst equation for d, : 7.85, which can be substitutedinto the second equationto solve for d : .:
-19.3 * 0.0333333(7.85 : ) -19.5617
242
TUFACTORIZATION Bothd, andd, canbe substituted into thethirdequation to give : 70.0843 dt : 7 1.4- 0.1(7.85)+ 0.02713(-19.,5617) Thus.
: {;r*y {,/} This resultcan thenbe substituted into Eq. (10.3),IU]{.r} : {di:
-0 .r
13
-0 .2
I
xl
0 7.00333 -0.2e3333 | | 0 1 0 . 0 1 2 0) L0
;#*vl
whicl.rcanb e s o l v e db 1 h a c ks u b s t i t u t i o(see n Example9.3 for details)for the final solution:
I :. I
{d\ :
I t o,io''.. I
The LU factorizatioualgorithmrequiresthe sametotal flops as for Gausselimination. The only differenceis that a little lesseftbrt is expendedin the factorizationphasesincethe operationsare not appliedto the riglrt-handside.Conversely,the substitutionphasetakesa little more eflbrt.
lO.2.l MATTABFunction:ru MMLAB syntax:
has a built-in function lu that generatestheLU factorization.It has the general
lL,t-t)
=
lu (X)
where I and u are the lower triangularand uppertriangularmatrices,respectively,derived fron-rthe LIJ factorizalionof the matrix -x.Note that this function usespartial pivotingt0 avoid division by zero.The following exampleshowshow it can be employedto generate both the factorization and a solution fbr the same problem that was solved in Examp l e s 1 0 . 1a n d 1 0 . 2 . E X A M P L E1 0 . 3
t U F o c t o r i z o t i o nw i t h M A T L A B Problem Stotement. Use MATLAB to compute LheL(l factorization and find the solutionfor the samelinear systemanalyzedin Examples10.I and 10.2:
-0r f .r
I or Inr
-_02llri
1 2 . 8 s1 7 -o.ill,, |l: l-ro.: I _o1 ro ll*, I
ltt.ql
GAUSSELIMINATION AS IU FACTORIZATION
Solution. The coetlicientmatrix and the right-hand-side vectorcan be enteredin standard fashion as >> A = t3 -.1 .2;.I >r b = 17.85; -19.3;
1 .3;.3 'lI.4l;
-.2
I0);
Next, the LU factorizationcan be computedwith
L-
1.0000 0.0333 0 . 1 ( 10 0
0 1.0000 -t.r.a27I
0 0 1.0000
3.01r00 a 0
0.1000 1.0033 0
-0.2000 0.2933 t].ai2a
This is the sameresultthat we oblainedby hand in Example 10.1. We can testthat it is correct by computingthe original matrix as >> L*LT
3. C)001r' -0. iL'00 1.0000 0.1004 0.2000 0.3000
-0.2000 0.3000 10.0000
To generate the solution. we flrst cornpute >> d = L\b d= ?.8500 19 . 5..11 70.0843 And then Llsethis result to compute the solution >> x = U\d
3.0000 2.5000 7.0000 These results conlbrm to those obtained by hand in Example 10.2.
244
IU FACTORIZATION
FACTORIZATION I O.3 CHOTESKY Recall from Chap. 8 tlrirta s.vmmetricnratrix is one whereeij : aji for all i andj. Inother rvords,[Al : IA)r. Such systemsoccLlrcommonly in hoth mathematicaland engineering/ scienceproblemcontexts. Specialsolution techniquesare availablefbr such systems.They offer computational advantagesbecauseonly half the storageis neededand only half the computation timeis requiredfor their solution. One crf the most popular approachesinvolves Choleskt .factoriz.ntion(alsocalle/ Choleskydecornposition). This algorithmis basedon the fact thata syrnmc'tric matrix can be decomposed. as in
tr o.u)
l A l :l u fw l
That is. the resultingtriirn-eular factorsare the transposeof eachother. The termsof Eq. ( 10.l4) ciln be rnultipliedout and setequalto eachother.Thefactotization can be generatted efTicientlyby recurrencerelations.For the lth row:
(l0.rt
1l
ai1 -
upltl11
|
1._l
forj-i+1,...,n
'
EXAMPLE 10.4
(r0.r6)
tlii
CholeskyFoctorizotion ProblemStotement.Compute theCholesky factorization for thesymmetric matrix I A t -l?rt
1.5 s5 I 5.5 22s I 1 5 5 225 919)
Ib
I r-5
Solution. Forthefirstrow(i : l). Eq.(10.15) is ernployedtocompute u11:Jan:16:2.44949 Then.Eq.(10.l6) canbeusedto deten.nine otz q
tt
-
Llll at3
.r l.r -
L ll l
l-5
:6.123124 2.44949 5-5 :22.45366 2.44949
For-thesecondrow (l : 2):
: ,f a.. - rrl, : ,/55 1r.2
: 4.1833 (6.12372q2
EXAM
FACTORIZATION I 0.3 CHOLESKY 423 -
245
225- 6.| 23124(22.15366) :10.9165 4 .I 8 3 3
Ul2ut3 Ll22
F o r t h e t h i r dr o w ( i : 3 ) :
u . r : , fo . .- , i t * u l . : @ : 6 . 1 l o l o l Thus, the Choleskyfactorizationyields
tul :
r 1 JJq4q t-
| L
6 I 11711
4.1833
11 457661 |
2 0 . 9 1 6 5| 6 .I l 0 l 0 l I
The validity ofthis factorizationcanbe verified by substitutingit and its transposeinto Eq. (10.l4) to seeif theirproductyieldsthe originalmatrix [Al. This is leflforan exercise.
After obtaining the factorization,it can be used to determinea solution fbr a righthand-sidevector {b} in a mannersimilar to LU factorization.First, an intermediatevector {d} is createdby solving IUlr {d} : {b}
(10.17)
Then,the final solutioncan be obtainedby solving
l u l { x }: l d } lO.3.l
(r0.r8)
MATTAB Funcfion: cuot
MATLAB has a built-in function chol that generatesthe Choleskyfactorization.It hasthe generalsyntax, t/ - chol (x) where y is an uppertriangularmatrix so that t/' * t/ = r. The following exampleshowshow it can be employedto generateboth the factorizationand a solutionfor the samematrix that we looked at in the previousexample.
EXAMPLE I O.5 Cholesky Foctorizotionwith MATLAB Problem Stotement. Use MATLAB to computethe Choleskyfactorizationfor the same matrix we analyzedin Example10.4.
[6 t A l: l l s 155
ls ssl ss 22sl 2 2 5 9 1 9J
Also obtain a solutionfor a right-hand-sidevector that is the sum of the rows of [A]. Note that for this case.the answerwill be a vector of ones.
246
TUFACTORIZATION
The matrix is entered in standard fashion as
Solution. :'> A =
f 6 l5
55;
15 55 225;
55 225 919);
A right-hand-side vector that is the sum of the rows of [A] can be generatedas >> b =
fsum(A(1, :))/'
sum\tt(2,:))
;
sum(A(3,:)ll
16 295 7259 Next, the Cholesky tactorization can be computed with >> U = chol(A)
u= 6 . r23t 4.1813 0
2 .4495 0 0
22 .45,31 2[).91{,.5 6.1101
We can test that this is correct by computing the original matrix as >> U' *IJ
6.0000 15.0000 55.0000
15.0000 55.0000 225.0000
55.0000 225.0000 979.0000
To generate the solution. we flrst conrpute >> d = A'\b d= 3r.0269 25.0998 6.1101 And then use this result ti'r compute the solution
l0.l Dete of equatio tion, and( versionof 10.2 Use E q s (. 1 0 . 10.3 Use systemac
10.r1-3xr xrj
>>x=A\y
Then,mul minethat I 10.4Use Prob.10.3 thesystem
r.0000 1.0000 1.0000
lblr :
I O.4 MATTABIEFTDIVISION We previouslyintroducedleft division without any explanationof how it works.Nowthu we have some backgroundon matrix solution techniques,we can provide a simplifid -t^.,^-:^ti
^^
^f
:*-
^-^-^ri.,-
10.5Solv factorizat 2xr -3xr -
247
PROBLEMS
When we implement left division with the backslashoperator,MATLAB invokes a highly sophisticatedalgorithm to obtain a solution. ln essence,MATLAB examinesthe structureof the coefficient matrix and then implementsan optimal method to obtain the solution.Although the detailsof the algorithmarebeyondour scope,a simplified overview can be outlined. First, MAILAB checks to see whether [A] is in a format where a solution can be obtained without full Gausselimination. These include systemsthat are (a) sparseand banded,(b) triangutar(or easily transformedinto triangularform), or (c) symmetric.If any of thesecasesare detected,the solution is obtainedwith the efTicienttechniquesthat are availablefbr such systems.Some of the techniquesinclude bandedsolvers,back and forward substitution,and Choleskyfactorization. If none of thesesimplified solutionsare possibleand the matrix is square.'a general triangular factorizationis computed by Gausselimination with partial pivoting and the solution obtainedwith substitution. 2 It should be noted that in the event that solutionis obtainedwith an approach [Al is not square,a least-squares c allecJQR .fac t or i zati on.
PROBTEMS the total flops as a function of the number l0,l Determine equations n for the (a) factorization,(b) forward substituand(c)back substitutionphasesof the lU factorization of Gausselimination Usethe rules of matrix multiplication to prove that (10.7) and(10.8)follow from Eq. (10.6). Usenaive Gauss elimination to factor the following to the descriptionin Section 1 0 . 2 : according
27 l2rz- xt: - 6rz + 2x',: -61 .5 I x z + 5 a : . : - 2 1. 5 multiplythe resulting [L] and [{l matricesto deterthat[A] is produced. UselU factorizationto solvethe systemof equationsin 10.3.Showall the stepsin the computation.Also solve svstem for an alternativerisht-hand-side vector
Ibf = Lt2 18 - 6) Solvethe following system of equations using LU with partial pivoting: 2x1-6x2- 13:-38 -3.r1- -r2*7x3 : -34 -Rr, -L
-)v.
-
-)O
10.6 Developyour own M-file to determinerheLU factoization of a squarematrix without pafiial pivoting. That is, develop a function that is passedthe squarematrix and returns the triangular matrices [Ll and [U]. Test your function by using it to solve the systemin Prob. 10.3.Conlirm that your f'unctionis working properlyby verifying that [L][Ul : tA] and by using the builrin function 1u. 10.7 Confirm the validity of the Cholesky factorizationof the resultsinto Eq. (10.14)to Example10.4by substituting verify that the productof l L/17and l Ll yieldslA l. l0.tl (a) Perform a Choleskyfactorizationof the fbllowing symmetricsystemb1 hand:
tl ilill{i:}:t*l (b) Verify your hand calculation with the built-in cnol function. (c) Employ the resultsof the factorizationIL4 to determinethe solutionfor the right-hand-sidevector. 10.9 Develop your own M-file to determinethe Cholesky factorizationof a symmetric matrix without pivoting. That is, develop a function that is passedthe symmetric matrix and returnsthe matrix [tIl. Test your function by using it to solve the systemin Prob. 10.8 and use the builrin function is working
248
IU FACTORIZATION
10.10 Solve the following set of equationswith LU factorization: 3 x 1- 2 x 2 * . r r : - 1 0 Z.rt]_6xz-4x7:
factorizationto determineIU] sothat 10.13UseCholesky
LA): W)rlul: 44
[2 l-r
-1 0l 2 ,tl
Lo -r
2)
- - x 1- 2 t 2 * 5 . 1 3 : - 2 6 1().14 Computethe Choleskyfactorizationof 10.U (a) Determine the LU factorizationwithout pivoting by hand for the fbllowing matrix and check your resultsby validatingthat [L][U] : [A|.
1l [8 2 7 2l l3 L2 -r e_.j
0l
Lo o 4-l
Do your results make sensein terms of Eqs. (10.15)and 0 0 . 16 ) ?
(b) Employ the result of (a) to compute the determinant. (c) Repeat(a) and (b) using MAILAB. 10,12 Use the following LU factorizalion to (a) compute tl.redeterminantand (b) solve [Al{r} : {b} with {b]r:
L- r0 44 ,26). lAl:tLlfUl:
[ e 2o, s o l
IAl:10
[1 |I 0 . 6 6 6 7
I
L -0.3333 - 0 . 3 6 3 6 , ]
'
[3
L
-2
-^.lrr*l
1.3333
3.6364)
Motrix Inverseond Condition
CHAPTEROBJECTIVES The primary objectiveof this chapteris to show how to computethe matrix inverse and to illustratehow it can be usedto analyzecomplex linear systemsthat occur in engineeringand science.In addition,a methodto assessa matrix solution'ssensitivity to roundofTerror is described.Specificobjectivesand topicscoveredare o r r o r
t.l
Knowing how to determinethe matrix inversein an efficient mannerbasedon lU factorization. Understandinghow the matrix inversecan be usedto assessstimulus-response characteristicsof engineeringsystems. Understandingthe meaningof matrix andvectornormsandhow they arecomputed. Knowing how to use norms to computethe matrix conditionnumber. Understandinghow the magnitudeof the conditionnumbercan be usedto estimatethe precisionof solutionsof linear algebraicequations.
THEMATRIXINVERSE In our discussionof matrix operations(Section8.1.2), we introducedthe notion that if a matrix [A] is square,thereis anothermatrix [A]-r , called the inverseof [A], for which
t A l t A l - ': [ A ] - r t A l: t / l
(r1.1)
Now we will focus on how the inversecan be computednumerically.Then we will explore how it can be usedfor engineeringanalysis. I l.l.l
Colculoting the Inverse
The inversecan be computedin a column-by-columnfashionby generatingsolutionswith unit vectorsas the right-hand-sideconstants.For example,if the right-hand-sideconstant
250
MATRIXINVERSE AND CONDITION has a I in the tirst positionand zeroselsewhere, tt .h' t\ --
s)
the resultingsolutionwill be the first column of the matrix inverse.Similarly,if a unitvector with a I at the secondrow'is used [b]:{
f0l l}
{il,1}
tol
the resultwill be the secondcolumnof the matrix inverse. The bestway to implementsucha calculationis with LU factorization.Recallthatone of the greatstrengthsof LU factorizationis that it providesa very efficientmeansto evaluate multiple righrhand-sidevectors.Thus, it is ideal fbr evaluatingthe multiple unitvecton neededto computethe inverse. E X A M P L EI L l
MqtrixInversion Problem Stotement. Employ LU factorizationto determinethe matrix inversefor the systemfrom Exaurple10.l: -or -o)l I l t-""-l A l : 1 0 . 1 7 _ 0 . 31 10.3 -0.2 10 _.1
Recall that the lactorizationresultedin the following lower and uppertriangularmatrices:
[r
-o.r
-0.2 I
I
t
-0.293333 7.00333 tul:10 1 1 0 . 0 1 2_0j 0 L0
o
o'l
1 0l ttl:10.0333333 L 0 . 1 0 0 0 0 0- 0 . 0 2 7 1 3 0l J0
Solution. The tirst column of the matrix inversecan be determinedby perfbrmingthe forward-substitutionsolution procedurewith a unit vector (with I in the first row)asthe right-hand-sidevector.Thus, the lower triangularsystemcan be setup as (recallEq.[10,8])
[r*-0.1,,.ui] {l}:{i}
-0.03333 -0.10091. Thisvector and solvedwith forward substitr.rtion for {d}r : ll can then be used as the right-handside of the upper triangularsystem(recallEq. [0.3]):
[: l0
lo
-or
-02
l[.] 7 . 0 0 3 3 . -10 . 2 e 3 3 3l.l1. : l : l o
r o . o r 2I [o' , J
[
r
I
0.0.133 13
[-orooeJ
which can be solvedby back substitutionfor {.r}r : 10.-1-1249-0.00518 -0.01008j which is the first column of the u.ratrixinverse:
f o.zzz+so o-l l A l ' : l - o . o o s r so o l I
nntooR
n
n I
THEMATRIXINVERSE
'lllil:{:}
To determinethe secondcolumn, Eq. (10.8) is formulatedas
[ro
l | 0.0333333 L 0.100000 -0.027r 300
This can be solvedfor irl), and the resultsare usedwith Eq. (10.3)to determine{r}' : 10.004944 0.142903 0.002111,which is the secondcolumn of the matrix inverse:
[Al
':
ol o.zzz+e o.oo4s44 f -0.00sr8 0.142e030
| | L -0.010080.0027r00_j
Finally, the sameprocedurescan be implementedwith {b}/: l0 0 1l to solve fbr , which is thefinal columnof thematrixinverse: {r }7 : 10.006798 0.004l83 0.099881
t
03324so.oo4s44 0.006798l
[A]-' : | -0.00s18 0.1429030.004183 | L -0.01008 0.0021100.0998801
Thevalidityof thisresultcanbechecked by verifyingthat[A][A] ' : Ul.
| 1.t.2 Stimulus-Response Computotions As discussedin PT 3.1, many of the linear systemsof equationsarisingin engineeringand scienceare derived from conservationlaws. The mathematicalexpressionof these laws is some form of balanceequationto ensurethat a particularproperty-mass, force, heat, momentum,electrostaticpotential-is conserved.For a force balanceon a structure,the propertiesmight be horizontalor vertical componentsof the forcesacting on eachnode of the structure.For a mass balance,the propertiesmight be the mass in each reactor of a chemicalprocess.Other fields of engineeringand sciencewould yield similar examples. A singlebalanceequationcan be written for eachpart of the system,resultingin a set of equationsdefining the behavior of the property for the entire system.Theseequations are interrelated,or coupled,in that eachequationmay include one or more of the variables from the other equations.For many cases,thesesystemsare linear and, therefore,of the exactform dealt with in this chapter: [A]{x}: {b}
(11.4)
Now, fbr balanceequations,the terms of Eq. ( I 1.4) have a definite physical interpretation. For example,the elementsof {x} are the levels of the property being balancedfor eachpart of the system.In a fbrce balanceof a structure,they representthe horizontaland vertical forcesin eachmember.For the massbalance,they arethe massof chemicalin each reactor.In either case,they representthe system'sstateor response,which we are trying to determine. The right-hand-sidevector {b} containsthose elementsof the balancethat are independentof behavior of the system-that is, they are constants.In many problems,they representthe or externalstimuli that drive the system.
252
MATRIXINVERSE AND CONDITION Finally, the nratrix of coeflicients [lJ usui,illycontains the ;xrrameter.rthatexpres how the parts of fhe systernirte ract or are coupled. Consequently,Eq. ( I I .4) mi-qht h reexpressed as I I n t e r a c t i o n s l { r e s p o n: s e{ }s t i m u l i } As we know from plevious chapters,there are a variety of ways to solveEq.(11.4). However.using the matrix inverseyielclsatparticularlyinterestingresult.The fbrmalsolution can be expressed as
h : l: [ A l
]{1r}
or (r'ecallingour detlnitionof ntatrixmultiplicationfrom Section8.I .2) .\'t :
L t, L {lt I /)l -f {/l, D. -f rlt l /)r
.\'J :
I (l)t l)l
. \ ' . r:
lr l/rl l)t +
f
l, r, rr.. lr. -f rt.. l.r1 l, {l{. lrl +
L (lr-t /tt
Thus, we lind that the invertedmatrix itself',asidetiom providing a solution,hasextrernelyuseful properties.That is, eachof its elementsrepresentsthe responseof asingle part of the systemto a unit stimulusof any otherpart of the system. Notice that theselbrmulationsare Iinearand, theretbre,superpositionandproportionality hold. Supe.rltositirnmeansthat if a systemis subjectto severaldifTerentstimuli(the b's), the responses individuallyand the resultssummedto obtainatotal can be corr-rputed response.P roportionalin' meansthat multiplying the stimuli by a quantityresultsin there. sponseto thosestimuli beingmultipliedby the sarnequantity.ThLrs,the coefficientour is a p r o p o r t i o n a l i t y c o r l s t a n t t h a t g i v e s t h e v a l udeuoef .tro-a1u n i t l e v e l o f l r s . T h i s r e s u l t r s independentof the eft-ectsof b2 and 6: on ,r1, which are retlected in the coefficients no' trnda,.', respectively. Therefore,we can draw the generalconclusionthattheelement a,,toftheinvertedmatrixrepresentsthevalueofx,cluetoaunitquantityofb,. Usin-ethe exanrpleof the structure,elementrr,l' of the matrix inversewouldrepresent the fbrce in member i due to a unit externalforce at node1. Even for sntal.lsystems, such behaviorof individualstimulus-response interactions would not be intuitivelyobvious. As such.the matrix inverseprovidesa powerful technique1brunderstandingthe interrelationshipsof componentpartsof complicatedsystems.
E X A M P LI E I.2
A n o l y z i n g t h e B u n g e eJ u m p e rP r o b l e m Problem Stotement. At the beginningof chap. 8, we set up a probleminvolvingrhree individualssuspendedvertically connectedby bungeecords.We deriveda s)'stemof linear algebraicequationsbaseclon fbrce balancestilr eachjuniper,
-roo o I - rso r 0 0 r s O - s 0I
| L 0
--50 s0 I |
I,I _
-ss8.6 I
;;I- etiil
In Example 8.2, we used MATLAB to solve this system for the vertical positionsof the jumpers(ther's). In the presentexample,useMATLAB to computethe matrixinverseand
I I . 2 E R R OA RN A L Y S IASN D S Y S T E M CONDITION
253
Solution. Start up MAILAB and enterthe coefficienrmatrix: t150 -100 0; 100 150
>> K -
50;O -50 501;
The inversecan then be computedas >>
Kf
KI
=
=
inv(K)
0.0200 0.0200 0.0200
0.0200 0.0300 0.0300
0.0200 0.0300 0.0500
Each elementof the inverse,k, I of the invertedmatrix representsthe vertical change in position (in meters)of jumper I due to a unit changein fbrce (in Newtons) applied to Jumper/. First, observethat the numbersin the first column ("1: l) indicatethat the positionof all threejumpers would increaseby 0.02 m if the force on the first jumper was increased by 1 N. This makessense,becausethe additionalforce would only elongatethe first cord by that amount. ln contrast,the numbersin the secondcolumn (j :2) indicatethat applying a force of I N to the secondjumper would move the first jumper down by 0.02 m, but the second and third by 0.03 m. The 0.02-m elongationof the first jumper makes sensebecausethe first cord is subjectto an extra I N regardlessof whetherthe force is appliedto the first or secondjumper. However, for the secondjumper the elongationis now 0.03 m because along with the first cord, the secondcord alsoelongatesdue to the additionalforce.And of course,the third jumper showsthe identicaltranslationas the secondjumper as thereis no additionalforce on the third cord that connectsthem. As expected,the third column (..j:3) indicatesthat applying a force of I N to the third jumper resultsin the first and secondjumpers moving the samedistancesas occured when the force was appliedto the secondjumper. However,now becauseof the additional elongationof the third cord, the third jumper is moved fartherdownward. Superpositionand proportionalitycan be demonstratedby using the inverseto determine how much fartherthe third jumper would move downwardif additionalforcesof 10, 50, and 20 N were appliedto the first, second,and third jumpers,respectively.This can be done simply by using the appropriateelementsof the third row of the inverseto compute, A r - j : / . 3 r r A F r+ k r j a f ' 2 + k ; r A F r : 0 . 0 2 ( 1 0 )+ 0 . 0 3 ( 5 0 )+ 0 . 0 5 ( 2 0 ): 2 . J m
II,2
ERRORANALYSISAND SYSTEMCONDITION Aside tiom its engineeringand scientificapplications,the inversealsoprovidesa meansto discernwhether systemsare ill-conditioned.Three direct methodscan be devisedfor this purpose: l.
Scalethe matrix of coefficients[A] so that the largestelementin eachrow is l. Invert the scaledmatrix and if there are elementsof [A] | that are severalordersof magniq\/{fcm
254
MATRIXINVERSE AND CONDITION 2. 3.
Multiply the inverseby the original coefficientmatrix and assesswhethertheresultis closeto the identity matrix. If not, it indicatesill-conditioning. Invert the invertedmatrix and assesswhetherthe resultis sufficientlycloseto theorig. inal coefficientmatrix. If not, it againindicatesthat the systemis ill-conditioned.
Although thesemethodscan indicateill-conditioning,it would be pret'erable to obtail a singlenumberthat could serveas an indicatorof the problem.Attemptsto formulate such a matrix condition numberare basedon the mathematicalconceDtof the norm. | 1.2.1 Vector qnd Motrix
Norms
Anonn is a real-valuedfunction that providesa measureof the size or "length"of multi. componentmathenaticalentitiessuchas vectorsand matrices. A simple exampleis a vectorin three-dimensional Euclideanspace(Fig. l1.l) thatcan be representedas b
lFl:la
cl
where c, b, and c are the distancesalong the r, 1,,and z axes,respectively.The lengthof this vector-that is, the distancefrom the coordinate(0, 0, 0) to (a, b, c)-can besimpty computedas lFll. : {/o\
h, + f
where the nomenclaturell F ll,, indicatet that this length is referred to as the Euclideannorn
of Ln.
Similarly, for an n-dimensionalvector lX,l : would be computedas
t.
llXll": if \
i
i:r
F I G U R EI I . I Grophicoldepictton of o vectorin Euclideon spoce
l"r1 x2
.rnj. a Euclideannorm
ERROR ANALYSIS AND SYSTEM CONDITION
The conceptcan be extendedfurther to a matrix [A], as in ll.{l tinll/
\- \-
-
/-J
/r--t
"? I
(il.s)
which is given a specialname-the Frobenius notm. As with the other vector norms, it providesa singlevalueto quantifythe "size" of [A]. It slrouldbe notedthat therearealternativesto the EuclideanandFrobeniusnorms.For vectors,thereale alternativescalledp norms that can be representedgenerallyby /
,,
\ll
lt /iX ' t lt ll t - :l L ' 'l,\t -
",
1/'l
\,-l
I
/
We can seethat the Euclideannorm and the 2 norm,llX ll, , areidenticalfor vectors. Otherimportantexamplesare (p : 1) rYr
- \-r..r i= |
whichrepresents thenormasthesumof theabsolute valuesof theelements. Anotheris the maximum-magnitude or uniform-vector norm(p : oo). llXll-: max,l.ril whichdefinesthe normastheelementwith thelargestetbsolute vallle. Usinga similarapproach, normscanbe developed for matrices. For example. llAllr : .maxu)-- .r;,1 l - i '- , 1 i-l
That is, a summationof the absolutevaluesof the coetlicientsis performedfor eachcolumn, and the largestof thesesummationsis taken as the norm. This is called the columnsum ,tot'tn. A similar determinationcan be made for the rows. resultine in a unifbrm-matrix or row-sumnorm:
l l A l l . : , l.aut a^n f -
l,,,7l
1:1
It should be noted that, in contrastto vectors,the 2 norm and the Frobeniusnorm for a matrix are not the same.Whereasthe Frobeniusnorm llAllf can be easily deterrninedby Eq. ( t I .5),the matrix 2 norm llA llz is calculatedas l lA l l : : ( / r n ' " * ) ' ' " where4,,,"*is the largesteigenvalueof [A]r[A]. In AppendixA, we will learnmore about eigenvalues. For the time being,the importantpoint is that the llAll2. or spectralnorm, is the mirrimum nonl and,therefore.providesthe tightestlreAsureof size (Ortega,1972).
11.2.2 Motrix ConditionNumber Now that we have introducedthe conceptof the norm, we can use it to define C o n d l A l: l l A l l ' l l l - ' l l
MATRIXINVERSE AND CONDITION where Cond[A] is called the nntix conditiort number Note that for a matrix [A], this numberwill be greaterthan or equalto l. It can be shown (Ralstonand Rabinowitz,1978: Gerald and Wheatlev.1989)that
l^xll ,.. .ll^Ail = condlAl il"il Mil That is, the relative error of the norm of the computedsolutioncan be as large astherelative error of the norm of the coefficientsof [A] multiplied by the conditionnumber.Forexarnple,if the coefficientsof [A] are known to t-digit precision(i.e.,roundingerrorsare0n the order of 10-r) and Cond[Al : 10', the solution[X] may be valid to only r - cdigits (roundingelrorsry l0' -' ). E X A M P L EI 1 . 3
M o i r i x C o n d i t i o nE v o l u o t i o n Problem Stotement. The Hilbert matrix,which is notoriouslyill-conditioned,canberepresentedgenerallyas
[r
I
+
1 'i l
lt l,
I
4
,+r I
;
l,
I
ll l: : : : l1 l -
i
-
Lfl
I
-
nrl
- l
nll
ltl
I I
tJ
Use the row-surnnorm to estimatethe matrix conditionnumberfor the 3 x 3 Hilbertmatrix;
t-,1
lr i i'
t A' l : l l{t
tl
L3
I. \ rI II I
I
ll
5l
Solution. First.the matrix canbe normalizedso thatthe maximumelementin eachrowis1:
rAl:
il iJ
fr1 | Ll
Summing each of the rows gives L833, 2.1667, and 2.35. Thus, the third row hasthe Iargestsurnand therow-sumnorm is 31
ll.4ll-:l+;+;:2.35 +i
The inverseof the scaledmatrix can be computedas
I g
-18 10l l
r) b - b u l | _ . -e0 60 -l L 30
lAl-l:l-tr'
Note that the elementsof this natrix are larger than the original matrix. This is also reflected in its row-sum norrn.which is comoutedas
EXAM
AND SYSTEM CONDITION ANALYSIS I 1.2 ERROR
257
Thus. the condition numbercan be calculatedas Cond[,A]: 2.35(192\: 451.2 The fact that the condition number is much greaterthan unity suggeststhat the system is ill-conditioned.The extentofthe ill-conditioningcan be quantifiedby calculatingc log rounding could exhibit of the solution :2.65. digits Hence,the last threesignificant 451.2 errors. Note that such estimatesaln-rostalways overpredict the actual error. However, they are usefulin alertingyou to the possibilitythat roundoff elrors may be significant.
| 1.2.3 Norms ond Condition Number in MATLAB MATLAB has built-in functionsto computeboth norms and conditionnumbers: >> norm (X, p) ano ._ cond{X,p)
where x is the vector or matrix and p designatesthe type of norm or condition number(1 , z , inf , or ' f ro'). Notethatthe cond functionis equivalentto >> norm(X,p)
*
norm(inv(X)
,P)
setto 2' Also,notethatif p is omitted,it is automatically I 1.4 MotrixConditionEvoluotion with MATLAB EXAMPTE Problem Siotement. Use MAILAB to evaluateboth the norms and condition numbers for the scaledHilbert matrix previously analyzedin Example 11.3:
i2 r. rl l
['
l^
tAr:lr I + | t_-l
lr L'
I 4
tl sl
(c) As in Example I 1.3, first computethe row-sum versions(p : inf ). (b) Also compute the Frobenius(p :' fro' ) andthe spectral(p : 2)conditionnumbers. Solution: (a) First,enterthe matrix: >> A -
l\
\/2
I/3;I
2/3 Ii2;L
3/4 3/51;
Then, the row-sum norm and conditionnumbercan be computedas > > n o r m (A , i n f )
2.3504 > > c o n d (A , r n f )
258
MATRIXINVERSE AND CONDITION These results con'espond to those that were calculated by hand in Example 11.3. (b) The condition numbers based on the Frobenius and spectral norms are >> cond (A, ' fro'
)
368.0866 >> cond (A)
366.3503
"'i
"i.;'j\';1
Il.
As the name implies, indoor air pollution deals with air contamination Bockground. in enclosed spacessuch as homes, offices, and work areas.Supposethat you are studying 'N Guzzle, a truck-stop restaurantlocatedadjacent the ventilation system for Bubba's Gas to an eight-lane freeway. As depicted in Fig. I L2, the restaurantserving areaconsistsof two rooms for smoken and kids and one elongatedroom. Room I and section 3 have sourcesof carbon monoxide from smokers and a faulty grill, respectively. In addition, rooms I and 2 gain carbon monoxide from air intakes that unfortunately arepositioned alongside the freeway.
F I G U R EI I . 2 Theone-woyorrowsrepresent volumetric oirflows, Overheodview of roomsin o restouront. whereosiheMo-woyorrowsrepresent diffusive mixing.Thesmoker ond grillloodsodd corbon monoxide mossfo thesystem butnegllgible oirflow. 0,.= 150m3/hri I Qt = 50 m3lhr
2 (Kids'section)
ct = 2 mglm3
Q":20Q m
Qa: 1oo
I 25 m3lht 4 c+ t %_l_-l El I oi I
3lhr 1 ( S m o k i n gs e c t i o n )
-3 cu = 2 mglm
Smoker load (1000mg/hr) )
>
25 m3/hr J
I G r i l ll o a d
CASESTUDY trli':'l);
. .
i
.l
. .
continued l
,
Write steady-statemass balancesfor each room and solve the resulting linear algebraic equationsfor the concentrationof carbonmonoxide in eachroom. In addition, generate the matrix inverse and use it to analyze how the various sourcesaffect the kids' room. For example,determinewhat percentof the calbon monoxidein the kids' sectionis due to (l) the smokers,(2) the grill, and (3) the intakevents.In addition,computethe improvementin the kids' sectionconcentrationif the carbonmonoxide load is decreasedby banning smoking and fixing the grill. Finally, analyzehow the concentrationin the kids' area would changeif a screenis constructedso that the mixing between areas2 and 4 is decreasedto 5 mj/hr. Steady-statemass balancescan be written for eachroom. For example.the Solution. balancefor the smoking section(room l) is QncoQocr *f13(ca-c1) (Load) *(lnflow) (Outflow)f (Mixing)
0:lVrmore.*
Similar balancescan be written for the other rooms: 0 : Qt,ct,+ (Q' - Qa)c'+- Q,cz'l E2a@a- c2) 0 : wern * Qnct I EnGt - c:) * Ey(!+ - cs) - Qoct 0 : Qoc3* Ez+Gt- c+) * Ezq(cz- c+)' Qoc+ Substitutingthe parametersyields the final systemof equation:
}:'t-l l:t*i;!i,,;r]{;i MATLAB can be used to generate the solution. First, we can compute the inverse. Note that we use the "short g" format in order to obtain five significant digits of precision: >> fcrrrrat short g >> A= 1225 A -25 0 0 175 0 -125 225 0 274 -5C 25 -254 215); 0 >>
A L=f
lT
-
nV
(A,j
0.0449962 0.0034483 0.0049555
I.432 6e-005 0.0062069 0.000r3193
^
n
nAtQ-)aC
nnn 6aaG6
0.00055L12 0.0034483 0.0049655 n n n t 9 ) , -6
0.00010728 0.0034483 0.00096552
260
$
MATRIXINVERSE AND CONDITION '' i''
.,;,0:j{;..., "..;S,'.
{ continued . ; " ,.i,15,j: The solution can then be generatedas >> b=11400 r00 2000 0l , ; >>
c=Af *b
c= 8.0996 1)
?/q,
L6.897 16.483 Thus, we get the surprising result that the smoking section has the lowest carbon monoxidelevels! The highestconcentrationsoccur in rooms 3 and4 with section2having an intermediate level. These results take place because(a) carbon monoxide is conservative and (b) the only air exhaustsare out of sections2 and 4 (Q, and 0a). Room 3 is sobad becausenot only does it get the load from the faulty grill, but it also receives the effluent from room l. Although the foregoing is interesting, the real power of linear systems comesfron using the elementsof the matrix inverse to understandhow the parts of the systeminteract. For example, the elementsof the matrix inverse can be used to determinethe percentof the carbonmonoxide in the kids' sectiondue to eachsource: The smokers: : uitr Wr*ok* : 0.0034483(1000) : 3.4483 c2,smokers : 7o,moke,.
? 4/l'9.?
d
1L345
x l00vo : 2'7.93Vo
The grill: cz.snl: arytWsrin: 0.0034483(2000): 6.891 Vo";t:
6.897 12.345
x l00%o:55.877o
The intakes: : arl Q,c' + arrt Qt ct,: 0.0034483(200)2 + 0.0062069(50)2 c2,irrrakes : 1.37931* 0.62069: 2 ) V'"snrr r ' , , :' 12.345 The faulty grill is clearly the most significant source. The inverse can also be employed to determine the impact of proposedremediessuch as banning smoking and fixing the grill. Becausethe model is linear, superpositionholds and the results can be determined individually and summed: Lc2: a;tt Lw, ok",+ aur LW-*lr:0.0034483(-1000) + 0.0034483(-2000) : _ 3 . 4 4 8 3_ 6 . 8 9 6 6 : _ 1 0 . 3 4 5
il.I
I l0 -3
Check useaI
rl
ii
PROBTEMS
li i
continued Note that the samecomputation would be made in MATLAB as >> AI \2,I)* (-1000)+AI (2,3\* (-2000) -10.345 Implementingboth remedieswould reducethe concentrationby 10.345mg/m3.The result would bring the kids'room concentrationto 12.345 - 10.345 :2 mg/mt. This makes sense,becausein the absenceofthe smokerand grill loads,the only sourcesare the air intakeswhich areat 2 mg/m3. Becauseall the foregoingcalculationsinvolved changingthe fbrcing functions,it was not necessaryto recomputethe solution.However,if the mixing betweenthe kids' areaand zone 4 is decreased,the matrix is changed
-2s 0 f225 0 l tl J C l r 0 s r s s 0 0 | -22s 0 21s -s0 I I | -5 -250 0
L
2s5Jt
C4
:l {*}*l
The resultsfor this caseinvolve a new solution.Using MATLAB. the resultis Cl
c2
C4
l:{r*tr1
Therefore, this remedy would only improve the kids' area concentration by a paltry 0.265 mp/m3.
PROB[EMS D e t e r m i n e t h e n r a t r i x i n v e r . s e f o l tl o hw e ingsystem: ll.2 Detelnrinethematlixinversefbrthefollowingsystem l0.rr-F2.rz- -rj : 2'7 -3r1 -6.12*2r::-61.5 r 1* , r 2 * 5 x . j : - 2 1 . 5 yourresults by verifyingthat [A][A]-r : [1] Donot aplv0tlngstratcgy.
-8rr+
x2-2x3:-/Q 2x1-6x2- xr:-38 3r1 - -t2*'7x., : -34
ll.3 The following system of equationsis dcsignedto determine concentrations(the c's in g/m3) in a series of
262
MATRIXINVERSE AND CONDITION
(a) Dcterminethe condifionnumberbasedon therow-sum n o r m f o r t h ec a s ew h e r e . r l- 4 . . x z : 2 , a n d . r := 7 . (b] Usc MATLAB to cornputethc spcct.raland Frobenius l 5 c 1- 3 c 2cl :3800 conclitionnur.nbcrs. 3 c r+ l 8 c : 6 r ' .:11 2 6 9 11.10 Usc MATLAB to determine the spectralcondition -4.i t : * l 2 c - l: 2 3 5 0 nutnbertbr a I0-dinrcnsiontrlHilbclt nratrix.How manydig. (a) Dctcrnrincthe lnatri.\inverse. its ofprccisionareexpected to bc lostdueto ill-conditioningl Deterrr-rine thc solutionfor this systernfbr thc casewhere each {b) Usc the inverscto dcterminethc solution. (c) Detcrminchorv much the ratc of rnassinpLttto reactor3 elcnientofthe righrhand-side r,ector{b} consists ofthesum. in its low. In otlrerwords.solve lnust be increascdto inc[rcca l0 g/m3rise in the con- mationof the coefTicicnts for thc casewherc all thc unknownsshouldbe exactlyone.Com. c e n l r u t i ou n l r c a c t o rl . (d) How much will thc concentraticrnin reactor 3 be rc- pal'c thc resultins crrors with those cxpectcdbasedonthe duced if thc rate of mass input lo reactorsI and 2 is conditionnumbcr. l l . l l R e p e a tP r o b . l l . l 0 , b u t f b r t h e c a s e o f a s i x . reducedby 500 and 250 g/day,rcspeclively? ll..l Determinelhc rni"rtrixinversefbr thc systcnrdescribcd dinrensionalVandcnrondcmatrix {sce Prob. ll.9) wnere i n P r o b . 8 . 5 . U s e t h e m a t r i x i n v c r s et o d e t c r m i n et h e -fr : 4. It = 2, -I: : 7.-r-+: 10.xs : 3. and"16:5. concentrationin reactor ,5 if the inflow concentrationsare ll.12 The l,owerColoradoRivcr consistsof a series oftbur rescrvoirsas shownin Fie. Pl l. 12. changedto c1l : 20 and ci;.1: -50. Mass balancescan be written fbr each reservoir. ll.5 Dcterminethe matrix invcrselbr the systemdc'scribed and the lbllowing setof sirnultancous in Prob. 8.6. Use the matrix inverscto dcterrninethe fbrce in Iinearalgehraic equalions thc thrcc nrembers(,Fr. & and F-r) if thc vertical load at resuIts: node I is doubledto F1.,,: -2000 lb and ahorizontalload t3.122 0 0 0 of F3.1,- -500 lb is applieclto nodc -3. t3.422 t2.252 0 0 ll.6 Dctelmine f o r a n d l l A l l r .l l A l l1 . llAllcouplcd reactorsas a function of thc antountof nrassinput tcr e-achreactor(the righrhand sidesin g/day):
z [t rAt: I -e I ls -l L
l.l
o 0
t2.252 0
12.3'71 0 12.377 n.797
i-l
:tl_
Belbre detcrmining thc norms, scale thc rnatrix by making the nraxirnumclementin eachrow equalto one. :[ll.7 Determincthe Frobeniusarrdrorv-sumnorms fitr the systemsin Probs.I 1.2and I 1.3. where the right-hand-sidcvector consistsof the loadingsof l l.ll Use MATLAB to deternlinethc spectfalconditionnurn- chloride to eacho1-thcfbr"rrlakesalld {rl. c2,r'r, andc',= fie ber for the fbllowin-csystem.Do not nornralizethe systcm: rcsulting chloriclecoltcentratiousfor Lakcs Powell,Mead, Mohave.and Havasu.respectively. 1 s 16 r,-l (a) Use the matrix invelse to solve tbr the concentrationsin [t e 16 2s 361 eachof thc fbur lakes. l4 e 1 6 2 s 4 e l 3 6 (b) How much must the loading to Lake Powellbereduced I 2s 36 1e 641 firr thc chlolide concentrationof Lake Havasuto be75? l16 (c) Using the column-sumnorm. computethe condition L2.5 36 4e 61 8r .l nurnbcr and how many suspcctdigits would begenuCompute the condition rrumberbascclon the row-sur.nnorrn. atedby solvingthis systcnr. ll.9 Bcsidesthe Hilbert matrix. there arc olher matrices ll.l3 (a) Dcternrinethc rratrix invcrseandcondition num. that are inherentlyill-conditioned.Onc such case is the ber fbr thc fbllowing rnatrix: Vantlcnnotttlerzalnr, u'hich has thc fbllou,ing tbrm:
['i r 'I ll l-rl ,r'
l'1,. rl
[ ' 5t 6, ll 14 Lt8eJ
( b ) R c p e a(t a ) b u t c h a n c ea . , s l i s h t l vt o 9 . 1 .
ll.l4 Po unique (z Such poll f(x):
P
where th, way for r algebraic the coeff
PROBLEMS
Upper Colorado River
Lake Powell
Lake Havasu
F I G U R EP I I . I 2 TheLowerColorodoRiver
coefficientsof the fourth-orderpolynomialf(r) : p6" * pzx3t pzx2+ p$ + psthat passesthroughthe following five points:(200,0.746),(250,0.675),(300,0.616),(400,0.525), and (500, 0.457).Each of thesepairs can be substitutedinto (Pl 1.14) l p2x"-2+ . . + pn rx * pn Eq. (Pl1.l4) to yield a systemof five equationswith five thep's are constant coefficients.A straightforward unknowns(thep's). Use this approachto solve for the coeffor computingthe coefficients is to generate n linear ficients. In addition, determineand interpret the condition number. equationsthat we can solve simultaneouslyfor that we want to determine the coefficients. Suppose Polynomial interpolationconsistsof determiningthe (n - 1)th-orderpolynomial that fits n data points. polynomials havethe generalform,
,','"
t,
Iterotive Methods
CHAPTER OBJECTIVES The primary objectiveof this chapteris to acquaintyou with iterativernethodsfor solving simuitaneousequations.Specificobjectivesand topicscoveredare ' ' ' '
Understandingthe differencebetweenthe Gauss-Seideland Jacobimethods. Knowing how to assessdiagonaldominanceand knowing what it means. Recognizinghow relaxationcan be usedto improve the convergenceof iterative methods. Understandinghow to solve systemsof nonlinearequationswith successive substitutionand Newton-Raphson.
terative or approximatemethods provide an alternativeto the eliminationmethodi I describedto this point. Such approachesare similar to the techniqueswe developed to I I obtain the roots of a single equationin Chaps.5 and 6. Those approaches consistedof guessinga value and then using a systematicmethod to obtain a refined estimate 0f the root. Becausethe presentpart of the book dealswith a similar problem-obtaining thevaluesthat simultaneouslysatistya setol'equations-we might suspectthat suchapproximate methodscould be useful in this context. In this chapter,we will presentapproaches for solving both linear and nonlinearsimultaneousequations.
l2.l
IINEAR SYSTEMS: GAUSS-SEIDEL The Gauss-Seidelmethodis the most commonly used iterativemethod for solvinglineu algebraicequations.Assumethat we are given a setof il equations: lAl{"r} : {b} Supposethat for conciseness we limit ourselvesto a 3 x 3 setof equations.If thedragonal elementsare all nonzero,the first equationcan be solvedfor.r1, the secondfor x2,andthe
EXAM
12.1 LINEAR SYSTEMS: GAUSS-SEIDEL
265
third for x.r to yield
xi
, Dr-4ttX.
t:l
-clrr-{l ,-!
(t2.1a) atl
, i tl D) -AtqXi -d)t.t;
(12.llr)
a22
bt-ayrl-a:txl
(12.rc)
1133
whereT and j - I are the presentand previousiterations. To start the solution process,initial guessesmust be made for the x's. A simple approach is to assumethat they are all zero.Thesezeroscan be substitutedinto Eq. (12.1a). which can be used to calculatea new value for -r1 : bt latr. Then we substitutethis new valueof r1 alongwith the previousguessof zerofor-rr into Eq. (12.1b)to compute'anew value for x2. The processis repeatedfor Eq. (12.1c)to calculatea new estimatefor x3. Then we return to the first equationand repeatthe entire procedureuntil our solutionconverges closely enoughto the true values.Convergencecan be checkedusing the criterion that for .lli' Jri -r/'I e,,.': l--l lr;l :XAMPLE l2.l
x 1 0 0 %< r ,
rll rr
G o u s s - S e i d eMl e t h o d ProblemStqtemeni. Use the Gauss-Seidel methodto obtainthe solutionfor 3xr-0.1r2-0.2x3: 1.85 0.lxr* 7.r2-0.3xj: -19.3 0 . 3 1 1- 0 . 2 t 2 + l 0 r 3 71.4 Note that the solutionis ixlr :
|3
-2.5
i ).
Solution. First,solveeachof the equationsfor its unknownon the diagonal: 7.85+0.1,r2f0.1yl .f1
. !
'10 1
3 - 19.3- 0.lrr t 0.3.r,r 1 1 1.4- 0.3xr -l 0.2x2
:
By assuming that;r2andx3 arezero,Eq.(E12.1.1) canbe usedto compute ^t
-
7.85+0.r(0)+0.2(0)
:2.616661
( Er 2 r. . l ) (Er2.1.2) (E12.1.3)
266
METHODS ITERATIVE into Eq'(El2'l This value,alongwith the assumedvalueof -rr : 0, can be substituted to calculate
- 1 9 . 3- 0 . 1 ( 2 . 6 1 6 6 6 + 10). 3 ( 0 )-
,1
1qL\)L
The first iteration is completeclby substitutingthe calculatedvaluesfor 'rt and'r: into E q . ( E 1 2 . 1 . 3t )o y i e l d
1 1.4- 0.3(2.616661 ) + 0.2(-2.191524) : 7.005610 l0
a 1 -
For the seconditeration,the sameprocessis repeatedto compLlte
^l
-
I 0) + 0.2(.1'0056 7.85+ 0.r(-2.194524) :2.990551 3 - 1 9 . 3- 0 . 1 Q . 9 9 0 s 5+7 0) . 3 ( 7 . 0 0 s 6 1- 0 -) 1 4 q q 6 r 5 7 1 r.4 - 0.3(2.990s57 ) + 0.2(-2.499625) :1.000291 l0
couldh The method is, therefore,convergingon the true solution.Additional iterations th know applied to improve the answers.However, in an actualproblem, we would not theenorFr true answerapriori.Consequently,Eq. (12'2) providesa meansto estitnate for x1: example, ',
- 2'616661| .,00" : |2'990551
|
2.qe0ss7
:
12.5(/'
|
F o r x 2 a n c l ' r r , t h e e r r o r e s t i m a t o s i l t ' € € , , . 2 : l l 'o8tcl d/ oe ' ' 3 : A ' 0 J 6 c / cN' o t e t h a t ' a s w o (12'2) usu the casewhen determiningroots of a singleequation,fbrmulationssuchasEq' theyensun are met, when they Thus, ally provide a conservativeappraisalofconvergence. that the result is kttown to at leastthe tolerancespecifiedby e''
ud method,it is immediately As eachnew r value is computeclfor the Gauss-Seiclel is converging'th solution if the Thus, value. -r another in the next equationto cleter-n-rine bestavailableestimateswill be enrployed.An altemativeapproach,calledJttcobiiteratiwr' utilizes a somewhatdifferent tactic. Rather than using the latest availabler's, thistch u niqueusesEq. ( 12.I) to computea setof new .r's on the basisof a setof old x's.Thus, forthenefl retaiued are rather but used not immediately are new valuesaregenerated,they iteration. betweenthe Gauss-Seidelmethocland Jacobi iterationis depictedin The clifTerence Fig. 12.I . Although therearecefiaincaseswheretheJacobirnethodis useful,Gauss-Seidel! utilization of the bestavailableestimatesusuallymakesit the methodof preference'
267
G:A U S S - S E I D E L SYSTEMS I2.1 LINEAR
First iteration
x, :
I b2 - d: rrr
\:
(bz-
a 3 1 x . t-
- atzr2 *
arrx)/atr
a v . \ 1 \/ a . 2
- u27x.\fu'
ap:t1)/a1z
- ttyx2)/a1
Second iteration
i r : ( h r - n , . r , - c l , 1 r 1 ) / ,ir 1
xr:
(b2 - a2trt - ar.x.)faa,
,r. :
(bq -
a31r.t - ct.r.r2)/ajz
x, - (bt - azrxt - a.14)/azz
-rr - (b.r
tt.rx, - tt.ttr)f ar. (b)
@l
F I G U R EI 2 . I ond (b)theJocobiiferotive between(o)theGoussSeidel of thedifference C1opn.otdepicrron m e t h o dfso rs o l v l n sql m ui o n e o u si n e o ro l g e b r o iecq u o t i o n s
l2.l.l
Convergence ond Diogonol Dominonce
Note that the Gauss-Seidelmethodis similar in spirit to the techniqueof simplefixed-point iterationthat wls usedin Section6. I to solvelbr the roots of a singleequation.Recall that simple fixed-point iteration was sometimesnonconvergent.That is, as the iterationsprogressed,the answermoved fartherand fartherfrom the correctresult. methoclcan also diverge,becauseit is designedfor linear Although the Gauss-Seiclel is much more predictablethan for fixed-point iteration of converge to ability systems,its that if the following conditionholds,Gauss-Seidel shown lt be can nonlinearequations. will converge: II \i1t
lo;;l:> )
ul i-l
I
l0;;l
(r 2 . 3 )
i+i
That is, the absolutevalue of the diagonal coefficient in each of the equationsmust be in the equation.Such larger than the sum of the absolutevaluesof the other coelTicierrts systenisare said tobe tliugonalb dotninant.This cliterionis sufficientbut not necessary for convergence.That is, although the method may sometilneswork if Eq. (12.3) is not met, convergenceis guaranteedif the condition is satisfied.Fortunately.many englneering and scientific problems of practical imporlance fuhlll this requirement.Therefore. Gauss-Seidelrepresentsa feasibleapproachto solve many problemsin engineeringand science.
268
METHODS ITERATIVE 12.1.2 MATTAB M-file: Gaussseidel Before developing an algorithm, let us first recastGauss-Seidelin a form thiit is compltible with MATLAB's ability to pelfbrm matrix operations.This is done by expresstttg E q .( 1 2 . 1 a) s bt
_
b2
-
; : ..,i.* '
U.t AT
a12 , o-v )
Atl
Cltl
-
_
,,,;,0 all
4f | -.n.r'
ll2-l
;^,
^r^'
ntl
.-nl.l
xi,.* - !!*uc* rl33
a.l
conciselyin matrixform as Noticethatthesolutioncanbe expressed { : r }: { d } - l c l { : r } where
: tfl:';':) {,1} and
anlal 0 [ : l a z t l t t z z 0 tcl osz a,,t a.,t L f lctr
onlottf ay/a2 l 0 -l
Eq. (12.a)is listedin Fig. 12.2. An M-file to irnplernent | 2.1.3 Reloxolion Relaxationrepresentsa slight modificationof the Gauss-Seidelmethodthat is designedto usingEq. ( 12.1).thatvalueis After eachnew valueof r is compr"rted enhanceconvergence. averageof the resultsof the previousand the presentiterations: modifiedby a wei-ahted .r,l'*: lr,lt" + (l
),).rild
(12.5)
where ). is a weighting f'actorthat is assigneda value between0 and2. If ). : 1, (1 - i) is equalto 0 and the resultis unmodified.However.if I is setata value between0 and I . the result is a weightedaverageof the presentand tl.repreviousreIt is typicallyemployedto make Tlris type of modificationis calledwtderrela.ration. sr"rlts. by dampeningout oscillations. hasten conversence converge or to system a ltonconvergent For valuesof .]. from I to 2, extra weight is placed on the presentvalue. In this instance,thereis an implicit assumptionthat the new value is moving in the correctdirection toward the true solution but at too slow a rate.Thus, the addedweight of /. is intendedto improve the estimateby pushing it closer to the truth. Hence,this type of rnodification, which is calledot'errelaratiort,is designedto acceleratethe convelgenceof an alreadyconor SOR. vergentsystem.The approachis also calledsuccessiveot'errelaxcfiion,
LINEARSYSTEMS: GAUSS-SEIDEL
functiol-r x = GaussSeidel (A,b, es,maxtt) ? GaussSeidel: causs Seidel nethod eo x = Gaus:lSei.del (A, b) : causs SeicleI
vrithout % lnput: % A=coefficientnatrix Z b - rlghL hand side vector ( d e f a u l l - - u '.l0 0 0 0 1 2 ) ? es = stop criterion (default = 50) Z maxlt = rnax iteratrons Z output: Z x= soluLionYector
relaxation
{ dr ledSL 2 i I naroin.2,err.l r p . l t d i . g u m e n -s y o q u i r e d , r , F t . d if nargin<4 | isernpty (maxit ) , maxit=50; end if nargin<3 | isempty(es), es=0. 011001;erLd In,nl - si.ze(A) ; if m-=n, error('Matrix A nust be scluare'); end for i = 1:n = 0; c(i,i) = 0i x(i) end x=x'; Fnr
i
-
l.n
= C(1,1;n)iA(i,i) C(i,1:n) end for i = 1:n = b(i)/A(i,i); d(i) end i f ar
n.
while
(1) -
xold F^-
I
;
x; _
1,-
= d(i)-C(i,:)*x; x(i) -- 0 if x(i) - al.s((x(l) ea(i) encl end = rter+1; rter if max(ea)<-gS iter end
- xold(i))/x(i))
*
>= maxit,
end
break,
1-00;
I2.2 FIGURE Mfilefo implemenl MATLAB Gouss-Seidel
The choice of a proper value for ). is highly problenl-specificanclis often detern.rined empirically.For a singlesolutionof a setof equationsit is oftenunnecessary. However,if the system under study is to be solved repeatedly,the efficiency introdr-rced by a wise choiceof ), can be extremelyimportant.Good examplesarethe very largesystemsof linear algebraicequationsthat can occur when solvillg partialdifferentialequationsin a variety of engineeringand scientificproblemcontexts.
270
ITERATIVE METHODS
12.2
EXAM
NONTINEARSYSTEMS The following is a setof two sinrultalleous nonlinearequationswith two unknowns:
"f+"'"2:lg . r l + J . r t . r ; : - )/
(12.to) (r26r)
In contrastto linear systemswhich plot as straightlines (recallFig. 9.l), theseequations plot as curvesoD an .r2 VeTSUS .r1graph.As in Fig. 12.3,the solutionis the inte$ection0f the curves. Justas we did when we determinedrootsfor singlenonlinearequations, suchsystems of equationscan be expressed generallyas . l i ( , t 1-,r 2 . . . . r , ) : 0 ./:(xr,r:,.....r,,) O -
(12,7)
J , ( r t , . r 2 . . . . .{ , ) : 0 Therefbre,the solutionare the valuesof the r's that make the equationsequalto zero.
12.2.1 SuccessiveSubstitution A simple approachfbr solving Eq. (12.7) is to usethe samestrategythat was employed for fixed-point iterationand the Gauss-Seidelmethod.That is, eachone of the nonlinear equa. tions can be solved for one of the unknowns.Theseequatiollscan then be implemenled iteratively to computenew valueswhich (hopefully) will convergeon the solutions. Thil irpproach,which is calledsrcce.rsive substitution,is illustratedin tlre lbllowingexample.
FIGUREI2.3 Grcphicol depictionof the soluiic-,n of irvo simultoneous nor,lineorequoiions
. t 1+ J J t . \ ; = 5 /
I2.2 NONLINEAR SYSTEMS E X A M P L1 E 2.2
27r
s u c c e s s i v es u b s t i t u t i o nf o r q N o n l i n e o r s y s f e m Problem Stotement. Use successive substitutionto determinethe roots of Eq. (12.6). Note that a correctpair of roots is.rl _ 2 ancl,rr : 3 Initiate the computafionwith guesses o f , r 1: 1 . 5a n d . f , z: 3 . 5 . Solution. Equation(12.6a)can be solvedfbr , t r l: -
l0 - "rri ( Er 2 . :I.)
xz
and Eq. (12.6b)can be solvedfor 12 : )/ - J.rl.yi ^ l
(E12.2.2)
On the basisof the initial guesses,Eq. (E12.2.1)can be usedto determinea new value
ofxl:
1 0- ( 1 . - s ) 2 :2.21429 3.5 This resultand the initial valueof x2 - 3.5 can be substituted into Eq. (F,12.2.2)todeter_ mine a new valueof 't2: .yz: 5'7- 3(2.21429)(3.5)2 : -24.3i 516 Thus' the approachseemsto be diverging.This behavioris even more pronouncedon the seconditeration: I0 - 12.21429t2 Lt
-
-0.20910 - 2 4 . 3 75 1 6 :
rz : 57 - 3( -0.20910)(-24.37 5 tO2 :
429.709
ObvioLrsly,the approachis deteriorating. Now we will repeatthe conlputationbut with the original equationsset up in a diff'erent fbrmat. For example,an alternativesolutionof Eq. ( I 2.6a) is
.t' : ./iblT'r2 and of Eq. (12.6b)is
Now the resultsare more satistactory: r, : yTo - 1.5t35 :2.1:,915
5 7- 3 . 5 /--:2.86051
y 3(2.17e4s)
272
ITERATIVE METHODS tr:@:1.94053 51 - 2.86051 : 3.04955 3(1.%0s) Thus, the approachis convergingon the true valuesof "r1: 2 andxz : 3. The previous example illustrates the most serious shortcoming of successive substitution-that is, convergenceoften dependson the mannerin which the equations are formulated.Additionally, even in those instanceswhere convergenceis possible,divergencecan occur if the initial guessesare insufficiently close to the true solution.These criteria are so restrictivethat fixed-point iterationhas limited utility for solving nonlinear systems.
12.2.2 Newton-Rophson Justas fixed-pointiterationcan be usedto solve systemsof nonlinearequations,otheropen root location methodssuchas the Newton-Raphsonmethodcan be usedfor the sanepurpose.Recall that the Newton-Raphsonmethodwas predicatedon employingthe derivative (i.e., the slope) of a function to estimateits intercept with the axis of the independent variable-that is, the root. In Chap.6. we useda graphicalderivationto computethisestimate.An alternativeis to derive it fiom a first-orderTavlor seriesexpansion: ( x r * r ) : f ( x i ) * ( x i + r- x , \ f ' \ x i ) (128) "f wherex; is the initial guessat the root andx;11 is the point at which the slopeintercepts the .r axis.At this intercept,I (.ri*r) by definitionequalszeroandEq. (12.8)canbe reananged to yield l;-1 :
ri -
f (x')
tl29t
^l \xt )
which is the single-equationform of the Newton-Raphsonmethod. The multiequationfonn is derived in an identical fashion.However, a multivariable Taylor seriesmust be usedto accountfor the fact that more than one independentvariable contributesto the determinationof the root. For the two-variablecase,a first-orderTaylor seriescan be written for eachnonlinearequationas f t . iv t : . f i . i - | ( . t 1 , ; * ,-
",,,1H
f z . i + t : , f 2 . , * ( x r . i + r- * r . , l Y f
0 f u' I ( x 2 . 1 . ,-1 - r' "r ; ) ?xz ( x z . i + -r
dlt;
-\'r;) "-"
Exz
(12.l}a)
( 12.10i)
Justas for the single-equationversion,the root estimatecorrespondsto the valuesof .r1and .d2,where/y .;11and .[.111eQualzero.For this situation,Eq. ( 12.l0) canberearranged togive (12.ltal
(t2.ttu
EXAA
Ir
NONLINEARSYSTEMS I I
's Becauseall valuessubscriptedwith i areknown (they correspondto the latestguessor ap. 2 . 1 1 )i s a s e t o f t w o p r o x i m a t i o n ) , t h e o n l y u n k n o w n s o r e " r l , ;a - 1n-d1x 2 , i a 1 . T h u s , E q( 1 linear equations with two unknowns. Consequently,algebraic manipulations (e.g., Cramer'srule) can be employedto solvefor "
3f2., -
t, .J).t n
.{t.i+t :
-tl.i -
OX2
\J\,i a
dx2
(12.12a1
\fi,i \fz,i _ \fi,i \fzl 0x10x2 3x2'dx1 3f ,,, ]fz,i " " t^._ Jl.t
x2,i+t : x2,i -
"
t^,JZ,r
_
n
dXt
t,,_ Jt,r
q
r,Xt
(12.12b)
3fi,i 0fzl * 3fi,i 3fz.i \xt
\xz
3x2 \xy
The denominator of eachof theseequationsis formally referred to as the determinantof the Jacobian of the system. Equation (12.12) is the two-equationversion of the Newton-Raphsonmethod.As in the following example,it can be employediterativelyto home in on the rootsof two simultaneousequations. E X A M P 1L 2E. 3
N e w t o n - R o p h s o fno r o N o n l i n e q rS y s t e m Problem Stotement. Use the multiple-equationNewton-Raphsonmethod to determine of x1 : 1.5 andxz:3.5. rootsof Eq. (12.6).Initiatethe computationwith guesses Solution. First computethe partial derivativesand evaluatethem at the initial guessesof -r and y:
#
: 2 x r+ x z : 2 ( 1 . 5 t *3 . 5: 6 . 5
ofz'o -3xi 2 : 36.i5 : 3(3.5) 3xr
W #
:rr : 1.5 : 32.5 : | * 6xtxz:I + 6( r .5) ( 3.5)
for thefirst iterationis Thus.thedeterminant of theJacobian - 1.5(36.75) : 156.125 6.5(32.s) as at theinitialguesses Thevaluesofthe functionscanbe evaluated 2 . 5 . f r . o: ( l . S ) 2+ 1 . 5 ( 3 . 5 ) 1 0: . f z . :, . t3 . 5+ 3 (1 . 5 ) ( 3 . 5-) 25 l : 1 . 6 2 5 into Eq. (12.12)to give Thesevaluescanbe substituted
- r.62s -2.s(32.s) (r.s) 156.125 -^-?<_
2.03603
- (-2.s)(36.7 s) : r.62s(6.s) 156.t25
2.84388
Thus, the resultsare convergingto the true valuesofxl : 2 and12 : 3. The computation can be repeateduntil an acceptableaccuracyis obtained.
274
ITERATIVE METHODS When the multiequationNewton-Raphsonworks,it exhibitsthe samespeedyquadratic convergenceas the single-equationversion.However,just as with successivesubsfitution. it can diverge if the initial guessesare not sufficiently close to the true roots.Whereas graphicalmethodscould be employedto derive good guessesfor the single-equation case, no such simple procedureis availablefbr the multiequationversion.Although thereare for obtainingacceptablefirst estimates,often the initial guesses someadvancedapproaches must be obtainedon the basisof trial and error and knowledgeof the physicalsystembeing modeled. The two-equationNewton-Raphsonapproachcan be generalizedto solve n simultaneousequations.To do this, Eq. ( I 2.I I ) can be written for the ftth equationas frf
- ; - *., .\'t.r, dxt
I t
, d.l'r,i
l.--r'l.i+t dx2
ih.i
''f
+'
;--lrr.i dxil
rt :
- J (". 1 f
t
"'
t
itf r,,i
.tl.t-::- f <JIt
Jn.i
itJr.i
'l':.t-;t/J2
ofi.i oxil
where the first subscriptk representsthe equationor unknown and the secondsubscript denoteswhetherthe valueor function in questionis at the presentvalue(l) or at thenextvalue (t + 1). Noticethat the only unknownsin Eq. (12.13) arethe,rr.i+rtermson theleft-hand side. All other quantitiesare locatedat the presentvalue (i ) and, thus, are knownatany iteration.Consequently,the setof equationsgenerallyrepresented by Eq. ( 12.l3) (i.e.,with k:1,2,...,n) c o n s t i t u t e as s e t o f l i n e a r s i m u l t a n e o uesq u a t i o n st h a t c a n b e s o l v e d numericallyby the elimination methodselaboratedin previouschapters. Matrix notationcan be employedto expressEq. (12.13)conciselyas
'(r2. t4)
l J l [ r i _ r ]: - t l l + l J l { x i }
wherethe partial derivativesevaluatedat i are written as theJacobianmatri.rconsisting of the partial derivatives:
3ft., ()X1
UI:
0Jt.,
Dfi.i 0r,
OX?
3f2., ofzl 8-r;1 Dxz
oIz.i
:: 0.fn,i iJxr
: 0f,,i
d"h
3f, .i }xz
3xn
The initial and final valuesare expressedin vectorforrr as {x;}r:
l . r 1;
xz.i
x n .)i
and {.t,*r)r:L,rr.r+r
.x2.i+r
) r a , , +l t
Finally, the function valuesat I can be expressedas
{f }r : 1.fi., .fz.i
f,,i )
275
I2.2 NONLINEAR SYSTEMS
Equation(12.14)cirn be solvedusing a techniquesuch as Gausselimination.This processcan be repeatediteratively to obtaiu refined estirnatesin a f'ashionsimilar to the two-equationcasein Example12.3. Insightinto the solutioncan be obtainedby solvingEq. (12.l4) with nratrixinvelsion. Recall that the single-equationversionof the Newton-Raphsonmethodis ^/+l
-
^/
./('ri )
(12.16)
'f \x1l
If Eq. ( 12.14)is solvedby multiplyingit by the inverseof the Jacobian,the resultis {.ri*r}: {t,}- [.1]
'{.lr
12.11)
Clomparisonof Eqs. (12.16) and (12.17) clearly illustratesthe parallelsbetweenthe two equations. In essence, theJacobianis analogous to thederilativeofa rnultir"ariate tunction. Such matrix calculationscan be implernentedvery efficiently in MATLAB. We can ilhrstratethis by using MATLAB to duplicatethe calculationsfror.nExalr.rple 12.3.After defining the initial guesses!we can computethe Jacobianand the function valuesas >> x=[1.5;3.5]; >>
J=[2*x(1)+x(2)
x(1)
;3"x(2)^2
I+6*x(1)*x(2)
I
T-
6 . 5 t 10 0 36.7500 >> f=tx(1)
1.5000 32.s000 ^2+x(1) *x(2) -10 ;x(2) +3*x(1) *x \2)'2-51
I
f2.5u00 1.6250 Then, we can implementEq. (12.17) to yield the improved estimates >> x=x ,l\f X.-
2.0360 2.'-]/139 Althoughwe couldcontinuethe iterationsin the commandmode,a niceralternative is to expressthe algorithmas an M-file. As in Figure 12.4,this routineis passedan M-file that computesthe fbnction valuesand the Jacobianat a given value of "r.It then calls this function and implernents Eq. (12.l7) in an iterativefashion.The routineiteratesuntil an upper limit of iterations(maxit) or a specifiedpercentrelativeenor (es) is reached. We should note that there are two shortcomingsto the foregoing approach.First. Eq. (12.15)is sometimesinconvenientto evaluate.Theretbre,variationsof the NewtonRaphson approachhave been developedto circunrventtlris dilemma. As might be expected.most alrebasedon using finite-differenceapproxir.nations fbr the partialderivatives that compriseIJ]. The secondshortcomingof the multiequationNewton-Raphsonmethod is that excellentinitial guessesare usuallyrequiredto ensureconvergence. Becausethese are -somefimes difTicult or inconvenientfo obta.in.alternafiveapproachesthat are slower
I i
276
ITERATIVE METHODS
f u n c L i - o r r l x , f , e a , i t e r l = n s . ' , ' 5 m ut 1( f L l n c , x 0 , e s , m a x i t , . z a r a r g i n ) % newtmult: Newton-Raphson root zeroes nonlinear systems ? [x, f,ea,iter]=nev/tmult(func,x0,es,maxit,p1,p2, ... ), % uses the Newton-Raphson method to find the roots of % a system of nonlinear equations ? input: % func = name of function that returns f and J Z x0 - initial guess % es = desired percent relative (default = 0.0001?) error a maxit = maximum allowable (default = 50) iterations p1,p2,... = additional parameters used by function ? ?r output: ? x = \/ector of roots Z f = vector of frrnctions evaluated ar roors 2 ea = approxinate percent- relative (Z) error = number of iterations % iter rf nargin<2,erra]. ('at least 2 input arguments required'),end if nargin<3 |iserLpty(es),es=0.0001;end if nargin<4 |isempty(maxit) ,maxit=50;end icer = 0; x=x0; while (1) IJ, f] =func (x, varargin{ : } ) ; dx=J\ f ; x=x-dx; iter = irer - l; e a = 1 0 0 * m a x( a b s ( d x . / x ) ) ; if iter>=nrax.it lea<=es, break, end
end
FIGUREI2.4 MATIAB Mljleto implement Newton Rophson method fornoniineor syslems of equoiions
than Newton-Raphsonbut which have betterconvergencebehaviorhave beendeveloped One approachis to refbrrrruiatethe nonlinearsysteu.las a singlefunction:
_ \LF(x) _
l . f i G t , x z ,. . . . i r , , ) 1 2
;-l
whereI (x1,,{2,...,.r,,) is the ith rnemberof the originalsystemof Eq. (12.7).Thevalues of r that nlinimize this ftrnctionalso reprcserltthe solutionof the nonlinearsystem. Thfle. fore, nonlinearoptimizationtechniquescan be employedto obtain solutions.
I2.3 CASESTUDY
277
CHEMICAL REACTIONS Bockground' Nonlinearsystemsof equationsoccur frequently in the characterization of chemicalreactions.For example,the following chemicalreactionstakeplacein a closed sysrem:
2 A+ B
--->
C
<-_
(12.r8)
A+D*c (12.19) At equilibrium,theycanbe characterized by Kl :3cicu
(12.20)
Kz- 3-
(12.2t\
cac,t
where the nomenclatureci representsthe concentration of constituent i. If x, and x" are the number of molesof C that are produceddue to the first and secondreactions,respectively, formulatethe equilibrium,relationships as a pair of two simultaneousnonlinearequations. tt : 4_x K, = 3:7.x lo-2, 50, c6.s: 20,c,,o: 5, and cr,6: 10, employ the ll !o-4, .co.o: Newton-Raphsonmethodto solve theie equations.
Sofution. Usingrhe stoichiometry of Eqs.(t2.lg) and(12.19),the concenrrationsof eachconstituentcanbe represented in termsof -r.,and_r,as ca:cu.o-2x1
-
12
(r2.22)
cb:cb,\-xI
(12.23)
cc=cc,o*-tt*.rz
(12.24)
Cd=Cd,\-Xz
(r2.25) where the subscript0 designatesthe initial concentration ofeach constituent.Thesevalues can be substirutedinro Eqs. (12.20) and,(12.21)ro sive
Kr:
(c..0*xL+x2) (co.o- 2x1 - x)2{c6,0 - xr)
(c",0+xr*xz) (c",0* 2x1- x2)@a,o * x:) Given the parametervalues,theseare two nonlinear equationswith two unknowns.Thus, the solution to this problem involves determining the roots of f1@1,x) f 2 @ 1 ,x 2 ):
5*xr+xz (50-2xr-x)2(20-x)
-4x10-a
(50-2xt-x)(10-xz)
- 3.7 x 10-2
(12.26) (12.21)
278
ITERATIVE METHODS iiir";illr ir.1
.,.''
continued
In order to use Newton-Raphson,we must determinethe Jacobianby taking thepartial derivativesof Eqs. (12.26) and (12.27).Although this is certainly possible,evaluating the derivativesis time consuming.An alternativeis to representthem by finite diff'erences inr fashionsinilar to the approachusedlbr the modified secantmethodin Sec.6.3.Forexanple, the partial derivativescomprisingthe Jacobiancan be evaluatedas
dxt
.fi(xt + 6x1,x2)* fi(xr, xz) 3xr
-ofr : 3xz
.fi(xr,xz * 6x2)- J\$r, xz) 3.uz
3fz o.rr
x2)- .ft(xt,.rz) fz(xr + E.r1, 6xr
3fz oxz
fz(xt, xz + d,r2)- .fz\r, xz) 3x,
3fr
These relationships can then be expressedas an M-file to compute both the function valuesand the Jacobianas f unction lJ,lr =jf r-c;rct{x, var.rrgin) del=0.000001; c l f l d x l - ( r r( x ( I ) + c l e l * x ( I ) , x (2) )-rr (x (1). x (2 )) dfldx2= (r (x(1) ,x(2) +del"x(2) ) -u(x(1 ) ,x(2) ) c l f 2 d x 1 = ( . . . ( x( 1 ) + d e l * x ( l - ) , x ( 2 ) ) - . u ( x ( 1 ) , x ( 2 ) ) ( r . (' x ( L ) , x ( 2 ) + d e l * x ( 2 ) ) - v ( x ( f ) , x ( 2 ) ) df 2c1::2= J = f c l f 1 d x 1 d f L d x 2 ; d f 2 c i x i d f 2 r . l x ? , ;1 f 1=u(x(1),x{lt)); f2='1(x(1.),x(l));
t2.t (^) ) ,' (de1*x t 1) ) ; ) / (del.*x(2)); ) ,r(rlel*x(1,)); ) / (ciel*x(2));
(50
-
,a x x
-'!)
^
'2
i
120
O. T I t-o
I L
(b) Rep 12.2 Us< systemu
a-f f 1.f ?t f unct ion f =u (x, !') (5 + x + y) f ,/
system ul
x) - 0.0004;
| 0-r1 -3xt Xl
funct ion f-v (x, y ) f = (i. + x + .,') I
(50
*
-
*
':<,.../)
i
(1Ll -.71
a . a : , - i:
The function nevrtmul t (Fig. 12.4)can thenbe employedto determinethe rootsgiveninit i a l g u e s s eosf x , : x z : 3 : --> forii-.ai, shorL
3.3,r66c+000 2.a,-f 2l+000 f= -'l .I2B5e-01.7 8.5?7-r€,-01'1
a.223 7e 010 i Ler
= 4
e
12.3 Re1 12.4 Tht termine( reactors reactor(1
Solve tl E s: 5 V o 12.5 Ust and (b) v tem to a equation
PROBLEMS
continued a solutionof x, - 3.3366&Ddx, : 2.6772is obtained. Thesevalues After four iterations, into Eq. (12.22)through(12.25)to computethe equilibriumconcanthenbe substituted of thefourconstituents: centrations ca : 50 - 2(3.3366)- 2.6772: 40.6496 cn:20 - 3.3366: 16.6634 cc : 5 + 3.3366 + 2.6112: 11.0138 : ca lO 2.6772:7.3228
PROBTEMS methodto solvethe followine I (a)UsetheGauss-Seidel untilthepercentrelativeerror falls below s, : 5olc:
i1 I 4i;T]{i:
4l 25 l0-s
12.6 Of the following threesetsof linearequations,identify the set(s)that you could not solve using an iterativemethod such as Gauss-Seidel.Show using any number of iterations that is necessarythat your solution does not converge. Clearly stateyour convergencecriteria (how you know ir is n()1converglng).
(a) but use overrelaxationwith I : 1.2 Repeat Usethe Gauss-Seidelmethod to solve the followine untilthe percentrelative error falls below e" - 57o: Jrr+2;r:- jrj: 27 3 r t- 6 x 2 1 } s l : - 6 1 . 5 . r r* - t t * 5 x - r : - 2 1 . 5 Repeat Prob. 12.2but useJacobiiteration. Thefollowing systemof equationsis designedto de(the c's in g/m3)in a seriesof coupled ineconcentrations asa function of the amount of mass inout to each Itheright-handsidesin g/day):
Set One
Set lwo
Set Three
9;* 3y*:: l3 .r*_y*6:: B -3.r*4y*5::6 6;r+B::2 r*5.y ::5 -2:+2t--3::-3 ?r -'-- l 2 r + 5 r -- : : 6 A . rI 2 y - 2 z : 4
12.7 Determine the solution of the simultaneousnonlinear equatlons l':--l-+x+u./)
r'::3800 t- 3c:'r * l8r2 - 6cr : 1200 c2't l2ca: 2350 Ithis problem with the Gauss-Seidel method to Usethe Gauss-Seidelmethod (a) without relaxation (b)withrelaxation(,)": 1.2) to solve the following syst0 a toleranceof s. : 57o. If necessary,reanangethe to achievec'onvergence 2 " t -16 . r ; - " r 3 : - 3 8 - 3 r 1- . t : l ' 7 t . , : - 3 4 -Rr,t
r"-)r.--)O
)*5x-r':-r2 Use the Newton-Raphsonmethodand ernployinitial guesses of .r : _t': 1.2. 12.8 Determinethe solution of the simultaneousnonlinear equatlons: ^
)-)
-J-!
)il:,r2 (a) Graphically. (b) Successivesubstitutionusing initial guessesof r : (c)
puesses
: r,r:
ITERATIVE METHODS
C
Tcct-
>
l
I
o,
-cLr<
ccz-
2Cc:-
l
I
-ctzl
I
cc,t -
F
cc.s -
I
l
l
+' + +
-Cn<
-cu<
-cl-s<
cL6
Qt
L
FIGUREP12.9
12.9 Figure P12.9 depicts a chemical exchange process consistingof a seriesof reactorsin which a gasflowing from left to right is passedover a liquid flowing from right to left. The transferof a chemicalfrom the gasinto the liquid occurs at a ratethat is proportionalto the differencebetweenthe gas and liquid concentrationsin each reactor.At steadystate,a massbalancefor the first reactor can be written for the sas as
derivatives,which result in a system of linear algebraic equations.Use the Causs-Seidelmethod to solve for the temperaturesof the nodesin Fig. Pl2.10.
25"C Qccco- Qcccr+ D(cu-
25"C
ccr):0
and for the liquid as Qrctz - Qrcu I D(ccr - crr) : 0
Tn
r2)
T,,,
Tl
100'c where Q" and Q, are the gas and liquid flow rates, respectively, andD : the gas-liquidexchangerate.Similar balances can be written for the other reactors.Solvefor the concentrationsgiventhefollowingvalues:Q6 :2, Q t: l, D : 0.8, c c o : 1 0 0 c, m : 7 0 . 12.10 The steady-statedistribution of temperatureon a heated plate can be modeled by the Laplace equation:
o:***
dx'
100"c
dy-
If the plate is representedby a seriesof nodes(Fig. P I 2. I 0), centered finite differences can be substituted for the second
75.C
F I G U R EP I 2 . I O
/5
L
t
&
&,
Curve Fitting braic r the
4.1 oVERVIEW WhEr ls Curve Fiffing? Darais oftengivenfor discrere varues arong mayrequire esrimalesat pointsbetweenthe discretevalues. " "11111111; chapters "o*,.1.1you
13 througtrto oescribe-techniques to fit curves to such data to esti-ates. In addi-rion,you rnay require a sirn., ri:" lftaln ilterm"diute plified version of a complicared funcrion. One *uy to ooil;;;;oure values of the : ,::., funct'ionat a numberof discretevalues alon^g . the range. of inr"r"r,. rn.n. a simplerfunction i: may be derived to fir thesevalues.Both of theseapplicationsare known.ascurre trtting. ,,;;
other'onthebasisof rhearnountof errorassqciared ;rh ;;;. ;ii;;;ffi;;;;,.,*'r,,. exhibits a significant degree of erroror"scatter,,, ,t. ,t u,"gfi, t."j.rr". , singre curvethat representsthe generaltrend of the data. Becauseany individualdadpoint *"V l. i"."r.cr. we make no efiort to ln^tersect every point. Rather.the curve is de_ srgnedto follow the pattern of the points .takenas a
'
,!! l\l
s.\,
e
tJi,i';,"):,::!::1'j-"H',narureis"uudt,o,t' squares ,rgrriiion (Fig.pT4. tar).rluuru! wucrs the rue oaul datals is Known knownto ro be very very Dre_ pre_ . fc1nd,. where cise.
the_ basicapproachis to fit.a curve or u ,".1., of curves rhat passdirectly through each of the points. such data usually orii,inatesfrom tables. Examples are valuesForthe density of water or for the heaica_ pacity of gasesas a function of temperature. The esti_ mation of valuesbetweenwell-known discretepoints is called inrerpolarion(Fig. pT4. lb and e. Curve Fittingond Engineeringond Science. your ftrst exposure[o curve fitting may have been to deter_ mtne lntermediatevalues from mbulated data_for instance, from interest tables for *";;;; ;;: or from steam fables for thermodynamics. Tri.r. the remainder of your career,you will ,rnroughout irequentoccasionto estimateintermediate values fave trom such tabJes.
282
4 CURVEFITTING PART
f(x)
(c) F I G U R EP T 4 . I (b)lineor regression, fivedoto points:io) leosl-squores fo fit o "best"curvethrough Threeottempts i n l e r p oo i o n o n o ( c ) c - , r
f l e o f r r t e t o oo l o n .
Although many of the widely usedengineeringand scientificpropertieshavebeentab' ulated,there are a greatmany more that are not availablein this convenientform. Special casesand new problemcontextsoften requirethat you measureyour own dataand develop your own predictive relationships.Two types of applicationsare generally encountered when fitting experimentaldata:trend analysisand hypothesistesting. Trend analysis representsthe processof using the pattern of the data to make predictions. For caseswherethe datais measuredwith high precision,you might utilize interporegression. lating polynomials.Imprecisedatais often analyzedwith least-squares variable. dependent values of the Trend analyslsmay be used to predict or forecast or interpolation observed data This can involve extrapolationbeyond the limits of the within the range of the data. All fields of engineeringand scienceinvolve problemsof this type. A second application of experimentalcurve fitting is hypothesistesting. Here,an existingmathematicalmodel is comparedwith measureddata.If the model coefficientsare
4.2 PARIORGANIZATION to determinevaluesthat bestflt the observeddata.On the ulrknown,it nray be necessary are alreadyavailable,it may be appropriother hand,if estimatesof the model coetTicients ate to comparepledicteclvaluesof the nrodelwith observedvaluesto testthe adequacyof the model. Ofien. alternativenodels are comparedand the "best" one is selectedon the basisof empiricalobservations. curvefitting is irnIn additionto the foregoingengineering and scientificapplications, portantin other nurnericalmethodssuchas integrationand the approxirnatesolutionof dil'Finally,culve-fittingtechniqtres ferentialequations. canbe usedto derivesimplefunctions functioDs. to approximatecornplicated
4.2
PARTORGANIZATION that is. hol'r'to deAfter a brief review of statistics.Chap. l3 focuseson linear regre.ssion; terminethe "best" straightline througha setof uncertaindatapoints.Besidesdiscussing how to calculatethe slopeand interceptof this straightline. we also presentquantitativc and visualmethodsfbr evaluatingthe validity of the lesults.In acldition,we describeser'fbr the linearizationof nonlinearequations. eral approaches of polynomialandmultiplelinearreglessiou. Chapter/4 beginswith brief discussions Pctlynomialregres.sirtn dealswith developinga bc'stfit clf parabolas.cubics,or higher-order polynomials.This is followed by a clescriptionof multiple linear regression,which is designedtbr the casewhere the dependentvariable), is a linear functionof two or more variablesr1, 12,. irrdependent , xtn.This approachhas specialutility for evaluatingexon a nunlberof differentfactors. perimentaldatawherethe variableof interestis dependent we illustratehow polynomialand multiple regressionare After multiple regression, model.Among otherthings,this will allow both subsetsof a generullinear least-squares ofregressionand discussits generalstatisus to introducea concisematrir representation tical properties.Finally.the last sectionsof Chap. l4 ale del'otedtononlinearregression. fit of a nonlinearequaticlnto data. This approachis designedto computea least-squares In Clnp. /5, the alternativeclrrve-tittillg techniquecaTledinterpolutionis describecl. valuesbetween As discussedpreviously,interpolationis usedfbr estimatingintermediate precisedatapclints.ln Chap.15,polynomialsarederivedfor this purpose.We introducethe to connect basicconceptof polynomialinterpolationby usingstraightlinesand parabolas procedureior fitting an rrth-ordelpolynomill.Two points.Then.we developa generalized thesepolynomialsin equationfbrm. The first, called fbrmatsare presentcdfbr expressing orderof the polynopolt'nontial,is preferablewhenthe appropriate Newton'sinterpolotirrg mial is unknown. The second,called the Lagrange interpolotingpoltrutmial, has advantagesrvhenthe properolder is known befbrehand. Finally, Chup. l6 presentsan alternativetechniquefor fitting precisedatapo.ints.This lits polynomialsto databut in a piecewisefashion. calleclsplineirrterytolation, techniclrre, As such.it is particularlywell suitedfbr fitting datathat is genelallysmoothbut exhibits The chapterenclswith an overviewof how piecewiseinterpolationis abruptlocalclranges. imolementedin MATLAB.
i !
':.
Lineor R"gression
CHAPTEROBJECTIVES The primary objectiveof this chapteris to introduceyou to how least-squares regressioncan be usedto fit a straightline to measureddata.Specificobjectivesand topics coveredare ' . ' . '
Familiarizing yourselfwith somebasicdescriptivestatisticsand the normal distribution. Knowing how to computethe slopeand interceptof a best-fit straightline with linear regression. Knowing how to computeand understandthe meaningof the coefficientof determinationand the standarderror of the estimate. Understandinghow to usetransformationsto linearizenonlinearequationssothat they can be fit with linear regression. Knowing how to implementlinear regressionwith MATLAB.
YOU'VE GOT A PROBTEM | we noted thaf a free-falling object such as a bungeejumper is subjecttothe I n Chap. , upward tbrce of air resistance.As a first approximation,we assumedthat this forcewas I I p r o p o r t i o n at ol t h es q u a r eo f v e l o c i t ya s i n Fu : cdu2
(13l)
where Fy: the upward fbrce of air resistance[N : kg m/sr]. c,t : a drag coefficient (kg/m),and u : velocity[m/si. Expressionssuchas Eq. ( I 3.1) come from the lield of fluid mechanics.Althoughsuch relationihipsderive in part fl-orntheor,v.experimentsplay a critical role in their formuiainawind is susperrded riorr.One suchexperinrentis depictcdin Fig. /j. l. An irrcJiridual
285
REGRESSION LINEAR
*
{\'
----+
FIGURE I3.I Wind tunnelexperimenlfo mecsurehow the force of o r reslstonce dependson veocily
1600 1200 z t\
800
40 u,m/s F I G U R EI 3 . 2 Plotof forceversus wind velocifu for on obieclsusoended in o w nd lunnel
TABTEl3.t u, m/s F,N
Experimentol doto for force(N) ond velocity(m/s)fromo wind tunnel exoeriment.
to 25
2A 7A
30 380
4A 550
50 610
60 l22A
70 830
BO I 450
tunnel (any volunteers?)and the force measuredfor various levels of wind velocity. The resultmight be as listedin Table l3.l . The relationshipcan be visualizedby plotting force versusvelocity.As in Fig. 13.2, severalfeaturesof the relationshipbear mention. First, the points indicate that the force increasesas velocity increases.Second,the points do not increasesmoothly,but exhibit rather significant scatter,particularlyat the higher velocities.Finally, althoughit may not be obvious,the relationshipbetweenforce and velocity may not be linear.This conclusion becomesmore apparentif we assumethat force is zero for zero velocity.
ri
286
LINEAR REGRESSION
In Chaps.l3 and 14,we will explorehow to flt a "besf' line or curveto suchdata.In so cloing,we will illustratehow relationships like Eq. ( 13.I ) arisefrom experimental data.
r 3.r
STATISTICS REVIEW Before describingleast-squares regression,we will first review some basic concepts frorn the field of statistics.These include the mean. standarddeviation,residualsumof the squares,and the normal distribution.In addition.we describehow simple descriptive statisticsand distributionscan be generatedin MATLAB. If you are familiar with these subjects,feel tiee to skip the following pagesand proceeddirectly to Sec. 13.2.If you are unf'amiliarwith theseconceptsor rrrein need of a review, the fbllowing materialis designedas a brief introduction.
I 3. t .l DescriptiveStotistics Supposethat in the courseof an engineeringstudy,severalmeasurements were madeof a particularquantity.For example,Table | 3.2 contains24 readingsof the coefllcientof thermal expansionofa structuralsteel.Takenat facevalue,the dataprovidesa linritedamount of information-that is, that the valuesrangefrom a minimum of 6.395 to a maximumof 6.775. Additional insight can be gained by summarizingthe data in one or more wellchosenstatisticsthat convey as much informationas possibleaboutspecificcharacteristics ( l) thelocaof the dataset.Thesedescriptivestatistics aremostotien selectedto l'epresent tion of the centerof the distributionof the dataand (2) the degreeof spreadof the dataset. Meosure of Locotion. The mostcomlron measureof centraltendencyis the arithmetic mean.The urithntetic ntean (l) of a sampleis defined as the sunr of the individualdata points(,r;) dividedby the numberof points(n), or
\-... (1 3 . 2 )
n
wherethe summation(and all the sr.rcceeding summaticlns in this section)is from I = I throughn. There are severalalternativesto the arithmeticrnean.The medianis the midpointof a groupofdata. It is calculatedby first puttingthe datain ascending order.Ifthe numberof measurements is odd.the rnedianis the middlevalue.If the numberis even.it is thearithmetic mean of the two middle values.The medianis sometimescalled the 50th percentile. The mode is the value that occursmost fiequenlly.The conceptusually hasdirectutility only when dealingwith discreteor coalselyroundeddata.For continuousvariablessuch as the data in Table 13.2.the conceptis not very practical.For example,thereare actually TABLE13.2 6 495 6 665 6755 6 56_s
Meosuremenis of the coefficient of thermolexponsionof siructurol steel 6 )i n l { i n . ' ' F ) ] . [(x to 6 595 6 505 6 625 65r5
6 615 6 435 6 . 7 15 6 555
o 635 6 625 6 575 6 395
6 485 6715 6.655 6775
6 55; 6655 6 605 668_5
ii
I3.I STATISTICS REVIEW
287
four modesfor this data:6.555.6.625,6.655,and6.715,which all occurtwice.If the numbers had not beenroundedto 3 decimal digits, it would be unlikely that any of the values would even have repeatedtwice. However,if continuousdataare groupedinto equispaced intervals.it can be an informative statistic.We will return to the mode when we describe h i s t o g r a mlsa t e ri n t h i ss e c t i o n . Meosures of Spreod. The simplestmeasureof spreadis the range, the differencebetweenthe largestand the smallestvalue.Although it is certainlyeasyto determine,it is not considereda very reliable measurebecauseit is highly sensitiveto the samplesize and is v e r y s e n s i t i v teo e x t r e l n ev a l u e s . The rnostcomlnon measureof spreadfor a sampleis thestandarddeviation(s,,) about the mean: ( 1 3 .)3 where S' is the total sum of the squaresof the residualsbetweenthe data points and the mean,or
$: Itr', - ir2
(13.4)
Thus, if the individual measurements are spreadout widely aroundthe mean,s, (and,consequently.s, ) will be large.If they aregroupedtightly, the standarddeviationwill be small. The spreadcan also be representedby the squareof the standarddeviation,which is called the variance: .
r r '
frv; - \:)l
------------l-
n-
|
(1 3 . 5 )
Note that the denominatorin borhEqs.(13.3)and (13.5)is ru- 1. The quantityru- I is ref'enedto as the tlegreesof freedont.Hences, and s,, are said to be basedon rr - I degreesof fieedom. This nomenclaturederivesfrom the fact that the sum of the quantities uponwhichgisbased(i.e.,f-yt,y-)2,...,,)r-y,)iszero.Consequently,ifyis known and rr - I of the valuesare specified,the remainingvalueis fixed. Thus, only n - 1 of the valuesare said to be freely determined.Anotherjustification for diviclingby n - 1 is the fact that there is no suchthing as the spreadof a singledatapoint. For the casewhere n : 7, Eqs.(13.3)and (13.5)yield a meaningless resulrof infinity. We should note that an alternative,more convenientforrnula is availableto compute the variance: -2 ,)r,:
- (Dt',)' l, f u,? n-l
(I3.6)
This versiondoesnot requireprecomputationof ! andyieldsan identicalresultasEq. ( 13.5). A final statisticthat has utilrty in quantifying the spreadof data is the coefficient of variation (c.v.).This statisticis the ratio of the standarddeviationto the mean.As such,it providesa normalizedmeasureof the spread.It is often multiplied by 100 so that it can be expressed in the form of a percent: s). x 1007o (1 3 . 7 )
t
ll
t
288
EXAMPLE l3.i
LINEARREGRESSION
S i m p l eS t o t i s t i cosf o S o m p l e Problem Stotement. Computethemean,median,variance,standarddeviation,andcoefficient of variation for the datain Table 13.2. Solution. The data can be assembledin tabularform and the necessarysumscomputed
""HJt;Jl3"un t" compured as[Eq.(13.2)], t -:
158'4 :6.6 24
Becausethere are an even number of values,the median is computed as the arithmetic meanof the middletwo values:(6.605+ 6.615)12:6.61. As in Table 13.3,the sum of the squaresof the residualsis 0.217000,which canbe usedto computethe standarddeviation [Eq. (13.3)]: sr, :
0.217000 *z + - L : 0 . 0 9 7 1 3 3
TABTE13.3
Dotoond summotions for compuiingsimpledescriptive stotistics forthe ,l3.2. coefficients of thermolexoonsionfromToble (Jl -.yr
I 2 3 4 5 6 7 8 9 t0 tl )2 t3 l4 t5 t6 l7 IB t9 2A 2l 22 23 aA
T
6 395 6 435 6 485 6 495 6 505 6 515 6 555 6.555 6 565 6 575 5 595 6 605 6.61 5 6 625 6 625 6.635 6.655 6 655 6 665 6.685 6.715 6 . 7 15 6755 6.775
I s8zdo
o 04243 4.02723 0 0t323 0.01103 0 00903
o.oo723 0.00203 0.00203 0 0 0 12 3 0 00063 0 00003 0.00002
0.oao22 0.00062 0.00062 o.oat22 0 00302 0.00302 a 00422 a.ao722 o 41322 a o)322 o.02402 0 03062
67Tn6
40 896 41 449 42 055 42.)85 42.315 42.445 42568 42968 43 099 43.23) 43 494 43.62b 43 758 4 38 9 1 4 3B 9 l 44 023 44 285 44.289 44422 44.689 45091 45091 45630 45901 u4) 0J/
I3.I STATISTICS REVIEW
289
the variance[Eq. (13.5)]; s,?: (0.097133)2:0.009435 and the coefficientof variation[Eq. (13.7)]: c.v.:_
0.097133 x1007a:1.47Vo 6.6
The validity of Eq. (13.6)can alsobe verifiedby computing a
\:
- (158.400)' 1045.651 /24 : 0.009435 24-l
13.1.2 The Normol Distribution Another characteristicthat bearson the presentdiscussionis the datadistribution-that is, the shapewith which the data is spreadaroundthe mean.A histogramprovidesa simple visual representationof the distribution.A histograntis constructedby sorting the measurenrentsinto intervals,or bins. The units of measurementareplotted on the abscissaand the frequencyof occurrenceof eachinterval is plotted on the ordinate. As an example,a histogramcan be createdfor the data from Table 13.2.The result (Fig. l3.a) suggeststhat most of the datais groupedcloseto the meanvalue of 6.6. Notice also,that now that we have groupedthe data,we can seethat the bin with the most values is from 6.6 to 6.64.Although we could say that the mode is the midpoint of this b|n,6.62, it is more common to report the most frequent range as the modal class interval. If we have a very large set of data, the histogramoften can be approximatedby a smooth curve. The symmetric,bell-shapedcurve superimposedon Fig. 13.3 is one such characteristicshape-the normal distribution.Given enoughadditionalmeasurements, the histogramfor this particularcasecould eventuallyapproachthe normal distribution.
FIGUREI3.3 A histogromused to depict the distribution of doto. a. rh^ ^"-h^, ^{ l^r^ ^^inis increcses,ihe n:srr-a,i- ofronnnn.rn.fpq 1l^pq-66+\ hcll..t-cned.r',"" .i rli 'i"ll]r[io ,r. br.ion.
OJ 0)
L2
290
LINEAR REGRESSION
The conceptsof the rrrean,standarddeviation,residualsum of the squares,andnormal distribution all have greatrelevanceto engineeringand science.A very simpleexample is their useto quantify the confidencethat can be ascribedto a particularmeasurement. If a quantity is norrnallydistributed,the rangedefinedby.v - s., to y * .1, will encompass approximately68Vcof the total rneasurements. Similarly, the rangedefined by tr - 2s, to )r + 2.s, will encornpassapproximately95Vo. For example,for the datain Table 13.2,we calculatedin Example13.1that) = 6.6 and s., :0.097133. Basedon our analysis.we can tentativelymake the statement that approximately95Voof thereadingsshouldfall between6.405'734and6.794266.Becauseit is so far outsidethesebounds,ifsomeonetold us that they had rneasureda valueof7.35,we would suspectthat the measurementmight be erroneous.
13.1.3 DescriptiveStotisticsin MATLAB StandardMATLAB has severalfunctionsto computedescriptivestatistics.rFor example, the arithmeticmeanis computedasmean(x). If x is a vector,the function returnsthemean of the vector's values.If it is a matrix, it returns a row vector containingthe arithmetic mean of eachcolumn of >r.The following is the result of using meanand the otherstatistical functionsto analyzea column vector s that holds the datafrom Table 13.2: g >> format short (s),mode >> mean (s),median
(s)
h h
6.61, 6.555 >> min(s),max(s) :ne
6.395 o,
t/a
>> range=max(s)
min(s)
O.3B >> var(s),std(s) :nc
0.0094348 anc
-
0.097133 I MATI-AB also ofiers a Statistics Toolbox that provides a wide range of common statistical tasks,from random rrrrmher
oeneration
fn .'rrrve fiffinq
fo desiqn
29r
I3.I STATISTICS REVIEW
Theseresultsare consistentwith thoseobtainedpreviouslyin Example 13.1.Note that althoughthere are four valuesthat occur twice, the mode function only returnsone of the values:6.55,5. MATLAB can also be usedto generatea histogrambasedon the hist function. The hist functionhasthe syntax ln, xl
= hisL (Y, x)
where n : the number of elementsin each bin, x : a vector specifyingthe midpoint of eachbin, and y is the vector being analyzed.For the datafiom TabIe 13.2,the result is >> hist(s)
3 6.4I4 6.452 6.49 6.528 6.566 6.604 6.642 6.68 6.718 6.7s5 The resultinghistogramdepictedin Fig. 13.4is similarto the one we generated by handin Fig. 13.3.Note that all the argumentsand outputswith the exceptionof y are optional.For example,his'* (y) without output argumentsjust producesa histogrambar plot with l0 bins determinedautomaticallybasedon the ranqeof valuesin r.'.
FIGUREI3.4 Hisfogrcmgeneroiedwith the MATLABhist funciion
6.5
6.55
6.6
6.65
6_7
6.75
6.8
6.85
292
LINEAR REGRESSION
13.2
TINEARTEAST.SQUARES REGRESSION Where substantialerror is associatedwith data,the best curve-fitting strategyis to derive an approximatingfunction that fits the shapeor generaltrend of the datawithoutnecessarily matching the individual points. One approachto do this is to visually inspect the plotted data and then sketch a "best" line through the points. Although such"eyeball" calculaapproacheshave commonsenseappealand are valid for "back-ot'-the-envelope" tions, they are deficient becausethey are arbitrary.That is, unlessthe points defineaperfert would straightline (in which case,interpolationwould be appropriate),differentanalysts draw different lines. To removethis subjectivity.somecriterion must be devisedto establisha basisforthe fit. One way to do this is to derivea curve that minimizesthe discrepancybetweenthedata points and the curve. To do this, we must first quantify the discrepancy.The simplestexamp l e i s f i t t i n g a s t r a i g h t l i n e t o a s e t opf a i r e d o b s e r v a t i o(nr sr ,:) l ) . ( x z , y ) , . . . , ( . r , , ) n ) , The mathematicalexpressionfor the straightline is (13.8)
y : a o* z t l x1 e
where a6 and a 1 are coefficients representingthe intercept and the slope,respectively, and e is the error, or rcsidual, between the model and the observations,which can be representedby rearrangingEq. (13.8)as (13.9)
e:!-as-a1x
Thus, the residualis the discrepancybetweenthe true valueof y andthe approximate value, a0 + atx, predictedby the linear equation.
I 3.2. I Criteriq for o "Best" Fit One strategyfor fitting a "best" line through the data would be to minimize the sumofthe residualerrorsfor all the availabledata.as in
i
L't r:l
",
:- i ,r, - as- a1x;) /-'rl
(l3.lo)
i:l
wheren : total numberof points.However,this is an inadequatecriterion,asillustratedby F i g . l 3 . 5 a , w h i c h d e p i c t s t h e f i t oaf s t r a i g h t l i n e t o t w o p o i n t s . O b v i o u s l y , t h e b e s t f i t i s t h e line connectingthe points.However.any straightline passingthroughthe midpointofthe connectingline (excepta perfectly vertical line) resultsin a minimum valueof Eq.(13,10) equal to zero becausepositive and negativeerors cancel. One way to remove the effect of the signs might be to minimize the sumof theab solutevaluesof the discrepancies. as in iL,t "' r,t, i=l
-:
i , ' , ' , - a o-
Lt.'t i:l
alxil
(l3ll)
Figure 13.5bdemonstrates why this criterionis also inadequate.For the four pointsshown, any straightline falling within the dashedlines will minimize the sum of the absolute values ofthe residuals.Thus, this criterion also doesnot yield a uniquebestfit.
il I 3.2 LINEARLEAST-SQUARES REGRESSION
293
1il lrl li
il
ttltt' -l,tt't7'j-'''--
oo2ote'
,l il
ii ,i
(c)
FIGUREI3.5 F x n m n l e s n [ s o n - e 6 1 1 t o s r{69 . b e s ' r i , t h o t o r c : n g l e n r r n t c f n r . e , t , e s s i o n l , - r lm ^ m ; z e s t h e s u m
o f t h er e s i d u o l(sb, )m i n i m i z et hs es u mo f t h eo b s o l u tveo l u e os f i h er e s i d u o losn, d ( c )m i n i m i z e s l h em o x i m u m e r r o ro f o n y i n d i v i d u o p lo i n t .
A third strategyfor fitting a bestline is theminitna-rcriterion.In this technique,the line is chosenthat minimizes the rnaximumdistancethat an individual point falls from the line. As depictedin Fig. 13.5c,this strategyis ill-suited forregression becauseit gives undue influence to an outlier-that is, a single point with a large error. It should be noted that the mininax principle is sometimeswell-suitedfor fitting a simple function to a complicatedfunction (Carnahan,Luther, and Wilkes, 1969). A strategythat overcomesthe shortcomingsof the aforementionedapproachesis to minimize the sum of the squaresof the residuals: iltI
: f- (v; - as- arxi)z S, : 5-"i L-t /--':' ;-1
;-l
(13.12)
,l il
294
TINEAR EGRESSION
This criterion, which is called leastsquares,has a numberof advantages, includingthatit yielclsa uniqueline lor a given setof data.Beforediscussing thesepr-o-pertles, we will presenta techniquefor cleterminingthe varuc-sof (h andat that minimize Eq. ( 13.r 2).
13.2.2 leost-SquoresFir of o Srroight Line To determine varuesfor .,s and at, Eq. (13.12)is differentiated with respectto each unknowncoeflicient: i)s'
: -? \- r,,
un -
tt 1x;)
ilLti
t't A^:
: - z2 \ - r r , , - ( / { ) - a 1 ' r , ) rl , Lll.\'l
Note that we have simplified the summationsymbols;unlessotherwiseindicated, all summationsare fiorn r : I to rl. Settingtheseclerivativesequalto zero will result in a minimurn S,..Ifthis is done,the equationscan be expressed as
o:f.n;-f,,u-f,,1,, - f ,n,.,-, - f ,,,*i o : f r;.r', .
L
Now, realizingthat I a0 : noy, we can expressthe equationsas a setof two simultaneous lirrearequationswith two unknowns(aoandnr).
tr
(1,, * (f
O \ s - . . , \t r , , + ( \ -
",)
,, : f .r,
'\
\LJ'ri)tt1
r-
:Lr,.t''
Thesearecalledthe nonnal equation.t. They canbe solvedsimultaneously for , _ ,rI.r,.r'; I.r, I.r.i
ltT.'? - (I ',)'
(13.r5)
This resultcan rhenbe usedin conjunctionwith Eq. (r3.r3) to solvefor ao: i - atx wherev andi arethe meansof .vand.r,respectively. E X A M P L E1 3 . 2
L i n e o rR e g r e s s i o n ProblemStotement. Fit a straightline ro rhevaluesin Table i3.1. Solution. In this application,fbrce is the clependent variable(,v) and velocityis the independentvariable (.r). The data can be set up in tabulirrfbrm and the necessarvsums
REGRESSION I 3.2 LINEARLEAST-SAUARES TABTE13.4
295
neededto compulethe best-fit linefor the doto Dotoond summotions
Toble, ]3,,I from xi
I 2 3 4 5 6 7 B
:
I
li
to 2A 30 4A 50 60 7A BO
25 7A 380 550 610 t 224 830 1, 4 5 0
q I ?<
JOU
x! r00 4AO 900 r,600 2 500 3,600 4,904 6,404 2a 404
xJi
254 t,400 I 1,400 22AAO 30500 73 204 5 8 ,r 0 0 I 16,000 3r2850
The meanscan be computedas 360 _ i-_-45 8
_ }:
5,135 : 641.815 g
withEqs.(13.15) and(13.16) as The slopeand the interceptcan then becalculated
o' :
8 ( 3 1 2 , 8 s- 03) 6 0 ( sr 3. 5 ) :19.41024 8(20.400)-(360P
* 19.47024(45) : -234.2857 ao : 641.875 fit is Usingforceandvelocityin placeofy and;, theleast-squares F : -2342851-t 19.41024u Theline,alongwiththedata,is shownin Fig. 13.6.
FIGUREI3.6 ineto thedotofromToble1 3 t leosf-squores fitof o strclght
1600 1200
z
800
296
LINEAR REGRESSION
Notice that althoughthe line fits the datawell, the zero interceptmeansthat theequa" tion predicts physically unrealisticnegativeforces at low velocities.In Section 13.3,we will show how transformationscan be employedto derivean altemativebesrfit line thatis more physically realistic.
| 3.2.3 Quontificotion of Error of lineor Regression Any line other than the one computedin Example 13.2resultsin a largersurnof thesquares of the residuals.Thus, the line is uniqueand in termsof our chosencriterion is a "best"line through the points. A number of additional propertiesof this fit can be elucidated by examiningmore closely the way in which residualswere computed.Recall that thesumof the squaresis definedas [Eq. (13.12)) n
,t,: f (li -ao-arxi)2 -i:t
(13.17)
Notice the similarity betweenthis equationand Eq. (13.4)
s,:ftlr-l)2
(13.18)
In Eq. (13.18),the squareofthe residualrepresented the squareofthe discrepancybetween the dataand a singleestimateof the measureof centraltendency-the mean.In Eq.(13.17), the squareof the residualrepresentsthe squareof the verticaldistancebetweenthedataand anothermeasureof centraltendency-the straightline (Fig. 13.7). The analogy can be extendedfurther for caseswhere (1) the spreadof the points around the line is of similar magnitude along the entire range of the data and (2) thedistribution of thesepoints about the line is normal. It can be demonstratedthat if thesecriteria
FIGURE I3.7 Theresiduol in lineorregression represents theverticol dlstonce o dotopointondihe between s t r o i g hlti n e .
Measurement
-------*o
a0+ afii
I 3.2 LINEARLEAST-SQUARES REGRESSION
297
are met, least-squares regressionwill provide the best (i.e.,the most likely) estimatesof ag and a1 (Draperand Smith, 1981).This is called the maximumlikelihoodprinciple in statistics. In addition,ifthese criteria are met, a "standarddeviation"for the regressionline can be determinedas [comparewith Eq. (13.3)] J],/r
-
s,
(13.19)
n-2
where s_"7,is called the standard error of the estimate.The subscriptnotation "y /x" designates that the error is for a predicted value of y correspondingto a particular value of x. Also, noticethat we now divide by n - 2 becausetwo data-derivedestimates----as anda1were used to compute S,; thus, we have lost two degreesof freedom.As with our discussion of the standarddeviation,anotherjustificationfor dividing by n - 2 is that thereis no such thing as the "spreadof data" arounda straightline connectingtwo points.Thus, for the casewheren : 2,F,q.(13.19)yieldsa meaningless resultof infinity. Just as was the case with the standarddeviation, the standarderror of the estimate quantifies the spreadof the data.However, syTxQuantifiesthe spreadaround the regression line as shown in Fig. 13.8b in contrastto the standarddeviation s,. that quantified the spreadaround the mean (Fig. 13.8a). Theseconceptscan be usedto quantify the "goodness"of our fit. This is particularly useful for comparisonof severalregressions(Fig. 13.9).To do this, we returnto the original data and determine the total sum of the squaresaround the mean for the dependent variable(in our case,y). As was the casefor Eq. ( 13.18),this quantityis designatedg . This is the magnitude of the residual error associatedwith the dependent variable prior to regression.After performing the regression,we can compute Sr, the sum of the squaresof the residualsaround the regressionline with Eq. (13.17).This characterizes the residual
F I G U R EI 3 . 8 Regression dotoshowing(o)thespreodof thedotooroundthemeonof thedependent vorioble ond (b)lhespreodof thedotooroundthebest-fit line.Thereduction in thespreodin goingfrom (o)to (b),os ind;cored by t[e be'lsl^'oped cu'uesot therigl^t,represenls theimprovement dueto linecregression.
tbl
298
LINEARREGRESSION
o o
I3.9 FIGURE E r o r ' . p l eos f i n e o e g r e s s i o w n r t n ( o )s m o l 'o n d { b )
'orge
residroe l 'rors
error that remains after the regression.It is, therefore,sometimescalled the unexplained sum of the squares.The differencebetweenthe two quantities,S, - $ , quantifiestheimprovement or error reduction due to describing the data in terms of a straight line rather than as an averagevalue. Becausethe magnitudeof this quantity is scale-dependent, the differenceis normalizedto S, to vield ,'2 :
S'-S,
s,
where 12 is called the coefficient of determination and r is the correlation coefficient . /-. \: J r)). For a peri-ectfit, t. : 0 and 12 : l, signifying that the line explains100%ofthe An variability of the data. For 12 : 0, S, : Sr and the fit representsno improvement, alternativeformulation for r that is more convenientfor computerimolementationis r:
nf(xiri) - (Ir')(Iy,)
- (I r,)' (L,,)'J,I r,?
EXAMPLE 13.3 Estimotion of Errorsfor theLineorLeqsl-Souores Fit ProblemStqtement. Computethetotalstandard deviation. thestandard errorof theestimate,andthecorrelation coefficientfor thefit in Example13.2. lrlyl'?: .
Thedatacanbe setup in tabularform andthenecessary sumscomputed asin
299
I 3.2 LINEARLEAST-SQUARES REGRESSION
TABTE13.5
Dotoond summotions neededto computethe goodness-of-fit stotistics f o r t h ed o t of r o mT o b l e1 3 . , ] . li
l
2 3 4 5 6 7 B
t0 20 30 4A 50 60 7A BO
25 7A 380 550 610 t 224 830 I ,454
eo i atxi
-39 58 t55t2 349 82 544.52 735.23 9 3 39 3 I,t2863 1 , 3 23 3
JOU
L
(yi - t)2
380,535 327,441 68,579 8,441 1,0r6 3 3 42 2 9 3 53 9 r 6 53 , 0 6 6 t ,808,297
(!i - ao - a$)2
4,171 7,245 9il 30 I6,699 B ], 8 3 7 8 9 ,I B O 16,444 216,1tB
The standarddeviationis [Eq. (13.3)]
: 508.26 and the standarderrorof the estimateis [Eq. (13.19)]
: 189.79 Thus, becausesr,/.r( s,,,the linear regressionmodel has merit. The extentof the improvement is quantifiedby [Eq. ( 13.20)l I
1 . 8 0 8 , 2 -9 72 1 6 . 1 1 8 : 0.880-5 l . 808.297
or r : \/0.88[5 : 0.9383. Theseresultsindicatethat 88.05clcof the orisinal uncertaintv hasbeenexplainedby the linearmodel.
Before proceeding,a word of cautionis in order.Although the coefficientof determiyou should be careful not to ascribe nation provides a handy measureof goodness-of'-fit, more meaningto it than is warranted.Justbecauser'2 is "close" to I doesnot meanthat the "good." For example,it is possibleto obtaina relativelyhigh valueof r'l fit is necessarily when the underlying relationshipbetween-y and ;r is not even linear. Draper and Smith ( 198l) provide guidanceand additionalmaterialregardingassessment of resultsfor linear regression.ln addition,at the rninimum,you shouldalwaysinspecta plot of the dataalong with your regressioncurve. A n i c e e x a r n p l e w a s d e v e l o p e d b y A n s c o m b e ( 1 9 7 3 ) . A1s3i n. lF0i,gh.e c a m e u p w i t h four datasetsconsistingof I I datapointseach.Although their graphsare very different,all have the samebest-fit equation,.)': 3 + 0.5r, and the samecoefficientof determination,
300
LINEAR REGRESSION
t3
t:)
10
10
10
F T G U Rt E 3.lO Anscombe's fourdotosets olongwithihebestfit Jine, ,y: 3 + O.5x
I3.3
TINEARIZATION OF NONTINEARRETATIONSHIPS Linear regressionprovidesa powerful techniquefor fitting a bestIine to data.However,it is predicatedon the fact that the relationshipbetweenthe dependentand independentvariables is linear. This is not always the case,and the first step in any regressionanalysis shouldbe to plot and visually inspectthe datato ascertainwhethera linear model applies. In some cases,techniquessuch as polynomial regression,which is describedin Chap.14, are appropriate.For others,transformationscan be usedto expressthe datain a form that w i t h l i n e a rr e g r e s s i o n . iscompatible One exampleis the exltonentialmodel: ! : olreP"
(l3.22)
where a1 and B1 are constants.This model is usedin many lields of engineeringandscienceto characterizequantitiesthat increase(positiveB1) or decrease(negativeB1) at a rate that is directly proportionalto their own magnitude.For example,populationgrowthor radioactivedecaycan exhibit suchbehavior.As depictedin Fig. 13.11a,the equationrepresentsa nonlinearrelationship(for fu l0) between,vand x. Another exampleof a nonlinearmodel is the simplepower equation: )- :
a2xP)
( 13.23)
wherea2 andB2 areconstantcoefficients.This model haswide applicabilityin all fieldsof engineeringand science.It is very frequently used to fit experimentaldata when the underlyingmodel is not known. As depictedin Fig. 13.11b,the equation(for B2 l0)is nonlinear.
FIGU {o)Th (fJore
I 3.3 LINEARIZATIO ON F NONLINEAR ELATIONSHIPS
:-:
(dl
GI
301
azxQl
(f)
HGURE r3.r I (b)thepowerequolion,ond (c)thesoiurotron-growth-role Ports(d),{e),ond equotion. equotion, lolTheerponentiol versions lhctresui tromsimpletronstormotions. oi theseequolions lllorelineorized
A third exampleof a nonlinearmodel is the saturation-growth-rateequation: J
.\':0.r_ D r+ , r
(1.1.2+)
where a3 and B3 are constantcoefficients.This model, which is particularly well-suited for characterizingpopulation growth rate under limiting conditions, also representsa nonlinearrelationshipbetween-y and r (Fig. l3.1lc) that levelsoff, or "saturates,"as x increases.It has many applications,particularlyin biologically relatedareasof both engineeringand science. Nonlinear regressiontechniquesare availableto fit theseequationsto experimental datadirectly.However,a simpleralternativeis to usemathematicalmanipulationsto transform the equationsinto a linear form. Then linear regressioncan be employed to fit the equationsto data.
302
LINEARREGRESSION For example,Eq. (13.22)can be linearizedby taking its naturallogarithmto yield
(13.2t
l n y : l n c v r* f r x
of Thus, a plot of ln y versusr will yield a straightline with a slopeof B1 and an intercept ln o1 (Fig. 13.|ld). Equation(13.23) is linearizedby taking its base-l0logarithm to give
(r3.26)
logy:loga2*B2logx
Thus, a plot of log y versuslog x will yield a straightline with a slope of B2 andaninter' cept of log cv2fig. l3.Ile). Note that any base logarithm can be used to linearizethis model. However, as done here,the base-10logarithmis most commonly employed. Equation(13.24)is linearizedby inverting it to give
I
1 I
qS
ful +'--
(13.21)
a3x
Thus, a plot of 1/y versusl/x will be linear,with a slopeof Bzluz and an interceptof 1/o3 ( F i e .1 3 . 1 l / ) . In their transformedforms, thesemodels can be fit with linear regressionto evaluate the constantcoefficients. They can then be transformedback to their original stateandused for predictivepurposes.The following illustratesthis procedurefor the power model.
13.4 EXAMPLE
FittingDotowith the PowerEquotion transProblemStqtement. Fit Eq. (13.23)to thedatain Table13.1usinga logarithmic formation. Solution. The datacan be set up in tabularform and the necessarysumscomputedasin Table 13.6. The meanscan be computedas
t: ?Tr :2'5644
,:Yy:1.5i5't TABTE| 3.6
neededto fit the powermodelto thedotofrom Dotoond summotions r l l
l o b l el 3 . l
xi
I 220 330 440 550 660 774 BBO
I
t0
li
25 7A 380 550 610 l22A 830 I 450
log ri
IogJi
000 301 477 642 699 L778 I .845 L903
1 398 t B45 2.584 2.744 2.785 3.086 2.919 3.t6r
tlouo
lu)
)
(log r;)2
I 000 I 693 2.182 2 567 2.886 3.162 3.404 3.622 /0 ) 1a
log x; logy;
r 398 2.AAl 3Bll 4 390 4.732 5.488 5 386 6016
ilbn
I3.3 TINEARIZATION OF NONTINEARRELATIONSHIPS
303
log v 1000
100 log.r
) 1600 1200 800 400 0 60
40
80
(b)
F I G U R IE3 . I 2 firof c po*ermodello thedotofromToblel3 I {oJlhe fitof theli'onsfoi'med Leost squores doto (b)Thepowerequotion fitolongwiththedolo.
The slopeand the interceptcan then be c a l c u l a t ew d i t h E q s .( 1 3 . 1 5 ) a n d( 1 3 . 1 6 )a s al :
8 ( 3 - r . 6 2 2-) I 2 . 6 0 6 1 2 0 l. 5 )
8 ( 1 0 . 5 1-6 r( 1 2 . 6 0 6 ) 2
:1.9842
c t l : 2 . 5 6 4 4- 1 . 9 8 1 2 ( 1 . 5 1 5 :1 ) - 0 . 5 6 2 0 The least-sqLrares fit is l o g r ' : - 0 . 5 6 2 0 t l . 9 8 4 2 l o gr The fit along with the datais shown in Fig. 13.12a.
ri
304
LINEAR REGRESSION
We can also display the fit usingthe untransformedcoordinates.To do this,thecoeffi:0.2741 andB. =1.9942. cierrtsof the power nrodelare determinedoScy2: 19 05610 Using fbrceand velocityin placeof ,r'andr, the least-squares fit is e8a2 F :0.2J4lut This equation,alongwith the data,is shownin Fig. 13.12b.
The fits in Example 13.4 (Fig. 13.12)should be comparedwith the one obtained previouslyin Example13.2(Fig. 13.6)usinglinearregression on the untransformeddata Although both resultswould appearto be acceptable,the transformedresulthastheadvantage that it doesnot yield negativeforce predictionsat low velocities.Further,it is known from the discipline of fluid mechanicsthat the drag force on an object moving through a flLridis often well describedby a model with velocity squared.Thus, knowledgefromthe field you are studying often has a large bearingon the choice of the appropriate mode, e q u a t i o ny o u u s ef o r c u r v eI ' i t t i n g .
t3.3.1 GenercrlCommenlson Lineor Regression Beforeproceedingto curvilinearand multiple linear regression,we mr.rsternphasize theintroductory nature of the foregoing materialon linear regression.We have focusedonthe simple derivationand practicaluse of equationsto fit data.You shouldbe cognizantofthe fact that there are theoreticalaspectsof regressionthat are of practicalimportancebutare beyondthe scopeof this book. For example,somestatisticalassumptionsthat areinherent proceduresare in the linear least-squares l. 2. 3.
Eachx has a fixed valuel it is not randomand is known without error. The -yvaluesare independentrandomvariablesand all have the samevariance. The I' vslusr for a given x must be normally distributed.
Such assumptionsare relevant to the proper derivirtion and use of regression. For example,the first assumptionrneirnsthat (l) the "r valuesmust be en'or-tieeand(2)the regressionof -t'versusx is not the sameas r versus1. You are urgedto consultotherreferencessuchas Draperand Smith ( 1981)to appreciateaspectsand nuancesof regression that are beyondthe scopeof this book.
I3.4
C OMP U T E R A P P L IC A TIONS Linear regressionis so commonplacethat it can be implementedon most pocketcalculators. In this section,we will show how a simple M-file can be developedfo determine the slope and interceptas well as to createa plot of the dataand the best-fit line. We will also show lrow linearregression can be implementedwith the built-in po ly f i t f unction.
305
I 3.4 COMPUTER APPLICATIONS 13.4.1 MATLAB M-file: linresr
An algorithm for linear regressioncan be easily developed(Fig. 13.13).The required summationsare readily computedwith MATLAB's sum function. Theseare then used to computethe slopeandthe interceptwith Eqs.(13.15)and(13.16).The routinedisplaysthe interceptand slope,the coefficient of determination,and a plot of the best-fit line along with the measurements. A simple exampleof the use of this M-file would be to fit the force-velocitydatathat was analyzedin Example13.2: >> x = 110 20 30 40 50 60 70 B0l; > > y = 1 2 5 1 A 3 8 0 5 5 0 6 1 0 1 2 2 0 8 3 0 1 4 5 0 1t >>
(x,y)
linregr
0. BB05 r 9 . 4 1A 2 - 2 3 4 . 2 8 5 1
1600 1400 1200 1000 800 600 400 200 -200 10
80
It can just as easily be used to fit the power model (Exarnple 13.4)by applyingthe Iosl0 function to the data as in >> linregr
(log10 (x),1oql0
0.9481
7.9842
-0.5624
(y) )
306
LINEAR REGRESSION
3.5
2.5
1.5
t.t
1.4
t.o
1.8
FIGURE I3.I3 AnM{ilelo implement lineor regression. function Ia, 12 ] = linreqr (x,y) % linregr: linear regression curve fitting % 1., 12l = linregr(x,y): Least squares fit of straight ? line to data by solving the normal equations Z input: ? x = independent variable y= Z dependentvariable Z output: Z a = vector of slope, a(1), and incercept, % 12 = coefficient of determination
a(2)
n = length(x); if length(y)-=n, error(,x and y musf be same length,); end x = x(:); y = y(:); ? convert i_o column vectors S X = 3 s 1 1( tx ) ; S y = S u m ( y ) ; sx2 = srtn(x.*x); sxy - Sum(x.*y); sy2 = Sum(y.*y); a (1) = (n*sxy-sx*sy) / (n*sx2-sx,'2) ; a(2) = sy/n-a(1)*sx/n; T2 = ((n*sxy-sx*sy) /sqrt (n*sx2-sx^2) /sqrt (n*sy2- sy^2) ^Z ) ; % create plot of data and best fit lrne xp = lanspace (min (x) ,max\x\ ,2) ; yF a(l)*xp-a(2); plot(x,y,,o,, xp,yp) g r r - id o n
I3.5 CASESTUDY
307
13.4.2 MATTAB Functions: polyf ir ond polwal MAILAB has a built-in function polyf it that fits a least-squares nth-orderpolynomial to data.It can be applicdas in >> p
= polyfit
(x,
y,
n)
wherex andy arc the vectorsof the independentand the dependentvariables,respectively, olld n : the order of the polynomial.The function returnsa vectorp containingthe polynomial's coefficients.We should note that it representsthe polynomial using decreasing powers of -r as in the following representation: f ( x ) : I t t x " * p 2 x " - t+ . . . +
p n x* p n , r 1
Becausea straightline is a first-orderpolynomial,potyf ir (x,y , t) will returnthe slope and the interceptofthe best-fit straightline. t10 20 30 40 50 60 70 B0l; 125 7A 380 550 610 1220 830 14501; polyf it (x,y,1)
>>x >>y >>a
I9 . 4702 -23 4 .2857
Thus, the slopeis 19.4702and the inrercepris -234.2857. Another function, polyval, can then be used to compute a value using the coefficients.It has the seneralfbrmat: >> y
= polyval
(p,
x)
wherep : the polynomial coefficients,and y: >> y
the best-fit value at x. For example,
= polyval(a,45)
6 4 7 . B 75 A
, it}fS9..,."fl,.!l;q,"1' 1 rr ,
, . : d . .. . -
E N Z Y ME K IN E T IC S
Bockground. Enzymesact as catalyststo speedup the rateof chemicalreactionsin living cells.In mostcases,theyconvertonechemical,the substrate,into another,theproduct.TheMichaelis-Menten eqvationis commonlyusedto describesuchreactions: u--
u_[,s]
k"+ [s]
(13.28)
where u : the initial reaction velocity, u-: the maximum initial reaction velocity, [.i] : substrateconcentration, and ft" : a half-saturation constant.As in Fig. 13.14, the equation describes a saturating relationship which levels off with increasing [.S]. The graph also illustrates that the half-saturation constant correspondsto the substrateconcentration at which the velocity is half the maximum.
308
LINEAR REGRESSION
continued
Second-order Michaelis-Menten model 0.5u,,,
k"
FIGURE I3.T4 modelof enzyme kinetics. Twoversions of theMichoelis-Menien
Although the Michaelis-Menten model provides a nice starting point, it has beenrefined and extended to incorporate additional features of enzyme kinetics. One simple extension involves so-calledallosteic enzymes,where the binding of a substratemolecule at one site leads to enhancedbinding of subsequentmolecules at other sites.For caseswith two interacting bonding sites, the following second-orderversion often results in a bettu fit: 't):
-
u- [s]2 ft"2+ [S]2
(t3.29)
-
This model also describesa saturatingcurve but, as depictedin Fig. 13.14,the squared concentrationstend to make the shapemore sigmoid, or S-shaped. Supposethat you are provided with the following data:
tsl U
IB 0 . 13
1.3
o.a7
3 0.22
d
a7 F.
6 0 . 33 5
B9 0.35
036
Employ linear regressionto fit this data with linearized versions of Eqs. (13.28)and (13.29). Aside from estimating the model parameters,assessthe validity of the fits with both statistical measuresand graphs. Solution. model Equation(13.28),which is in the format of the saturation-growth-rate (Eq. 8.2D, can be linearizedby inverting it to give (recallEq. 13.27) k,
I
I3.5 CASESTUDY
309
;ryifiru::ri x.,. ii{. .i:.
continued
:i
The linreqr
function from Fig. 13.13 can then be used to determine the least-squaresfit: >> S=11.3 1.8 3 4.5 G 8 9l; >> v=f 0.07 0.Ll 0.2?, O.2ja A. j.l 5 0..iir 0.3t, 1; > > L a , r ' 2 l = L i n r e g r ( 1 . r . . - j.,1. , / v ) 16.4A22
0.19u2
12=
tl.9t4l The model coefficients can then be calculated as >> vm-Ira(2) \ rrr
i:
-':'
aRral
.{S=\.1n*J
( 1 j
ks= 3 r :. 2 : t6 ( ) Thus, the best-fit model is ,,_
5.25701S1
86.2260 + [s]
Although the highralue of 12might lead you to believe rhat this result is acceptable, inspection of the coefficients might raise doubts. For example, the maximum velocity 15'2510)is much greaterthan the highestobservedvelociry tb.sol. In addition, the halfsaturation tate (86.2260) is much bigger than the maximum substrate concentration (9). The problem is underscoredwhen the fit is plotted along with the clata.Figure l3.I5ta shows the transformedversion.Although the straightline iollows the upward trend, the dataclearly appearsto be curved.When the original equationis plotted along with the data in the untransformedversion (Fig. 13.15b),the fit is obviously unacceptable. The data is clearly leveling off at about 0.36 or 0.37. If this is correct. an eyeball estimatewould suggestthat u^ shouldbe about 0.36, and ft, shouldbe in the rung. of 2 to 3. Beyondthe visual evidence,the poornessof the fit is alsoreflected by statisticslike the coefficient of determination.For the untransformedcase,a much less acceptableresult of I - 0.6406 is obtained. The foregoing analysiscan be repeatedfor the second_order model. Equation (13.28) can also be linearizedby inverting it to give I I k? I +j_ u D, u. [Sl2 The linreqr function from Fig. 13.13can againbe usedto determine the least-squares fit:
l9..l7iiO i:l
=
? .. 4 4 9 )
3to
LINEARREGRESSION
#i*,i ; (a) Transformed model
0.4
0.5
14,r1 (b) Originalmodel
0.8 0.6 a 0.4
o.2 0 isl I3.I5 FIGURE Theplotin modelolonqwithdoto(points}. fit{line) Plots of leosl-squores of theMichoelis-Menten fit,ond{blshowshowlhefit lookswhenviewedin iheuntronsformed, {o)showsthetionsformed oilgrnorrorm.
The model coefficients can then be calculated as >> vm=I/a(2) 0.4083 >> ks-sqrt(vm*a(1)) 2 . BI?,1 Substituting these values into Eq. (13.29) gives
0.4083ts12 u:7.91r+[s:p Although we know that a high I doesnot guaranteeof a good fit, the fact thatit is very high (0.9929) is promising. In addition, the parametersvalues also seemconsistentwiththe trends in the data; that is, the ft- is slightly greaterthan the highest observedvelocityand stthcfrqfc
nnnnanfrofinn
/o\
I3.5 CASE STUDY
3ll
ffi'-ru conrinued i,lt+F.$fii:{:ifl (a)Transformedmodel
t/$12 ( b ) O r i g i n a lm o d e l
0.4 0.3 . 0.2 0.1 0
tsl FIGURE T3.I6 Michoelis-Menfen modelolonowithdoto Plots fit(line) of thesecond-order of leost-squores fit,ondib)shows iheuntronsformed, originol form thetrqnsformed Theplotin (o)shows {points).
graphically.As in Fig. 13.16a,the transformed The adequacyof the fit can be assessed results appear linear. When the original equation is plotted along with the data in the untransformedversion (Fig. 13.16b),the fit nicely follows the trend in the measurements. Beyond the graphs, the goodnessof the fit is also reflected by the fact that the coefficient of determination for the untransformedcasecan be computed as I : 0.9896. Based on our analysis,we can conclude that the second-ordermodel provides a good fit of this data set. This might suggestthat we are dealing with an allosteric enzyme. Beyond this specific result, there are a few other generalconclusionsthat can be drawn from this case study. First, we should never solely rely on statistics such as I as the sole basisofassessinggoodnessoffit. Second,regressionequationsshouldalwaysbe assessed graphically. And for caseswhere transformations are employed, a graph of the untransformed model and data should always be inspected. Finally, although transformationsmay yield a decent fit of the transformed data, this does not always translateinto an acceptablefit in the original format. The reasonthat this might occur is that minimizing squaredresidualsof transformeddata is not the sameas for the untransformed data. Linear regression assumesthat the scatter of points around the
3r2
LINEARREGRESSION ":;'i:-,'; " ':, '
'";;
'i
Lineariz on the f( the data.
, continued
best-fit line follows a Gaussiandistribution, and that the standarddeviation is the sameat every value of the dependentvariable. Theseassumptionsare rarely true after transforming data. As a consequence of the last conclusion,someanalystssuggestthat ratherthanusing linear transformations,nonlinear regressionshould be employed to fit curvilinear data.In this approach, a best-fit curve is developed that directly minimizes the untransformed residuals.We will describehow this is done in Chap. 14.
l3.l Given the data
9.5 l0.l 10.4 9.5 9.5
9.8 9.2 7.9 8.9 9.6
9.4 11 . 3 10.4 8.8 10.2
That is, detelminethe slope that resultsin the least-squares fit for a straightline with a zero intercept.Fit the following data with this model and display the result graphically,
t0.0 9.4 9.8 10.6 x 2 4 6 7 y4565 8.9
Determine (a) the mean, (b) median, (c) mode, (d) range, (e) standarddeviation, (f) variance,and (g) coefficient of variation. lJ,2 Construct a histogram from the data from Prob. 13.I. Use a rangefrom 7.5 to 11.5with intervalsof 0.5. 13.-l Given the data
28.65 28.65 27.65 29.25
26.55 29.65 28.15 2'7.65
26.65 2'/.8s 28.65 28.6s
27.65 27.05 28.45 27.65
27.35 28.25 3 r .65 28.s5
28.35 28.85 26.35 27.65
26.85 26.75 27.'75 21.2s
Determine (a) the mean, (b) median, (c) mode. (d) range, (e) standarddeviation, (f) variance,and (g) coefficient of variation. (h) Constructa histogram.Use a range from 26 to 32 with incrementsof 0.-5. (i) Assuming that the distribution is normal, and that your estimateof the standarddeviation is valid, computethe range (that is, the lower and the upper values) that encompasses68Vo<>fthe readings.Determinewhetherthis is a valid estimatefor the data in this problem. 13,.1Using the same approachas was empioyed to derive Eqs. (13.15)and (13.16),derivethe least-squales fit of the following model: \t-n1r'-lp
13.8 Fir naturalI 13.9 Tb areais n
f (hr) c (CFU/
PROBI.EMS
8.8 9.4 10.0 9.8 10.1
x 0 .1 y O.75
t0ill41720 876912
regressionto fit a straightlineto 13.5 Use least-squares
x02469 y56764
ll I
12 t5 710
l7 l2
19 t2
Along with the slopeand intercept,computethe standardenor of the estimate and the correlation coefficient. Plot thedata and the regressionline. Then repeatthe problem,but regress x versus}-that is, switch the variables.Interpretyour results. 13.6 The following datawas gatheredto determinetherelationship betweenpressureand temperatureof a fixedvolunteof I kg of nitrogcn.The volumeis l0 mr. T,'C p,Nlm2
-aO t 2A 0 40 B0 160 6 9 0 0 B1 0 0 9 3 0 0 1 0 , 5 0 0 I 1 , 2 0 0 t 2 9 0 0
Enrploy the ideal gas law pV - nRT to deterrninert onthe basis of this data. Note that for the 1aw,I must be expressed in kelvins. 13.7 Beyond the examples in Fig. l3.ll, there areother models that can be linearized usins transformations.FoI example, \) : a.^rep1r
The timr and the r estimate and (b) 2OOCF be cons impossi creases 13.10 F (Eq. l3 model:
When t results parame
(d').U
tion, de 13.11( related ber of (kg) git
w(kgl A (m')
Show t well. E
3r3
PROBLEMS this model and use it to estimatear and 6.r based thefollou,ingdata.Develop a plot of your fit along rvith data.
0 t 0 2 0 4 0 6 c )9 t 3 t5 17 tB a/5 125 145 125 085 055 035 028 0 rB
Fita powermodel to the datafiom Table 13.l, but use 13,8 logarithmsto perform the transformations. nafural 13,9Theconcentrationof E. c'oli bacteriain a swimnrins ueais monil.ored after a slorm:
4 ,{h4 B mL) ls9O I 320 c(CFU/l0O
t6 2A 24 900 650 560
Ihetimeis measuredin hoursfollowing the end of the storrn ardtheunit CFU is a "colony lbrrning unit." Use this data to (a) the concentrationat the end of the storm (t: 0) estimate and(b) the time at which the concentrationwill reach 200CFU/100mL. Note that yor.rrchoice of model should beconsistent with the fact that negativeconcenrationsale inpossible and that the bacteria concentrationah.r,aysdewith time. creases Rather than using the base-e exponential nrodel 13.10 { E qI.3 . 2 2 t u. c o m n l o na l t e r n a t i v ies to employ a base-10 model:
13.12 Determinean equationto pledict metabolismrateas a function of massbasedon the followinc data: Animol
Moss (kg)
Cow Humcn Sheep Hen Rcri Dove
400 7A 45 2 03 0 16
270 B2 50 4B 1 /<
497
l-1.1-1Fit an exponentialmodei to
04 800
x
v
08 975
I',2 I 500
1.6 I 950
2 2900
23 3600
Plot the data and the equationon both standardand semilogarithmicgraphpaper. 13.14 An investigatorhasreportedthe datatabulatedbelow fbr an experimentto determinethe growth rate of bacteria ft (per d) as a function of oxygen concentrationc (mg/L). It is known that such data can be modeled by the fbllowing equatiou: t a
, /< :
Krnaxc---; cr+c'
where c, and ftnra\are parameters. Use a transformation ro linearize this equation. Then use linear regression to estimate c. and ftn,.,xand predict the growth rate at c :2 mg/L,. c
Whenusedfbr curve iitting, this equatior'lyields identical results to the base-r,version, but the value of thL-exponent (lr) will diff'erfrom that estirnated parameter with Eq.13.22 (0,).Usethe base-IOversionto solve Prob. 13.9.In addidevelopa formulation to relateut ro frs tion, l3,ll On average.the surface area A of human beings is rclated to weight W and height H. Measurementson a nunberof individualsof height 180 cm and dil-lerentweishts (kg)-eivevaluesof A (m2) in the lbllowing rable:
Metqbolism (wofts)
k
0.5 tt
OB 2.4
t5 53
25 76
4 B9
13.15 Developan M-file function to computedescriptive stafisticsfor a vector of values.Have the function detennine and displaynumberof values.mean,median.mode.range, standarddeviation,r'ariiince.and coefficientof variation.In
addition, huvcit gencralc .rhisloglinr. Trrslil wilhlhcrl;tl;r lromProb.13.3.
13.16 Modify the lrnresr funcrionin Fig. 13.l3 so that it (a) computesand returnsthe standarderror of the estimate, and (b) usesthe sDbptor functionto alsodisplaya plot of W(kS)7A 75 77 BO 82 84 87 90 the resicluals(the predictedminLrsthe measured-r.)versus-r-. A ( r n 22l 1 0 2 t 2 2 t 5 2 2 A 2 2 2 2 2 3 2 2 6 2 3 0 13.17 Develop an M-file function to fit a power model. Have the function return the best-fit coefficient a2 and powerB, along with the 12fbr the untransfbrmedmodel. In Show that a power law A : oil' fits thesedata reasonablv addition,usethe .jubp l ot function to displaygraphsof both well.Evaluatethe conslantsa and b. and predict what the the transfbrmedand untransformedequationsalong with the orrf.^a.rpr ic fnr r Q5_Lo nersrrn data.Test with
3r4
LINEARREGRESSION
l3.ltl The tbllowing data shows the relationshipbetween the viscosityof SAE 70 oil and temperature. After taking the log of the data,use linear regressionto find the equationof the line thrt bestfits the dataand the r2 value.
are shown beloll,.Which model bestdescribesthedata(sl& tistically)?Explain your chorce.
T e m p e r o t u r e ,' € Viscosiry p,N.s/m2
s"
26.67 93 33 I 35 O0B5
I48 Bq 3I5.56 0 012 OOoO/5
13.19 You perform experimentsand determinethe fbllowing valuesoi heat capacityc at varioustemperaturesZ for a gas:
T50-30060 l25O I 280 c
I 350
I480
90 I 580
ilO l70A
Use regressionto determinea nrodel to predict c as a function of I. 13.20 lt is known that the tensile strengthof a plastic increasesas a tunction of the time it is heat treated.The following data is collected:
Time l0 )5 2A 25 40 50 55 60 Z5 TensifeStrength 5 2A I B 40 33 54 70 6A 78
Model A
135
Number of Model Porometers Fif
2
Model B
r05
ModelC i00
3
13.23 Below is data taken from a batch reactorofbactenal growth (after lag phasewas over). The bacteriaareallowed to grow as fast as possiblefor the first 2.5 hours,andthen they are inducedto producea recombinantprotein,thepro duction of which slows the bacterial growth significantly. The theoreticalgrowth of bacteriacan be described by
'4: dt
r,x
where X is the number of bacteria,and p is the specrfic growth rateof the bacteriaduring exponentialgrowth.Basd on tbe data,estimatethe specificgrowth rate of thebacteria duling the tirst 2 hoursof growth and during thenext4 houm of erowth.
Time,
(a) Fit a straight line to this data and use the equation to h0t23 determinethe tensilestrengthat a time of 32 min. ICells], (b) Repeat tl.reanalysis but for a straight line with a zercr S l L 0 1 0 0 0 3 3 2 t t 0 2 I 6 4 4 2 453 3 660 5 460 intercept. 13.21 The following data was taken from a stirred tank re13.24 A transportationcngineeringstudy was conductedto actor for the reactionA - B. Use the data to determinethe dctL'rminethe properdesignof bike lanes.Datawasgathered best possible estimatesfor ft,,, and E, fbr the following on bike-lanewidths and averagedistancebetweenbikesand kineticmodel: passingcars.The clatafrorn I I streetsis dA - ---:- : AT
A0le
A
where R is the gas constantand equals0.00198kcal/mol/K.
-dAldt (moles/Us) A (moles/t) r (K)
46A 960 2485 l600 200 150 50 20 2BA 320 45O 500
14.4<
Dislonce,m 24l524IB1.82.5 l2 312 l o n e W i d r hm , 2 9 2 . 1 2 . 3 2 1 I B 2 7 1 . 52 9 1 5
(a) Plot the clata. (b) Fit a straightline to the data with linearregression. Add this line to the plot. (c) If the minimum safeaveragedistancebetweenbikesand passingcars is consideredto be 2 m, determinethecor. points 13.22 Concentrationdatawas takenat l5 time for the minimum lanewidth. responding polyrnerizationreaction: 11.25 In water-resourccsengineering,the sizing of reserxA*,tB - A,B:. voirs dependson accurateestimatesof water flow in the We assumethe reaction occurs via a complex mechanism river that is being impounded.For some rivers, long-term consistingof many steps.Severalmodelshavebeenhypoth- historicalrecordso1'suchflow data are difficult to obtain.In esized,and the sum of the squaresof the residualshad been contrast,meteorolo-vical data on precipitationis often avail^^l^,,1^r^J
f^-
.L^
f:r^
^f
rL.-
'-..'J,-1.
^f
rli^
,l^.-
TL,.
t0 550
-,,-,,1..
determir This rela when or followin Procipifqtion, Flow, m3/s
(a) Plot (b) Fit r Supt (c) Use the 1 (d) rf th of th ratio use. 13.26 "t) 10.65cn alloy. Te tween stl Slroin, cm/cm Stress, N/<m2
The stres force in r value car the mast mast's le estimate 13.27 Tl that mee voltages V,V irA
(a) On cur an{ (b) Re
PROBLEMS
315
determine a relationshipbetween flow and precipitation. 13.28 An experiment is performed to determine the Voelonthisrelationship can then be usedto estimateflows for years gation of electricalconductingmaterialas a function of temwhen only precipitation measurementswere made. The perature.The resulting data is listed below. Predict the 7o following datais availabletbr a river that is to be dammed: elongationfor a temperatureof 400 'C. ?ncipit o t i o nc, m B B9 l 0 B 5 1 0 4 I lbw m3/s 1 4 6 1 6 7 1 53
1 3 9 . 71 2 7 9 4 r 1 6 B 9 9 1 232 195 )61 II I
*" cm/cm 0 0 0 3 2 0 0 0 4 5 0 0 0 5 5 0 0 0 r 6 0 0 0 8 5 0 0 0 0 5 5lrcss, N/cm2 4970 5170 .5500 3590 6900 t24A
thestress causedby wind can be computedasF/A, whereF : in themast and /. : mast'. cross-sectionalarea.This force value canthenbe substitutedinto Hooke's law to determine themast'sdeflection, Al- : strain x l, where L : the mast's length.If the wind force is 25,000 N, use the data to estimate thedeflectionof a 9-m mast. The following data was taken from an experiment 13,27 thatmeasuredthe cunent in a wire for various imposed voltages:
i,A
2 52
3 7B
4 lo7
5 t3
7la 1 93
475 600 127 153
16.6
(a)Plotthedata. 0) Fit a sffaight line to the data with linear regression. Superimpose this line on your plot. (c)Usethe besffit line to predict the annual water flow if theprecipitation is 120cnr. (d)If rhedrarnage areais I 100km:. estirnatewhat fraction of theprecipitationis lost via processessuch as evaporation,deep groundwaterinfiltration, and consumptive use. The mast of a sailboat has a cross-sectionalarea of 13,26 cmr and is constructedof an experimentalaluminum 10.65 Testswere performed to define the relationshipbeallo1. tween stressand strain.The test resultsare
v,v
Temperofure,oC 200 25a 300 375 425 T"Elongotion 75 86 87 lO ll3
27.5
(a)0n thebasisof a linear regressionof this data,determine cunentfor a voltage of 3.5 V. Plot the line and the data andevaluatethe fit. (b)Redothe regressionand force the interceptto be zero.
11.29 The populationp of a small community on the outskirts of a city grows rapidly over a 2O-yearperiod:
ta p
1OO
to 450
5 204
t5 950
20 2000
As an engineer working for a utility company, you must forecast the population -5years into the future in order to anticipate the demand for power. Employ an exponential model and linear regressionto make this prediction. 13,30 The velocity r of air flowing past a flat surface is measuredat severaldistancesl away from the surface.Fit a curve to this data assumingthat the velocity is zero at the surface(y : 0). Use your resultto determinethe shearstress (du/dy) at the surface.
J,h u, mls
0.002 A 2B7
0006 0 899
o0l2 r9r5
00rB 3048
oo24 4299
13,31 Andrade'secluationhas beenproposedas a model of the elfect of temperatureon viscosity: P:
PsB/r"
where p : dynamic viscosity of water ( l0-3 N's/m2), 4 : absolutetemperature(K), and D and B are parameters.Fit this model to the following data for water:
705r0203040 p I 7B7 I .519
I 3A7
I 002
O7 9 7 5
A.6529
" l
rol-,l,,
GenerolLineorLeost-Squores ond NonlineorR"gression
CHAPTEROBJECTIVES This chaptertakesthe conceptoffitting a straightline and extendsit to (d) fitting a polynomialand (D)fitting a variablethat is a linear function of two or more independent variables.We will then show how suchapplicationscan be generalizedand appliedto a broadergroup of problerns.Finally, we will illustratehow optimizationtechniquescanbe usedto implementnonlinearregression.Specilicobjectivesand topicscoveredare o o o o .
I4.I
Knowing how to implementpolynomial regression. Knowing how to implementmultiple linear regression. Understandingthe formulation of the generallinear least-squares model. Understandinghow the generallinear least-squares model can be solvedwith MATLAB usingeitherthe normalequationsor left division. Understandinghow to implementnonlinearregressionwith optimization techniques.
POTYNOMIAL REGRESSION In Chap.13, a procedurewas developedto derive the equationof a straightline usingthe least-squares criterion. Some data, although exhibiting a marked pattern such as seenin Fig.lrt.f.ispoorlyrepresentedbyastraightline.Forthesecases,acurvewouldbebetter suited to fit the data.As discussedin Chap. 13, one method to acconrplishthis objective is to use transformations.Another alternativeis to fit polynomialsto the datausingpolynotnial regression. procedurecan be readily extendedto fit the data to a higher-order The least-squares polynomial. For example,supposethat we fit a second-orderpolynomial or quadratic: ,y:uotal.r+
a z r :+ e
(14.1)
317
REGRESSION POLYNOMIAL
(b)
I4.I FIGURE ( b )l n d i c o i i otnh o io p o r o b o l o is regression forlineor leost-squores (o)Dotoihotis ill-suited preferoble. For this casethe sum of the squaresof the residualsis II
(y, - nu- atxi - orri)t
s, : t
\14.2)
i-l
fit, we take the derivativeof Eq. (14.2) with respectto To generatethe least-squares of the polynomial, as in coefficients eachof the unknown
as, d0t
:
at
: ;d0t aS, :--
rtu.
:
-/'
.\-r )
\.\',
-c'ltt -a,'t,
-t'1*;)
- us- d1x;-t^sa ) .xi ()l -L r
\/
. . 2l -
rt
\-\r
a2xl)
- a0 - a1x1 - u7.tl)
These equationscan be set equal to zero and reananged to develop the following set of normal equations:
z )z : L v i ( n ) a o +( I t ' ) a r + ( f - t ' ' a (I r') oo+ (I-t,t) n' + (t ti) a2: lxiYi (Ir,?) oo+ (T.",1)"' + (I-'1) tL2: lx!)'i
318
AND NONLINEARREGRESSION GENERAL LINEARTEAST.SQUARES where all summationsare from i : I through n. Note that the precedingthreeequations are linear and have three unknowns:a0, ctt, anda2. The coefficientsof the unknownscanbe calculateddirectly from the observeddata. For this case.we see that the problem of determininga least-squares second-order polynomial is equivalentto solving a systemof three simultaneouslinear equations. The two-dimensionalcasecan be easily extendedto an mth-orderpolynomial as in ! : a o t d l x f a 2 x 2+ . . . a a . x ' ' I e The foregoinganalysiscan be easily extendedto this more generalcase.Thus,wecan to recognize that determining the coefficients of an mth-order polynomial is equivalent enot solving a systemof m I I simultaneouslinear equations.For this case,the standard is formulatedas
s,
(14.3)
n - (m -l1)
This quantity is divided by n - (m I l) because(m * l) data-derivedcoefficientso0, at,..., an-were usedto computeSr; thus,we have lost z * I degreesof freedom. fot In addition to the standarderror, a coefficient of determination can also be computed polynomialregression with Eq. (13.20).
.l4.1 EXAMPLE
Polynomiol Regression polynomialto thedatain thefirst twocolumns ProblemStotement. Fit a second-order ofTable14.1. TABTEl4.l
fif. for on erroronolysisof the quodroticleost-sguores Compuiotions (Ji -J)-
v,
xi
0 I 2
2l 77 t3.6 272 40.9 6t t 152.6
j?
4 5
I
544 44 314.47 1 4 00 3 3.12 239.22 1272.1) 2 5 r 33 9
Soluiion. The following can be computedfrom the data: \'-. - r< n:6
I l'i : 152.6
Lx!:9tg I",)t :585'6
;-r<
\-'l-ss
L*?Y,: 2488'8
r-)5477
\L '-- 't J - l l s
-
/r ^t
r J
tg,- ao-arx,-aS)r 0.t 4332 r 00286 r 08r60 0 80487 0 6 19 5 9 0 09434 3.74657
]4, 1 POIYNOMIAL REGRESSION
319
'i, tr iil'i,l:hfx
Therefore,the simultaneouslinear equationsare
These equations can be solved to evaluate the coefficients. For example. using MMLAB: >> N = l6 15 55;15 55 225;55 225 9j9); >> r = 1152.6 535.6 24BB.Bl; >> a = l.i\r 2 .41 B6 2.3593 1.8507 Therefore. the least-squaresquadratic equation for this case is ! :
2 . 4 1 8 6* 2 . 3 5 9 3 r * 1 . 8 6 0 7 1 2
The standard error of the estimate based on the regression polynomial is tEq. (14.3)l | 1,.,.:
/
_1./+b) /
!o-rz+ l)
:lll75
""'-
The coefficientof determinationis r-
-
- 3.74651 2513.39 :0.99851 2513.39
and the comelationcoefficientis r :0.99925. F I G U R EI 4 . 2 Filof o secondorderpolvnomiol
320
LINEARLEAST-SAUARES GENERAL AND NONLINEARREGRESSION These results indicate that 99.851 percent of the original uncertaintyhas beenexplainedby the model.This resultsupportsthe conclusionthat the quadraticequation representsan excellentfit, as is alsoevidentfrom Fig. 14.2.
14.2 'T4UITIPIE TINEARREGRESSION Another useful extensionof linearregressionis the casewheref is a linear functionof two or more independentvariables.For example,1 might be a linear function of r, andxr. asin . y : r r o { c t 1 - t 1l a 2 x z a s Such an equationis particularly useful when fitting experimentaldata where the variable being studiedis often a function of two other variables.For this two-dimensionalcase.the regression"line" becomesa "plane"(Fig. 14.3). As with the previous cases,the "best" values of the coefficientsare determinedby formulating the sum of the squaresof the residuals: $ : i
()'i - ao-(r-tl.i - az.rt.i)?
and differentiatingwith respectto eachof the unknown coefficients: aS, ;-:-l) dao
as,
,:- : d L ll
as,:
;-
OQ)
_\-r. ( l ' i - 2^
11 )
- t t 1 1 - t 1 1 . x 1 . -1o 2 x 2 . i l
xt.i()i - (t0-
^ \--
-2
) .r2.,0"i
a1xl.i - azxz.i)
- ao - cttxt.i * azxz.i)
FIGUREI4.3 Grophicol depiction y iso lineor of multiple lineor regression where function of.r,ondrr.
_-__---!-_
//
(r4..1)
EXA
I4.2 MULTIPLE LINEARREGRESSION
32r
The coefficientsyielding the minimum sum of the squaresof the residualsare obtainedby settingthe partial derivativesequal to zero and expressingthe resultin matrix form as
I*'., Irr., Ifoo f n I " ' t . , I " t i . ' I x r . i r z . iI l a r I L I tr.r Ir,.,.rr., I .rt., _.1[ nt
l ;?;r:
(14.5)
EXAMPLE 14.2 Multiple Lineor Regression Problem Sfotement. The following data was calculated from the equation y : 5 + 4x1- 3r2'. .rt
x2
00s 2 2.525 r30
I
t0
463 7227
Use multiple linear regressionto fit this data. Solution. The summationsrequired to develop Eq. (1a.5) are computedin Table 14.2. Substitutingthem into Eq. (14.5)gives
16.5 rll [ 6 1 6 . 5 1 6 . 2 s4 8 | I 48 54) L t4
a0 Al A1
54 243.5 100
(14.6)
which can be solvedfor (10:5
at:4
aZ:-3
which is consistentwith the original equationfrom which the datawas derived.
The foregoing two-dimensionalcasecan be easily extendedto m dimensions,as in ) : a oI a 1 x 1* a 2 x 2 + . " + a r x ^ * e
TABTE I o:? c:Tp':olio1'.:"".:'i':ddey3lon '::::l fo1Elomele 1o li:. ::q,,oti:,i: la'2 ! 50 t0 9 0t 34 277 -a4
xr 2 25
0)
xz 0 I 2 3 6 2 t4
x!
xl
xrx2
.rilr
0 4 6.25 I t6 49
0 l 4 I 36 4
o 2 5 3 24 l4
,/a.l)
J4
T6
0 20 225 0 l2 t89 )77-
x2y 0 t0 IB 0 ]B \,4
lao
322
LINEARLEAST-SQUARES AND NONLINEARREGRESSION GENERAL where the standarderror is formulatedas
s n-(nt+D and the coefficientof determinationis computedas in Eq. ( 13.20). Aithough therernarybe certaincaseswherea variableis linearly relatedto two or mole other variables,multiple linear regressionhas additionalutility in the derivationof power equationsof the generalform x;t .' . x::i," )' : ctox';t SuchequationsAreextremelyusefulwhen fitting experimentaldata.To usemultiplelineat regression,the equationis transformedby taking its logarithm to yield l o g , y: l o g a 0* a 1l o g x l * a 2 l o g x 2 + ' ' ' + a n l o g t .
I4.3
GENERATTINEARLEASTSQUARES In the preceding pa-qes,we have introduced three types of regression:simple linear, polynomial, and multiple linear. In fact, all three belong to the following generallinear model: least-squares -l''' ) ' : a o z oI a r z t * a y Z z
I a n , z mI €
(14.?)
where :0, zl,..., z.marem * I basisfunctions.It can easilybe seenhow simplelinear and multiple linear regressionfall within this model-that is, ;s: l, zr :.rl,t2= is alsoincludedif the basisfunctionsate x2,. .., Znt:.xm. Further,polynomialregression : : :16m. : l,;r x,.tt x2,..., Zp1 s i m p f em o n o m i a lassi n : o "linear" to the model's dependenceon its refers only Note that the terminology polynomial regression,the functionsthem' parameters-that is, the a's. As in the caseof selvescan be highly nonlinear.For example,the :'s can be sinusoids.as in y : ao * a1 cos(a;r) I azsin(o_tx) Such a format is the basisof Fourier analysis. On the other hand, a simple-lookingmodel suchas ):
do(l -
e-ol')
is truly nonlinearbecauseit cannotbe manipulatedinto the format of Eq. ( 14.7). Equation(14.7) canbe expressedin matrix notationas (14.8)
[y]:fzl{a}+lel
values where[Zl is a matrix of the calculatedvaluesof the basisfunctionsat the measured of the independentvariables: zll t7l t-t-
. [:::
:*t-f
a,:,,l
EXA
14.3 GENERAL LINEARLEAST SAUARES
323
where m is the number of variablesin the model and n is the number of data points. Becattsen > m * 1, you shouldrecognizethat most of the time, [Z] is not a squaremarrix. The column vector {y} containsthe observedvaluesofthe dependentvariable: {-v}r :
L,vr !2.
-}nl
The column vector ia] containsthe unknown coefficients: {a)r :
la11 at
a , ,)
and the column vector {e} containsthe residuals: {e}r :
le1
e2
e u)
The sum of the squaresof the residualsfor this model can be definedas n
s,:Ilt, i-l
/
.
\2
-lo1r1,l
\
j=tt
(14.9)
/
This quantity can be minimized by taking its partial derivativewith respectto eachof the coefficientsand settingthe resultingequationequalto zero.The outcomeof this processis the normal equationsthat can be expressedconciselyin matrix form as
[ z ] r V \ { a } : { t z l r{ y } )
( 1 4r.0 )
It can be shownthat Eq. (14.l0) is, in fact,equivalenfto the normalequationsdeveloped previouslyfor simple linear,polynomial, and multiple linear regression. The coefficientof determinationand the standarderror can alsobe formulatedin terms of matrix algebra.Recallthat r: is definedas t
S,-S, -
s,$
| _ _S ,
Substitutingthe definitionsof S, and S, gives r--l-
I(vr - -urt' I(),r
- .ti t2
where! : the predictionof the least-squares fit. The residualsbetweenthe best-fit curve and the data,y; - j, can be expressedin vector form as
{y} - f"zlla} Matrix algebracan then be usedto manipulatethis vector to computeboth the coefficient of determinationand the standarderror of the estimateas illustratedin the following example.
EXAMPLE I4.3
P o l y n o m i oR l egressiow n ith MATLAB ProblemSiotement. RepeatExample14.1,but usematrixoperations asdescribedin this section. Solution. First,enterthe datato be fit >> x - t0 12 3 4 5l'; > > y = 1 2 . I 1 . 1 1 3 . 6 2 ' 7. 2 4 A . 9 6 1 . 1 1 ' ;
324
GENERATL\NEARIEASI.SQUARESAND NONI\NEAR RIGRESS\ON Next, create the [Z] matnx: >>
Z
=
[ones(slze(x)
)
x
x.^2]
100 111 1,24 139
7416 1 5
25
We can verify that LZf lzl resultsin the coefficientmatrix for the normal equations: >> z'*z 15 ss 225
6 15 55
55 225 979
This is the sameresultwe obtainedwith summationsin Example 14.I . We can solvefor the quadraticby implementingEq. (14.l0): coefficientsof the least-squares >> a =
\Z'*Z) \(Z'*y)
2 .47 B6 2.3593 1.8607
EX
In order to compute 12 and s"7-,, first compute the sum of the squares of the residuals: > > S r = s u m( ( y - Z * a ) . ^ 2 )
3 .'t 466 Then r2 can be computed as > > 1 2 = I - S r / s u m( ( y - m e a n ( y ) ) . ^ 2 ) 0.998s and sr,7,.can be computed as >> syx -
sqrt (Sr/ (lenqth (x) -lenqth
(a) ) )
1.1175
Our primary motivation for the foregoing has been to illustrate the unity among the three approaches and to show how they can all be expressed simply in the same matrix notation. It also sets the stage for the next section where we will gain some insights into the preferred strategies for solving Eq. (1a.10). The matrix notation will also have relevance when we tum to nonlinear regression in Section 14.5.
,I4.4
AND THEBACKSLASH OPERATOR QR FACTORIZATION
32s
14.4 QR FACTORIZATION AND THEBACKSIASHOPERATOR Generatinga bestfit by solving the normal equationsis widely usedand certainlyadequate for many curve-fittingapplicationsin engineeringand science.It must be mentioned,however,tlrat the normal equationscanbe ill-conditionedandhencesensitiveto roundoff errors. Two more advancedmethods,QRfactorization andsingular valuedecomposition,are more robustin this regard.Although the descriptionof thesemethodsis beyondthe scope of this text, we mention them herebecausethey can be implementedwith MATLAB. Further.QR factorizationis automaticallyusedin two simple ways within MATLAB. First, for caseswhere you want to fit a polynomial, the built-in polyf it function automatically usesQR factorizationto obtain its results. problemcan be directly solvedwith the backSecond,the generallinear least-squares slashoperator.Recallthat the generalmodelis formulatedas Eq. (14.8) 1 y y: l Z l { a l
( 1 4I. l )
In Section10.4,we usedleft divisionwith thebackslashoperatorto solvesystemsof linearalgebraicequationswherethe numberof equationsequalsthe numberof unknowns(n : n). For Eq. ( 14.8)as derivedfiom generalIeastsquares,the numberof equationsis greaterthan the number of unknowns (n.> m).Such systemsare said to be overdetermined.When MATLAB sensesthat you want to solvesuchsystemswith left division,it automaticallyuses QR factorizationto obtainthe solution.The following exampleillustrateshow this is done. E X A M P L1E4 . 4
l m p l e m e n t i n gP o l y n o m i o R l e g r e s s i ow n i t h p o , y i j t o n d L e f tD i v i s i o n Problem Stotemeni. RepeatExample 14.3,but use the builrin polyf ir functionand left division to calculatethe coefficients. Solution. As in Example 14.3,the datacan be enteredand usedto createthe lZl matrix as in >> x = >> y = >> z = The potyf
3 4 5l'; lA I2 1 2. I 1 . 1 1 3 . 6 2 1 . 2 4 A . 9 6 I . I l ' [ones (size (x)) x x..2); it
;
function can be used to compute the coefficients:
>> a = polyf it \x,V,2) 1.8607
2 .3593
2.41 B6
The sameresult can also be calculatedusing the backslash: >>a=Z\y 2 .41 86 2.3593 1 8607 As just stated, both these results are obtained autornatically with QR factorization.
326
AND NONLINEARREGRESSION GENERAL LINEARLEAST-SAUARES
I4.5
NONTINEARREGRESSION There are many casesin engineeringand sciencewhere nonlinearmodels must be fit to data.In the presentcontext.thesemodelsaredefinedas thosethat have a nonlineardependenceon their oarameters.For example. -v:ao(l
-r-at"r)*e
( 1 4 .21)
This equationcannotbe manipulatedso that it conformsto the generalform of Eq. (14.7). As with linear least squares,nonlinearregressionis basedon determiningthe values of the parametersthat minimize the sum of the squaresof the residuals.However,for the nonlinearcase,the solution must proceedin an iterativefashion. There are techniquesexpresslydesignedfor nonlinearregression.For example,the Gauss-Newtonmethod uses a Taylor seriesexpansionto expressthe original nonlinear equationin an approximate,linear form. Then least-squares theory can be usedto obtain new estimatesof the parametersthat move in the directionof minimizing the residual.Details on this approachare providedelsewhere(Chapraand Canale,2002). An alternativeis to useoptimizationtechniquesto directly determinethe least-squares fit. For example,Eq. (la.l2) can be expressedas an objectivefunction to computethe sum of the squares: n
f ( a q ,a 1 ) :
)-
[t, - a o \ - , - a t r ' 1 1 2
(t4.r3)
An optimizationroutine can then be usedto determinethe valuesof ae and a1 that minimize the function. As describedpreviouslyin Sec.7.3.1.MATLAB's fminsearch functioncanbe used for this purpose.It has the generalsyntax lx,
fvaLl
= fmlnsearch(fun, xa, options,pl,p2,
...)
wherex = a vector of the valuesof the parametersthat minimize the function f un, f va7 = the value of the function at the minimum. x0 : a vector of the initial guessesfor the parameters,options: a structurecontainingvaluesof the optimizationparametersas created with the optimset function (recall Section6.4), and pl, p2, etc.: additionalarguments that are passedto the objectivefunction. Note that if options is omitted,MATLAB uses default valuesthat are reasonablefor most problems.If you would like to passadditional arguments(p1. p2,...). but do not want to setthe aptions, useempty brackets[] asa place holder. E X A M P L E1 4 . 5
N o n l i n e o rR e g r e s s i ow n ith MATLAB Problem Stotemeni. Recallthat in Example13.4,we fit the power model to datafrom Table l3.l by linearizationusing logarithms.This yielded the model: F : 0.2741ute\a2 Repeat this exercise,but use nonlinear regression.Employ initial guessesof I for the coefficients.
I4.5 NONLINEAR REGRESSION
327
Solution. First, an M-file function must be createdto computethe sum of the squares. The following file, called f ssR. m,is set up for the power equation: function f = fSSR(a,xm,ym) yp = a(1)*xm.^a(2); f = s u m (( y m - y p ) . ^ 2 ) ; In commandmode, the datacan be enteredas >> x = t10 20 30 40 50 60 7O B0l; >> y - 125 7A 380 550 610 1220 830 14501; The minimization of the function is then implementedby >> fminsearch (@fSSR,[1, 1] , 2.5384
tL
x, y)
r.4359
The besrfit model is therefore F : 2.5384u1a35e Both the original transformedfit and the presentversion are displayedi-li'Frg.14.4. Note that althoughthe model coefficientsarevery different,it is difficult to judge which fit is superiorbasedon inspectionof the plot. This example illustrateshow different best-fit equationsresult when fitting the same This model using nonlinearregressionversuslinear regressionemployingtransformations. is becausethe former minimizes the residualsof the orisinal datawhereasthe latter minimizes the residualsof the transformeddata.
FIGUREI4.4 velocity ondunlronsformed modelfitsforforceversus dctofrom Comporison of tronsformed ODIE
J
,
ta
a
z {
800
Untransformed
4 40 u, m/s
, aa a
328
LINEARLEAST-SAUARES AND NONLINEARREGRESSION GENERAL
F IT T INS GIN U S OIDs Bockground. Engineersand scientistsoften deal with systemsthat oscillateor vibrate.As might be expected,sinusoidalfunctions play a fundamentalrole in modelingsuch problem contexts. In this discussion,we will usethe term sinusoidto representany waveformthatcanh describedas a sine or cosine.There is no clear-cut convention for choosing either function, and in any case,the results will be identical. For this chapter,we will use the cosine,which is expressedgenerally as * d) f (t) : As * Cv cos(a-rst
(14.14)
Thus, four parametersserve to characterizethe sinusoid (Fig. 14.5a). The mean valued sets the average height above the abscissa.The amplitude C, specifies the height of the
F I G U R EI 4 . 5 . o r t h i s c o sA f uonl c t i o n } :( A r )o * C , c o s ( a r o r * 9 ) F en , =1.7,C,=), { o }A p l o t o ft h e s i n u s o i d ao: 2n/l 1.5 s),ond 0 : r/3 rodions(b)An olternotive expression of thesomecurveis in Thethreecomponents of thisfunction oredepicted * B,sin{arol). I(r) : Ao + A) + A,cos{coor} of thethreecurvesin (b)yieldsthesingle {b),whereA, : 0.5 ond B, : -0 866.Thesummotion curvein {o).
cl
t,s at, fad
I4.6 CASESTUDY
329
continued oscillation. The angular frequency o.4characterizeshow often the cycles occur. Finally, the phase angle, or phase shift 0 parameterizesthe extent to which the sinusoid is shifted horizontally. It can be measuredas the distancein radiansfrom / : 0 to the point at which the cosine function begins a new cycle. The angular frequency (in radians/time) is related to frequency/(in cycles/time) by oo :2r.f
(14.15)
and frequency in turn is related to period I (in units of time) by I
(14.r6)
r-T
Although Eq. (14.14) is an adequatemathematicalcharacterizationof a sinusoid, it is awkward to work with from the standpoint of curve fitting becausethe phase shift is included in the argument of the cosine function. This means that it is not in the form of the general linear least-squaresmodel (Eq. U.7), and we must use nonlinear regressionto estimate the coefficients. This deficiency can be overcome by invoking the trigonometric identity: sin(p)] cos(p)- sin(cosr) Cr cos(abl + d) : C1[cos(rr-rst)
(r4.r7)
SubstitutingEq. (14.17)into Eq. (14.14)and collectingtermsgives (Fig. 14.5b) f (t) : Ao * Ar cos(a-lol)* 81 sin(arer)
( 1 4l.8 )
where Ar : Ct cos(0)
Br : -Cr sin(9)
(14.19)
Dividing the two partsof Eq. (14.19)gives / B'\ 0 : arctan| --| \ Arl
(14.20)
where if A r < 0, add z to 9. Squaringand summingthe two partsof Eq. ( 1a.19) leadsto
(14.2r) Thus,Eq. (14.18)representsan alternativeformulationof Eq. (14.14)that still requires four parametersbut that is cast in the format of a general linear model (Eq. 4.7). Thus, it can be simply applied as the basis for a least-squaresfit. The average monthly maximum air temperaturesfor Tucson, Aizona, have been tabulated as Month T'"C
.J lB9
F 2 1 . 1 23.3
AM 27.8
32.2
A J 3 7 . 2 3 6 .I
S 1.1 ^
OND 2e.4 23.3
r8.9
330
G E N E R ALLI N E A R LEAST-SQUAR AE NS D NONTINEAREGRESSION
continued Observe that the July value is missing. Assuming each month is 30 days long, fit a sinusoid to this data. Use the resulting equation to predict the value in mid-July. Solution.
In a similar fashion to Example 14.3, we can enter the data and createthe
lZl matrix: >> >> >> >>
vrO=2*pi/360; t=t15 45 75 105 135 165 225 255 285 315 3451'; r=118.9 2r.r 23.3 27.8 32.2 31 .2 36.1, 34.4 29.4 z= [ones(size(t)) cos(w0*L) sin(w0*t)];
23.3 18.91';
The coefficients for the least-squares fit can then be computed as >> a= lZ'*Z ) \ (Z'*T)
28.381 B -9 .2s59 ,2 . BA02 The statistics can also be determined as >> Sr=sum((T-Z*a)
.^2)
6 .4398 >> r2=I-Sr
/ s u m( ( T - m e a n ( T ) ) . ^ 2 )
0.9862 >> syx=sqrt (Sr/ (length(t) syx
-lengch(a)
) )
= 0.8972
A plot can be developed as >> tp=[0:360]; >> Tp=a (1)+a (2 ) *cos (w0*tp)+a (3 ) *sin (w0*tp) ; >> plot (t,T,'o', tp,Tp) The fit, which is displayed in Fig. 14.6, generally describes the data trends. Equations (14.20) and (14.21) can be used to express the best-fit sinusoid in themore descriptive format of Eq. (14.14): >> theta=at.an2 (-a (3 ) , a (2J ) *360 i (2*pi ) theta = L63.r6'76
331
CASESTUDY
continued
15
300
350
FIGUREI4.6 Leost-squores fitof o sinusoid to temperoture dotoforTucson, Arizono >> C1=sert (a(2 ) ^2+a (3) ^2)
9 .67 A2 Note that because it is more physically meaningful, the phase shift is expressed in days rather than in radians. Therefore, the final model is
/2n \ R : 2 8 . 3 8 7 8+ 9 . 6 7 0 2 c o|s* - t r + 1 6 3 . 1 6 7|6 ) \Jbu
/
We can see that the mean is 28.3878, and the amplitude is 9.6702. Hence, the maximum monthly temperaturein Tucson rangesfrom about 18.7 to 38.1 'C. The phaseshift tells us that the peak temperatureoccurs about 163 days prior to the start of the year, which translates to about day 197 (July 17). The value in mid-July (/ : 195 d) can thereforebe computed as > > T p J u l = a ( 1 ) + C 1 * c o s( w 0 * ( 1 9 5 + t h e t a ) ) TpJul = 38.0530
332
G E N E R ALTI N E A R LEAST-SQUAR AE NS D NONLINEAR EGRESSION
Estim and C
PROBLEMS 14.1 Fit a parabolato the data lrom Table 13.1.Determine the r2 for the fit and comment on the efficacy of the result. 14.2 Using the same approachas was employed to derive Eqs. (13.15)and (13.16),derive the least-squares fit of the following model: _ \ ' : . t l i r+ a 2 . t 2 + e That is, determinethe coefficientsthat reslllts in the leastsquaresfit for a second-orderpolynomial with a zero intercept. Test the approach by using it 1(Jfit the data fiom T a b l e1 3 . 1 . 14.3 Fit a cubic polynornialto the following data:
.r34 16 J
578 44 34
36
2.2
9 28
t) 3B
)2 4.6
Along with the coefficients.determinerr and -s,.7_.. lzl.4 Develop an M-file to implement polvnomial regression. Passthe M-file two vectorsholding the -r and ) values along with the desiredorder rn. Test it by solving Prob. 14.3. 14.5 For the data from Table P14.5.use polynornialrellession to derive a predictive equation for dissolved oxygen concentrationas a function of temperaturelor the casewhere the chlorideconcentrationis equalto zero.Employ a polynomial that is of sufficiently high order that the predictions match the numberof significantdigits displayedin the table. 14.6 Use multiple linear regressionto derive a predictive equationfbr dissolvedoxygen concentrationas a function of temperatureand chloridebasedon the dataliom TableP14.5. Use the equationto estimatethe concentrationof dissolved oxygenfora chlorideconcentration of 15 glL atT : 12"C. TABTE P14.5
Note that the true value is 9.09 mg/L. Computethepercent relative error for vour prediction. Explain possiblecauses fot the discrepancy. 14.7 As comparedwith the rnodelsfiom Probs.14.5and 14.6, a somewhatmore sophisticatedmodel that accounts fbr the eflect of both temperatureand chloride on dissolved oxygen saturationcan be hypothesizedas beingof theform o:
xrO1)223344 I x2 0 2 y 151 t7.e t27
T, "C
c:OglL
0 5 t0
45 2B t3
r5
0r
20 25
909 B.26
c=lOg/L
12.9 It3 r0 t 903 B.l7 746 A e5
I 1 2 2 2 I 256 205 35 I 297 454442
Computethe coefTicients. the standarderror of the estimate. and the conelation coefticient. 14.9 Thc lbltowing datawas collectedfbr the steadyflowof u a(erin a concretccilcularpipe: Experiment
Diomeler, m
l 2 3 4 5 6 7 B I
03 06 0.9 03 06 O9 0.3 06 09
Dissolved Oxygen (mglt) for Iemperolure ('C) ond Concenlrotion of Chloride (g/t)
Slope, m/m
0 00t 0 001 0 00r 001 00r 0 0t 005 005 005
Flow, m3/s
aa4 4.24 069 0 13 a82 238 0 . 3r 195 566
c=20g/L
114 t0 3 B96 BOB 735 573 6)O
Use multiple linear regressionto fit the fbllowing modelto this data: Q
14.rr
solar r P.
hG) *./r(c)
That is, a third-orderpolynomial in temperature anda linear relationshipin chloride is assumedto yield superiorresults. Use the general linear least-.squares approachto fit tlis model to the data in TablePl4.5. Use the resultingequadon to estimatethe dissolvedoxygenconcentrationfor a chloride concentrationof 15 g/L aI T : 12 'C. Note that thetrue valuc is 9.09 rng/L. Compute the percentrelativeenorfor your prediction. 14.8 Use multiple linear regressionto fit
Dissolvedoxygen concentrotionin woier os o function of lemoerofure ("C) ond chloride concentrotion(g/L).
t p(t)
- a11D"'5"'
where Q : flow, D : diameter,and S: slope. l4.ll) Three disease-canyingorganisms decay exponentially in scawateraccordingto the following model:
where maxin radiati UtE n Iru, b"
r5( P9(
14.1 and fi
User follor .rU,
,a
14.1. teriz exan
U(
wher subs The
lsl, C C C t l t( 5(
rii ,ii
PROBLEMS
the initial concentrationof cach organisrn(A, B, E$imate C)siventhe followins mcaslrremcnts: and
r
05
, , I, \ A
l AA
'2 )-)
3 .t
f\t
4
5
6
7
9
2
1.9
1.7
1.4
Ll
(a) Use a transfbrnrationto linearizethe nrodeland evaluate thc paramcters.Display the data and the rnodelfit on a grapn. (b) Pcrforrnthc sanreevaluationas in (a) bul usenonlinear regrcsslon. 14.14 Givcn thc data
l4.ll Thefollowing model is uscdto representthe efl'cctol fadiation on the photosynthesis ratc of aquaticplants: r solar
y
I P=P,,, ,-1;*t I ,,,, where P : the photosynthesis rate 1mg m 3d-l), P,, : the photosynthcsisratc (mg m-id l.), 1 : solar maximum (lrE nr 2s l). and 1,,,/: optimal solar radiation radiation rs-l). Usc nonlinearrcgressionto evaluateP,, and lpEm based 1,u, on the fbllowing data: r
5a\
P99
80 r30 200 254 350 454 550 7AA )77 2A2 248 229 219 173 142 72
5 l7
l0 24
25 37
35 4A 45 40 40 42
30 37
50 4l
legressionto {lt (a) a straightlinc, (b) a usc least-squares power equation,(c) a saturation-growth-rate equirtion.and (d) a parabola.For (b) and (c), employ transfbnnationsto Iincarizcthe data.Plot the dataalongwith all the curves.Is arly one of fhe curve-ssuperior?If so,jLrstify. 14.15 The following datarepresentsthe bacterialgrorvthin a liquid cultureover of numberof days:
Doy
a
Amounl x to linearrze l{,12In Prob. I3.8 we used transtbrmations 106 fit thefollowing moclel: and )'
15 20 31 33
4
B
t2
t6
20
67 38 7467 8274 9l69 rot 6C l]2 58
- n1't g'-B ,r'
Find a best-flt equation to the data trend. Tr)r several nonlinearregressionto estimateaa arrdp.r basedon the possibilities-polynomial,logarithmic, and exponential. Use data.Developa plot ol your fit along with thc da1a. Determinethc bestequationto predict the amouut.of bactefollowing riu atier 30 days. Ii, a2 04 06 oe r3 r5 t7 IB 14.16 Derive the lcast-squares flt ol the fbllowing model:
1 4 . 7 5) 2 5 I 4 5 1 2 5 0 8 5 0 5 5 0 3 5 0 2 8 0 1 8 11.13 Enzymaticrcactionsarc usetl extcusively to characterize biologicallyrnediatedreactions.The following is rn of a model that is usedto fit sr-rch reactions: example k,,,IS]3 ''-K+[.tll
.f': l]1,\:lazx2+e. That is, determinethe coefficientsthat resultsin the leasl squares tit tbr a second-orderpolyuomial with a zero intercept.Test thc approachby using it to fil the data fron.r T a b l e1 3 . 1 . 1 4 . 1 7 D y n a m i cv i s c o s i t yo f w a t e rp ( 1 0 3 N . s / m 2 ; i s l c I('C)in the ftrllowinglranner: lateclto telnperaturc
u,here u0: the initial rate of the reaction (M/s). [S] : tne concentration(M). and ft,,,and K are paramcters. T substrate Ihefollowingdata can be tit with this modcl: lL
lsl,M
cc r 005 0l 0-\ ,5
rc 50 r00
uo,M/s 6 O Z Bx l O ') 7.595x la 6 C ) 6 3r l 0 3 5ZBBxlOr' 1 . 7 3 7x l O ' 2423v.)a5 2430xlC' 2431 x la' 2 , 1 3 1. t O 5
0 5 i.787 I5t9
t0 I3Q7
20 I002
30 4a a 7,)75 0 6529
(a) Plot this data. (b) Use linearintcrpolatior.r to predictp at T :7 .5 "C.. (c) Usepolynornialregressionto lit a parabolato thc data in order lo make the si.rmeprediction. 14.18 Use the following setof pressure-volume datato find the bestpossiblcvirial constants(A, and Ar) fbr the following equationof state.R : 82.05rnL atm/gnlol K. and I :
-r03K. PV
At
At
I i It
334
P (orm)
V (mt)
GENERAL LINEARLEAST-SAUARES AND NONLINEARREGRESSION
O 9B5
2saoo
I t0B 22,204
I 363 rB000
collectedto quantify this relationship.Developbest-fitequa-
163t tions fbr both the thinking and braking components.Use r 5 0 0 0 these equationsto estimatethe total stoppingdistancefor a car travclingat ll0 km/h.
l.l.l9 Environmentalscientistsand engineersdealing with thc impacts of acid rain must detcrmine the value of the ion productof water K,, as a lunction of temperature.Scientists have sr.rggested the following equation to model this relationshio: - l o 9 1 6K .
a
0 t0 2A 30 4A
* D log,o 7,, I cT,, I d
t0 : l0 r5 l0-r5 t0-' t0- '
45 8.5 l2 3
60 75 90 ltt 145 )67 2 l . o 3 2 . 9 4 76
t2A 224 847
fime, h 0 2 4 5 7 I l2 t5 2A 2224 pH 76 72 7 6.5 7.5 72 8.99t BS 79 7
1.1.22The solar radiation fbr Tucson. Arizona. has been tabulatedas
K, I . 1 6 4x 2950x 6846x 1467x 2 . 9 2 9x
30 56 5.0
ll.2l The pH in a reactor varies over the courseof a day. Use least-squares regressionto fit a sinusoidto the following data. Use your fit to determinethe mean, amplitude,and time of maximumpH.
where I, : absolutetemperature(K), and a, b, c, and d are parameters.Employ the fbllowing data and rcgressionto estimate the parameters:
r (K)
Speed,km/h Thinking,m B r o k i n g ,m
Time, mo F OND I Rodiotion, w/m2 1 4 4 t B B ) / \ 3 r r 3 5 r 3 5 93 0 82 8 72 6 4 2 r l r 5 9 r 3 r
11.20 The distancerequircd to stop an automobileconsists of both thinking and braking components,eachof which is a function of its speed.The fbllowing cxperimentaldata was
Assumingeach month is 30 days long, fit a sinusoidto this data. Use the resulting equationto predict the radiatronrn mid-August.
.'.: r"it'liii
Polynomiol Interpolotion
CHAPTEROBJECTIVES The primary objectiveof this chapteris to introduceyou to polynomial interpolation. Specific objectivesand topics coveredare ' ' ' ' ' ' '
Recognizingthat evaluatingpolynonrialcoefficientswith simultaneous equations is an ill-conditionedproblem. Knowing how to evaluatepolynomial coetficientsand interpolatewith M A T L A B ' s p o l y f i t a n dp o 1 7 v a 1 f u n c t i o n s . Knowing how to pelfbrm an interpolationwith Newton's polynomial. Knowing how to pertbrm an interpolationwith a Lagrangepolynomial. Knowing how to solve an inverseinterpolationproblem by recastingit as a roots problem. Appreciatingthe dangersof extrapolation. Recognizingthat higher-orderpolynomialscan manif-estlargeoscillations.
YOU'VE GOT A PROBTEM f we want to improve the velocitypredictionfor the free-fallingbungeejumper, we might I expandour model to accountfor other fhctorsbeyondmassand the drag coefTicient.As I I was previouslymentionedin Section2.7,the drag coefficientcan itself be formulatedas a function of other factors such as the areaof the jumper and characteristicssuch as the air's density and viscosity. Air density and viscosity are commonly presentedin tabular form as a function of temperature.For example.Table 15.1is reprintedfrom a popularfluid mechanicstextbook (Whire, 1999). Supposethat you desiredthe densityat a temperaturenot includedin the table.In such a case,you would have to interpolate.That is, you would have to estimatethe value at the
336
POLYNOMIAL INTERPOLATION
TABTEl5.t T,'C /A
0 2A 50 100 150 200 254 300 400 500
Density(p),dynomicviscosily(p), ond kinemotic viscosity(u)os o functiono[ (f) ot I otm os reportedby White (1999). temperoiure 1t, N . s/m2
p,kg/m3
5t x 7l x B Ox 95 x 2 . 1 7x 2 . 3 8x 2 . 5 7x 2 75 X
52 29 2A 09 4.946 0 835 0 746 0 675 06r6 0 525 a.457
/
YJ
lO 5 lO 5 l0 5 lO r lO-5 l0 5 lO-r IU'
x l0 r X ]U
355 X
U'
-
v , m2ls 0.99x l0 r 1 33 x l0-r j l . 5 Ox l 0 l /9 x 105 230xl0j 2 85 x l0-5 3 . 4 5x l 0 - j 4 O Bx l O - j 4 75 x lA': 6 20 x lO-j 777 xla5
desiredtemperaturebasedon the densitiesthat bracketit. The simplestapproachis t0 determine the equationfor the straightlirre connectingthe two adjacentvaluesandusethls equation to estimatethe density at the desiredintennediatetemperature.Althoughsuch linear interpolationis perfectlyadequatein many cases,error can be introducedwhenthe data exhibits significantcurvature.In this chapter,we will explore a numberof different approachesfor obtainingadequateestimatesfor suchsituations.
I5.I
INTRODUCTION TO INTERPOTATION You will frequently have occasionto estimateintermediatevalues betweenprecisedata points. The most common method usedfor this purposeis polynonial interpolation. The generalformula for an (a - l)th-order polynomialcan be written as ,f(x):ar*azxla3x2
+..'+erx"t
(15.1)
For n datapoints,thereis one andonly onepolynomialof order (n - l) that passes through all the points. For example,there is only one straightline (i.e.. a first-orderpolynomial) that connectstwo points(Fig. 15.1a).Similarly,only one parabolaconnectsa setofthree points (Fig. l5.lr). Polltnomialinterpolationconsistsof determiningthe unique@ - l)thorder polynomial that fits n data points. This polynomial then provides a formulato computeintermediatevalues. Before proceeding.we should note that MATLAB reprcsentspolynomialcoefficients in a different mannerthan Eq. (15.1).Ratherthan usingincreasingpowersof .r, it usesdecreasingpowers as in I '+'.. f (x): plx" I p:.t'
1 1 t , ,1 t I p , ,
(15.4
To be consistentwith MATLAB. we will adontthis schemein the followins section.
EXA
15. I INTRODUCTION TO INTERPOLATION
337
FIGUREI5.I Exomplesof nierpolotingpolynomiols:(o) firsl-order (lineor)connecfingtwo poinis, { b ) s e c o r d - o r d e ' { o u o d ' o l o: 6' o o ' o b o l i c )c o n n e c r i n tgL r e ep o i n t s o n d { c )r h i r d - o r d e{ cr u b ' c l conneciinoiour ooints.
l5.l.l
Defermining
Polynomiol Coefficients
A straightforwardway for computingthe coefficientsof Eq. (15.2)is basedon the fact that n datapoints are requiredto determinethe n coefficients.As in the following example,this allows us to generaten linear algebraicequationsthat we can solve simultaneouslyfor the coefficients.
E X A M P L1E5 . . l
D e t e r m i n i n gP o l y n o m i oC l oefficientw s i t h S i m u l t o n e o uE s quotions Problem Stotemeni. Supposethat we want to determinethe coefficientsof the parabola, \ pl,thatpassesthroughthelastthreedensityvaluesfrom l 5T. la: b l e .f (x): pt.t21 p2-+
rr : 300 "f(rr) : 0.6t6 xz:400 f(x):0.525 . r : : 5 0 0 . f( x ) : 0 . 4 5 ' 7 Eachof thesepairscanbe substituted into Eq. (15.2)to yield a systemof threeequations; 0.616: pt(30q2 + p2Q00)+ p3 0.525: p1@00)z + p2(400)+ p3 0.451: pr(500)2-t p2600)-l pt or in matrixform:
3oo rl I p,| [ o.oro I oo.ooo I 1 6 0 . 0 0o 0ot l l n z | : l o . s z s ] 4 I p, 1250.000s00 lJ I
J
10.4s7|
Thus, the problem reducesto solving three simultaneouslinear algebraicequationsfor the three unknown coefficients.A simple MATLAB sessioncan be used to obtain the
338
INTERPOLATION POLYNOMIAL solution: >>
format long - 190000 3tl0 1;160000 404 b = [0.6r6 4.525 0.451 )';
>> A >>
I;250000
500
1l;
>>P=A\b
0.00000115000000 -0.00171s00000000 r.42100ct00000000 Thus, the parabolathat passesexactly throughthe threepoints is f ( . x ): 0 . 0 0 0 0 0 1 1 5 . r 2- 0 . 0 0 1 7 1 5 .+r 1 . 0 2 1 This polynomial then providesa meansto determineintermediatepoints.For example, the value of densityat a temperatureof 350 oC can be calculatedas / ( 3 5 0 ) : 0 . 0 0 0 0 0 1 1 5 ( 3 5 0 )- 2 0 . 0 0 1 7 t 5 ( 3 5 0+) 1 . 0 2 7: 0 . 5 6 1 6 2 5
Although the approachin Example 15.1 provides an easy way to performinterpolation, it has a seriousdeficiency.To understandthis flaw, notice that the coefficientmatrix in Examplel5.l has a decidedstructure.This can be seenclearlyby expressing it ingeneral terms:
l-"i -rr I I lr.'.tr) ll
t'.-l Lxi ,r.r I J
i\l:{ilii}
Coefficient rnatricesof this tbrm arc ref-eflrd to as Vctndennondenntrices. Suchmatrices are very ill-conditioned.That is, their solutionsarevery sensitiveto round-offenon. This can be illustratedby using MATLAB to computethe conditionnumberfor thecoefficient matrixfrom Examole15.1as >> cond (A)
5.8912e+a)0t-, This condition number, which is quite large for a 3 x 3 matrix, implies that about six digits of the solution would be questionable. The ill-conditioning becomes even worse asthe number of simultaneous equations becomes larger. As a consequence, there are altemative approaches that do not nranifbst rhis shortcoming. In this chapter, we will also describe two altematives tlrat are well-suited for computer implementation: the Newton and the Lagrange polynomials. Before doing this, however. we will first briefly review how the coefficients of the interpolating polynomial can be estimated directlv with MAILAB's builrin functions.
I 5.2 NEWTONINTERPOLATING POLYNOMIAL
339
15.1.2 MATTAB Functions: polyf it dlrd potyval Recall from Section 13.4.2,that the polyf it function can be usedto perform polynomial regression.In such applications,the number of data points is greaterthan the number of fit line doesnot necessarily coefficientsbeing estimated.Consequently,the least-squares passthrough any of the points,but ratherfollows the generaltrend of the data. For the casewherethe numberof datapointsequalsthe numberof coefficients,po ly f it performs interpolation.That is, it returnsthe coefficientsof the polynomial that pass directly through the datapoints. For example,it can be usedto determinethe coefficients of the parabolathat passesthroughthe last threedensityvaluesfrom Table 15.1: >>
format
longt
>> r: [300 400 500]; >> density = i0.616 0.525 0.451); >> p = polyfit
(T, density,2
)
O . O O O O O l l 5 O O O O I ] O_ O . O O 1 7 1 5 O O O O O O O OI . 0 2 ' I O O O O O T ] O O O O O We can then use the polyval >> d = polyval
function to perform an interpolation as in
(p, 350)
d,
0.56762500000000 These results agree with those obtainedpreviously in Example l5.l with simultaneous equations.
15.2 NEWTONINTERPOTATING POTYNOMIAT There are a variety of alternativeforms for expressingan interpolatingpolynomial beyond the familiar format of Eq. (15.2). Newton's interpolatingpolynomial is among the most popular and useful forms. Before presentingthe generalequation,we will introducethe first- and second-orderversionsbecauseof their simple visual interpretation. 15.2.1 lineor Interpolotion The simplestform of interpolationis to connecttwo datapoints with a straightline. This technique,called linear interpolation,is depictedgraphicallyin Fig. 15.2. Using similar triansles.
fi(r) - f (rt) ^
4l
f(x)-f(xt) x2-xl
( 15.,1)
which can be rearrangedto yield
fr?):f(x)+
f(xr)-
f(,rr)
x2*xl
(1 s . s )
which is the Newton linear-interpolation formula. The notation /r (-r) designatesthat this is a first-order interpolatingpolynomial. Notice that besidesrepresentingthe slope of the --rr) is a finite-difference line the the term
340
INTERPOLATION POLYNOMIAL
._*\\
f(xz) d
.li(r)
---_-------:-,/ I rii t
,'
,.4
-/
-/
./
"f(,xr)
T5.2 FIGURE used triongles theslmilor indiccrte oreos Theshoded of ineorinierpolofion. C.phicoldepiction i o d e r i v ei h e N e w t o n l l n e o ri n l e r p o l o l i ofno r m u l ol F q { 1 5 . 5 ) ]
approximationof the first derivative{recallEq. (a.20)1.In general,the smallertheinterval betweenthe data points, the better the approximation.This is due to the fact that,asthe interval decreases,a continuousfunction will be better approximatedby a straightline. This characteristicis demonstratedin the following example'
'I5.2 L i n e o rI n t e r p o l o t i o n EXAMPLE Problem Stotement. E,stimatethe naturallogarithmof 2 using linear intelpolation.First, Then, perform the computationby interpolatingbetweenln I :0 and ln 6: 1.791759. the Note that ( 4 I .386294). I ln ln to repeatthe procedure,but use a smallerinterval liom true valueof ln 2 is 0.6931412. Solution. We useEq. ( 15.5)from :r1: 1 . 7 9715 e- 0
.fi (2:o ) + ff
I to rz : 6 to give
(2- 1): 0' 358351e
w h i c h r e p r e s e n t s a n e r r o reo,f: 4 S . 3 V c . U s i n g t h e s m a l l e r i n t e r v a l f r o m x: l I t o ; r : = 4 yields - 0 t.-r86294
lre\ :0+
t) :0.4620e81
POLYNOMIAL I 5.2 NEWTONINTERPOLATING
341
J(.;r) 2
05r
FIGURE I5.3 Two lrneorinterpololions lo esiimoleln 2 Note how the smollerinlervolprovideso better esiimote.
Thus. usirrg the shorter interval reducesthe percentrelative error to €t :33.3Vo. B
| 5.2.2 GluqdroticInterpolotion The error in Exaniple 15.2resultedfrom approximatinga curve with a straightline. Consequently,a strategyfor improving the estimateis to introducesomecurvatureinto the line with a connectingthe points.If threedatapointsare available,this can be accomplished second-orderpolynomial (alsocalled a quadraticpolynomial or a parabola).A particularly c o n v c n i e nltb r r nl b r t l r i sp u l p o s ei s l : ( x ) : b t * b z \ - : r r ) - | D : ( . t- . r r ) ( x - : ; z )
(15.6)
A simple procedurecan be used to determinethe values of the coefTicients.For b1, Eq. (15.6.twith -r :.rl can be usedto conrpLlte b1: .f (.xy)
(15.7)
Equation(15.7)can be substitutedinto Eq. (15.6),which can be evaluatedat r : -r2 for .f (x) - l'(,tr) , (15.8) br:"{2
- .\'l
into Eq. (15.6),which can be evaluatedat Finally,Eqs.(15.7)and (15.8)can be substituted for -t : -r-.rand solved(after some algebraicrrranipulation.s) .f(-t:)- /(-r:) h--
1l
-
f(.rz)- l(.rr)
.{2
{.1 - .rt
(I 5.9)
342
POLYNOMIAL INTERPOLATION Notice that, as was the casewith linear interpolation,b2 still representsthe slopeofthe line connectingpoints xy and "r2.Thus, the first two terns of Eq. (15.6) are equivalent to linear interpolationbetweenx, and rr, as specifiedpreviouslyin Eq. (15.5).The lastterm, b:(-r - xr)(.r -.r:2), introducesthe second-ordercurvature into the formula. Before illustrating how to use Eq. (15.6), we should examinethe form of the coefficient &.,.It is very similar to the finite-differenceapproximationof the secondderivative introducedpreviously inEq. (4.27').Thus, Eq. (15.6) is beginningto manifesta structure that is very similar to the Taylor seriesexpansion.That is, terms are addedsequentially t0 captureincreasinglyhigher-ordercurvature.
.l5.3 EXAMPLE
Q u o d r o t i cI n i e r p o l o t i o n Problem Stotement. Employ a second-orcler Newton polynomial to estimateln 2 with the samethreeooints usedin Example 15.2: rl:1
"/(xt):0
.\z:4
J$):1.38629.1
r-r : 6
-f(.r:) : l.'791159
S o l u t i o n . A p p l y i n gE q . ( 1 5 . 7 )y i e l d s bt:o Equation( 15.8)gives b 1:
|.386294- 0 _:0.4610981 +- |
F I G U R EI 5 . 4 T h eu s eo f q u o d r o t ircn t e r p o o t i o l one s tm o l eI n 2 . T h el i n e o irn l e r p o l o i ifor no m , r : I t o4 i s o l s oi n c l u d efdo rc o m p o r i s o n . | \.r) 2
fl.r) : In.t
Ouadraticestimate Linearestimate
343
I 5.2 NEWTONINTERPOLATING POLYNOMIAL a n d E q . ( 1 5 . 9 )y i e l d s - 1.386294 1.791759
6-4 6- I
- 0.462098r : -0.0518731
Substitutingthesevaluesinto Eq. (15.6)yieldsthe quadraticformula . f z @ ): 0
+ 0 . 4 6 2 0 9 8 1 ( -r l ) - 0 . 0 - 5 1 8 7 3 1 -( x l ) ( : r - 4 )
which can be evaluated At ,r : 2 for fz(2) : 0.5658444, which representsa relative error of e, : 18.47o.Thus, the curvatureintroducedby the quadraticformuia (Fig. 15.4) improves the interpolation cornparedwith the result obtained using straight lines in Example15.2and Fig. 15.3.
| 5.2.3 Generol
Form of Newfon's
Interpolofing
Pofynomiqfs
The precedinganalysiscan be generalizedto fit an (zz- l)th-order polynornialto n data points.The (n - I )th-orderpolynomialis . f " _ t f ) : b r 1 -b z j - x r ) * . . . * b u \ - ; r r ) ( x - x ) . . . ( x * - t , _ r )
(15.10)
As was done previously with linear and quadraticinterpoliition,datapointscan be usedto e v a l u a t et h e c o e f f i c i e n tbs1 . b 2 , . . . , b , . F o r a n ( n - 1 ) t h - o r d epr o l y n o m i a ln, d a t ap o i n t s a r e r e q u i r e dl r:r , l ' ( - r r ) 1 , l x z , J Q ) 1 , . . . , [ - r ,f ,( x , 1 1 .W e u s e t h e s e d a t a p o i n t s a n d t h e following eqnationsto evaluatethe coefficients: b1:fQ1)
( 1 5 .l r)
6' : .f'[r2,.r]
(1s.12)
f u : f f v , x z ,x t l :
(r5.r3)
b , ,: f l x , r ,x n - 1 ,. . . , " r z , , r r ]
(r-5.14)
where the bracketedtunction evaluationsare finite divided differences.For exanrole.tlre first finite divided differenceis representedgenerallyas r,
,
J l ^ i . r"i l :
-I k i t - f ( r l ) -t, _ _yl
( 1 s1. 5 )
The secondfinite divided difference.whichrepresents the difference of two first divided differences,is expressedgenerallyas
f lx;, .q, x1,l:
f [ x ; ,. q J- f [ r i . x r l
(15.16)
ri - .rt
Similarlv. the nth finite divided differenceis .\n -
( rs . 1 7 ) "Yl
344
POLYNOMIAL INTERPOLATION
xi
f(x)
First
Second
';:rl it::ti=;[] .:..l ii,j.':^:),a4t;'^'
xl
Third r flxo, xt,x2,xl
FIGURE I5.5 Grophicol depiction is of iherecursive noture of finitedivided Thisrepresentotion differences. ,eferred to oso divided difference toble.
Thesedifferencescan be usedto evaluatethe coefficientsin Eqs.(15.11)through(15.14), which can then be substitutedinto Eq. (15.10)to yield the generalform of Newton'sinterpolating polynomial: f,-rG):.f
( x t ) + ( r - x t ) f l x z . . r r l * ( " r- x r ) ( x - x ) f l . 4 , x 2 , x 1 l + . " + ( x - x r ) ( x - . r z ) . . . ( r - x n - t ) . f l x n , t r n - t , . . . , x z , x r l( 1 5 . 1 8 )
We should note that it is not necessarythat the data points used in Eq. (15.18) be equally spacedor that the abscissavaluesnecessarilybe in ascendingorder,asillustrated in the following exarnple.Also. noticehow Eqs. (15.15)through(15.17)arerecursivethat is, higher-order differences are computed by taking differences of lower-orderdifferences(Fig. 15.5).This property will be exploitedwhen we developan efficientM-filein Section15.2.4to implementthe method. E X A M P L E1 5 . 4
N e w t o n I n t e r p o l o t i n gP o l y n o m i o l Problem Siotement. In Example 15.3,data pointsat xt : l, xz:4, and,r3= ( 1ry91s = used to estimate ln 2 with a parabola. Now, adding a fourth point [xa : 5; ,f (.r,{) L6094381,estimateln 2 with a third-orderNewton's interpolatingpolynomial. Solution. The third-orderpolynomial,Eq. (15.10)with n : 4, is f z @ ) : b r * b z @- x r ) * b z ? - x r ) ( , r - x ) * b q ( x- , r r ) ( x - x 2 ) ( x - x 3 ) The first divided differencesfor the problem ale [Eq. (15.15)] 1.386294- O - :0.4620981 : -"f["r:.rr] 4 _,
f [4, x2]: fr... .,/ L^41
- _ l --
^-{l
- r.386294 r.791159
:0.2027326 6-4 - 1.19r'759 1.609438 :0.1823216 5-6
I 5.2 NEWTONINTERPOLATING POLYNOMIAL
345
The seconddivideddifferences are [Eq. (15.16)] /[.t3, .t2, -r1] :
. f f x a , x 3 ,x 2 ) :
- 0.4620981 o.2021326 : - 0 . 0 5 1 8 7 3I 1 6-l 0.1823216- 0.2027326
: -0.02041100
5-4
The third divideddiff'erence is [Eq. (15.17)with n :4] J ' [ x + , r t . - t 2 ,x 1 ] :
-0.02041 r00- (-0.051873 1l) J -
: 0.007865529
|
Thus. the divided differencetable is
f (xi) j
0 386294 791759 609438
4 6 5
t) e d
n
a 462A9Bt 0.2427326 o l823216
-cc5lBZ3il - 0 0 2 0 4 1r 0 0
00a786552a
T h e r e s u l t sf o r / ( x r ) , f [ x z , x t ] , . f [ x t , x z , x 1 ] , a n d f l x q , x z , x 2 , . t l l r e p r e s e ntth e coefficientsb1,b2,b3, andb',, respectively, of Eq. (l-5.10).Thus,the interpolatingcubicis -f:(r) :0
+ 0 . 4 6 2 0 9 8 1 (-x l ) - 0 . 0 5 1 8 7 3 1 l (-" t l ) ( x - 4 ) + 0.007865529(x- l)(x - 4)(r - 6)
which can be used to evaluate.fzQ) :0.6287686, which representsa relative error of €t :9.3Vo. The completecubicpolynomialis shownin Fig. 15.6.
FIGURE I5.6 The use of cubic interoolofion lo estimoten 2
f(x) z
346
INTERPOLATION POLYNOMIAL
15,2.4
MATTAB M-file: Newtinr
It is straightforwardto developan M-file to implementNewtoninterpolation.As in Fig.15.7, the first step is to compute the finite divided differencesand storethem in an array.Thedifferencesare then usedin conjunctionwith Eq. (15.18)to per-formthe interpolation. An exampleof a sessionusing the function would be to duplicatethe calculation we just performedin Exarnple15.3: >>
form.et = tl
>> x
Longt 4
6
5l
FIGUREI5.7 An M fl e lo imp ement Newton inferpolotion.
= Newtint(x,y,xx) function yint polynomial I'Jewton interpolal-ing % Newtint: = llewtrnt(x,y,xx): Uses an (n - 1)-order Newton ? yint ,x, yl polynrmial inr erpofdtrng "o bosed or n dala poinrs lyrnt) % to determine a vafue of the dependent variable at a given value of the independenc variable, xx. % % input: :-.lanor F va ' ' rh'c 1Fr x%
z
v-
%
xx=
r.lanonrlanf
value
z
Intrlyn^
? output yint %
rr:ri
:h-l
o
independent
of
;-
a-'o-
fdf
variable
at
which
C-IaLeO
i
=
inierpolated
!} .omnrl e rlre finir-e
value
divided
of
dependent
differences
variable
in
the
form of
a
% difference table n = length(x); : r
1Fr
ifnlv,-:-
Fr-^1
!
| s
rli
z
mr
c
ho
camo
etnt
hri:
6ncl
b = zeros (n,n) ; ? assign dependenL variables to the first column of b. = y(:); b(:,1) % the (:) ensur:es that y is a column vector for j - 2,n for i = 1:n j+1- (b(i+1,j-1)-b(i,j-1))/(x(i+j-r)-x(i)) b(i,j) ; end end ? use Lhe finite 1.ts
,
1
divided
-
= b(1,i); for j = 1;n-1 xr xt"{xx x(jr); = yrnt+b(1, j+1) *xt; yint
yint
end
differences
to
inLerpolate
I 5.3 LAGRANGE INTERPOLATING POLYNOMIA.
347
> > y = t o S f( x ) ; >> Newtint(x,y,2)
0.6287685"t890841
I5.3
TAGRANGEINTERPOTATING POTYNOMIAT Supposewe formulatea linear interpolatingpolynomialas the weightedaverageof the two valuesthat we are connectingby a straightline:
./(.r): Lt J @r)-t Lzf Qz)
(rs.r9)
where the Z's are the weightingcoefficients.It is logical that the first weightingcoefficient is the straightline that is equal to I at x1 and 0 at x2:
xt-x2 Similarly,the secondcoefficientis the straightline that is equalto I at "r2and0 at,r1: r
tr-trI 12 -
.rl
Substitutingthesecoefticientsinto Eq. 15.19yields the straightline that connectsthe p o i n t s( F i g . 1 5 . 8 ) : .. : "/r(x)
.r-X2' xl-X2
^ J 6) -l
.\'-.Xr X2-Xt
(1s.20)
J',6)
where the nomenclatnrefi(l) designatesthat this is a iirst-order polynomial. Equation (15.20) is refered to as the LinearLagrangeinterpolatingpolynomial. The samestrategycan be employedto fit a parabolathroughthreepoints.For this case threeparabolaswould be usedwith eachone passingthroughone of the points and equaling zero at the othertwo. Their sum would thenrepresentthe uniqueparabolathat connects the threepoints. Such a second-orderLagrangeinterpolatingpolynomial can be written as At.t
-
(.r-rr)(.r-xr) ( x r x z.). (- -- r l- x : ). J r i l r - r
+
(.t - . \ ' r ) ( r - . 1 r ) --.,-=1./
(r3-x1)(xj-r2)
/,r r-.l , \
-
(x-x1)(x-x3)
( . { : - x r ) ( . r :- - r t )
j
tt2t
( r5 . 2 1 )
N o t i c e l r o w t h e f l r s t t e r n r i s e q u a l t o / ( . r 1 ) aat n 1d 1isequaltozeroat.r2and.r-l.Theother terms work in a similar fashion. Both the first- and second-orderversionsas well as higher-orderLagrangepolynomials can be representedconciselyas
(1s.22)
348
INTERPOLATION POLYNOMIAL
LtfU)
FIGUREI5.8 A v i s u o ld e p i c t i o no f t h e r o f i o n oe b e h l n dL o g r o n g ei n t e r p o l o i i npgo l y n o m i o l sT.h ef i g u r es h o w s t h e f i r soi r d e r c o s e E o c h o ft h e t r " o l e r m s oE [ q ( 1 5 2 0 ) p o s s e s t h r o u o h o n elohfe p o i n t s o n d line is zero ot the other.The summoiionof ihe two fermsmusi,iherefore,b"ethe uniquestroiqhl thot connectsihe h^/opornts.
where tl
_-
! I j=l
y. ^1
r L;(.r): | | '
-fl
(1523) _
| . r/
.IFI
where ,r : the numberof datapoints and fl designatesthe "product of."
,|5.5 Interpoloting Polynomiol EXAMPLE Logronge polynomialof thefirst andsecond ProblemStotement. Use a Lagrangeinterpolating data: orderto evaluate thedensityof unusedmotoroil at T : 15'C basedonthefollowing rr :0
f(xr):3.85
r::20
l(.r:):0.gtlO
.{::40
f(rt):0.212
polynomial Solution. Theflrst-order tEq.(15.20)lcanbe usedto obtaintheestimateal ,r: I5: t 5 - 2-0 t 5 - o" 0 . 8 0 0 : 1 . 5 6 2 5 -1.85+'" 'f r( . r ) : ' " 0-20 t0-0
349
POLYNOMIA. I 5.3 LAGRANGEINTERPOLATING
polynomial is developed as[Eq.( 15.21)] In a similarfashion, thesecond-order -
/.{r):
(15-20)tl5-40.r^^_ (15-0)(15-40)^^^^ - i . 6 1+ - t ] . 6 t . , U (20- 0)(20- 40) (0 - 20)(0- 40) +
(l5-O)fl5-20).,_,_ U.212:1.J316875 (40-0)(40-20)
| 5.3.1 MATTAB M-file: Lasranse It is straightforward to developan M-file basedon Eqs.(15.22)and (15.23).As in (x) and the Fig. 15.9,the functionis passedtwo vectorscontainingthe independent (y) variables. variablewhereyou It is alsopassed thevalueof theindependent dependent (xx). Theorderof thepolynomialis basedon thelengthof thex vector wantto interpolate If n valuesarepassed, an (n - 1)thorderpolynomialis fit. thatis passed.
FIGUREI5.9 inlerpolction AnMJilelo implement Logronge y nL - Lag"6ngerx,\,xxl trrccion polynomial Z Lagrange: Lagrange interpolatingr = Lagr:ange(x,y,xx) : Uses an (n - 1)-order yint 2 polynomial based on n data points ? Lagrange inLerpolating (yint) at to determine a value of the dependent variable % xx. Z a given value of the independent variable, Z input: x - independent variable Z y= % dePendentvariable xx = value of independent variable at whlch the % is calculated % inter:polation ? output: = interpolated yint value of dependenc variable ? n = lenqth(x); if lenqth(y)-=n, -
_
error('x
and y must be same length');
n.
= 1:n ; Produce - I'i for j = 1;n if i -= i product = product* (xx-x(:)
for: i
end end s = s+prc,duct; end = s; yint
) / (x(i) -x(j)
) ;
end
3so
POLYNOMIAIINTERPOLATION An exampleof a sessionusing the function would be to predict the densityof airat I atm pressureat a temperatureof 15 "C basedon the first lour values from Table15.1. Becausefour valuesare passedto the function, a third-orderpolynomial would be implementedby the r-aqrange functionto gile: >>
format
longr
>>T=t-40024541;
>> d = 1r.52 1.29 i.2 >>
densitY
o-ns
L a Q fr a n g e
t.091; (T, d, 15 )
i I \
I .22II2B
I5.4
-
41 22222 2
INVERSE INTERPOLATION As the nomenclatureimplies, the .f ('.r)and.t valuesin most interpolationcontextsarethe dependentand independentvariables,respectively.As a consequence,the valuesof the r's are typically unilbrmly spaced.A simple exampleis a table of valuesderivedfor the t u n c t i o nf ( r ) : l/x: x
f@)
2 05
3 0 3333
4 025
5 o2
6 0.1667
7 0 . t4 2 9
Now supposethat you must use the samedata,but you are given a value for ./(r) and must determinethe correspondingvalue of x. For instance,for the dataabove,suppose that you were askedto determinethe value of I that correspondedto l (-r) : 0.3. For thiscase, becausethe lunction is availableand easyto manipulate,the correctanswercan be determ i n e dd i r e c t l )& S- { : 1 1 0 . 3: 3 . 3 3 3 3 . Such a problem is called inverse interpolatiort For a rnore conrplicatedcase,you might be temptedto switchthe /(;r) and.r values[i.e..merelyplot x versus/(x)] anduse an approachlike Newton or Lagrangeinterpolationto determinethe result.Unfortunately, when you reversethe variables,thereis no guaranteethat the valuesalong the new abscissa [the /(.r)'s] will be evenly spaced.In fact. in many cases,the valueswill be "telescoped." That is, they will have the appearanceof a logarithmic scalewith some adjacentpoints bunchedtogetherand othersspreadout widely. For example,for /(r) : l lx theresultis
f (x) x
o t429 7
o 1667 6
02 54
425
03333 32
05
Suchnonunitbrm spacingon the abscissaoften leadsto oscillationsin the resultinginterpolatingpolynomial. This can occur even for lower-orderpolynomials.An alternative strategyis to fit an nth-orderinterpolatingpolynomial,/, (r), to the original data[i.e.,with f (r) versus.rl. In most cases,becausethe.r's are evenly spaced,this polynomialwill not be ill-conditioned.The answerto your problem then amountsto finding the valueof .rthat makesthis polynomial equalto the given /(,r). Thus. the interpolationproblemreduces to
I 5.5 EXTRAPOTATION AND OSCILLATIONS
351
For example,for the problernjust outlined,a simple approachwould be to fit a quadraticpolynomialto thethreepoints:(2, 0.5).(3, 0.3333),and(4, 0.25).The resr.rlt wouldbe / : ( r ) : 0 . 0 4 1 6 6 7 r r- 0 . 3 7 5 r+ 1 . 0 8 3 3 : l The answer to the inverse interpolation problem of finding the .r correspondingto l (.r) : 0.3 would therelbreinvol'r,edeterminingthe root of 0 . 3 : 0 . 0 4 1 6 ' 71 2 - 0 . 3 1 5 x+ 1. 0 8 3 3 3 For this simplecase,the quadraticfbrmulacan be usedto calculate
0.375+
(-0.37s)2- 4Q.041661)0.78333 : 5.704158 3.295842 2(0.041667)
Thus, the secondroot, 3.296,is a good approximationof the true value of 3.333.If additional accuracywere desired,a third- or tburth-orderpolynomial along with one of the root-locationmethodsfrom Chaps.5 or 6 couid be employed.
I5.5
EXTRAPOTATION AND OSCITLATIONS Before leaving this chapter,there are twcl issuesrelatedto polynornialinterpolationthat must be addressed. Theseare extrapolationand oscillations. 15.5.1 Extropolofion Ertapolation is the processof estimatinga value of ./ (,r) that lies outsidethe rangeof the k n o w n b a s ep o i n t s ,r r , - r 2 , . . . . . t , , . A s d e p i c t e di n F i g . 1 5 .1 0 , t h e o p e n - e n d enda t u r eo f
F I G U R IE5 . I O lllustrolion of thepossible divergence of on extropoloted prediction. Theextropolofion isbosed through on fittinq o porobolo thLfirstthreeknownpoinfs .f (x) ,.
I n t e r p o l a t i o n. ' E x t r a p o l a t i o n -
r, Extrapolation I 0 rof1 interpolati tnteroolattno Il .
polynomial
I I I I I I t>
l"r
3s2
POLYNOMIAL INTERPOLATION extrapolationrepresentsa step into the unknown becausethe processextendsthe curve beyondthe known region.As such,the true curve could easilydivergeliom the prediction. Extreme care should, therefore,be exercisedwhenever a case arises where one mu$ extrapolate.
E X A M P L E1 5 . 6
D o n g e r so f E x t r o p o l o l i o n Problem Stqtement. This exampleis patternedafteroneoriginallydevelopedby Forsythe, Malcolm, andMoler.rThe populationin millions of the United Statesfrom 1920to 2000can be tabulatedas Dote )S2O I 930 l94O I 950 I 960 l97O I 980 I 990 2000 Populalion 10646 123.08 13212 15227 tBA67 2A505 22723 24946 28142
Fit a seventh-orderpolynomial to the first 8 points (1920 to 1990).Use it to cornputethe population in 2000 by extrapolationand compareyour prediction with the actualresult. Solution. First,the datacanbe enteredas >> t = L\924:10:19901; >> pop = 1105.45 I23.08 132.I2 L52.27 180.61 205.05 227.23 2 4 a. r 6 l ; The polyf it function can be usedto computethe coefficients >> p - polyfrt
(t, pop,7 )
However,when this is implemented.the following messageis displayed: VJarnlncJ: Polynomial po'nt s or
is
badLy
condil-ioned.
rr2l conter ino
Remove repeated drid scdr jng ds desct ibeo
data in
HELD
POLYFIT.
We can follow MATLAB's suggestionby scalingand centeringthe datavaluesas in >> rs = (r - 1955)/35; Now po11,fit works without an errormessage: r . ' >p = p o l y f i t ( t s , p o p , 1 ) ; We can then use the polynomial coefficientsalong with the Fo1yr731function to predict fhe populationin 2000 as :'> polyval (p, (2000-f955)i 35) 175.0800 which is much lower that the true value of 281.42.Insight into the problem can be gained by generatinga plot of the dataand the polynomial, > > t t = l i n s p a c e ( I 9 2 A , 2 0 0 A ); >> pp = polyval (p, (tt-1955) /35); >,
nloi
(r.{)oD.,o,,tt,pp)
E}
I 5.5 EXTRAPOLATION AND OSCILLATIONS
353
r\
250
200
150
1920 1930 1940 1950 1960 1970 1980 1990 2000 F I G U R EI 5 . I I U s eo t o s e v e ^ t l ' o ' d e ' p o l y ^ o nrioortr o i . eo p r e d i c t i oon[ U . S .p o o u l oo n , " 2 O O O b o s e oo n d o t oi r o ml 9 2 Of h r o u q h 1990.
,,.*1;;?,[i;1]ii*:;:;*::TJ:il:1Jl:J,",'n::ilT::;: polynomialplungesto the erioneouspredictionin 2000. extrapolation,the seventh-order
| 5.5.2 Oscillotions Although "more is better" in many contexts,it is absolutelynot true for polynomial interpolation.Higher-orderpolynomialstend to be very ill-conditioned-that is, they tend to be highly sensitiveto round-off eror. The following exampleillustratesthis point nicely. E X A M P L lE5 . Z
D o n g e r so f H i g h e r - O r d e rP o l y n o m i o Il n t e r p o l o t i o n Problem Stotement. In 1901,Carl Rungepublisheda studyon the dangersof higherorder polynomial interpolation.He looked at the following simple-lookingfunction: I rt l l\t -: - '
l *25x2
(t5.24)
which is now called Runge'sfunction. He took equidistantlyspaceddata points from this function over the interval t-l, ll. He then used interpolatingpolynomials of increasing order and found that ashe took more points,the polynomialsandthe original curve differed considerably.Further,the situationdeterioratedgreatly as the order was increased.Duplicate Runge'sresult by using the polyf it and polyval functionsto fit fourth- and tenthorder polynomials to 5 and 1l equally spacedpoints generatedwith Eq. (15.2$. Create
354
POLYNOMIAL INTERPOLATION
Solution. The five equally spaceddatapointscan be generatedas in >> x
-
linspace(.-I,L,5);
>> Y
=
I./(I+25*x.^2);
Next, a more finally spacedvector of rx valuescan be computedso that we cancreate a snooth plot of the results: >> xx = linspace(-1,1); Recall that l inspace automaticallycreates100 points if the desirednumberof points is not specified.The po ly f r t function can be usedto generatethe coefficientsof thefounhorder polynomial, and the polval function can be usedto generatethe polynomialinterpolation at the finely spacedvaluesof xx: = po11'f it >> 1.4 = polyval >> p
(x,y , 4) ; (p, xx) ;
Finally. we can generatevaluesfor Runge'sfunction itself and plot them alongwiththe polynomial fit and the sampleddata: >> yr - I.i (1+25*xx."2); >> plot
(x,y,'
--'
o',xx,y4,xx,yt:,'
)
As in Fig. 15.12,the polynomialdoesa poorjob of followingRunge'stunction. Continuingwith the analysis.the tenth-orderpolynomial can be generatedandplotted with >>
X
=
>>
y
=
linspace( 1,1,11); I . / (I+25*x. ^2) ;
FIGURE I5.I2 funct]on Cornporison o[ Runges line)wiiho fourth-order polynomiol fitto 5 poinis {doshed sompleo lrom lne iunclron.
/t
r\
It
l! t;
|t ,t
i
r\ t\
r\ i\ ta\
t5.l (
tta
x "f(r) -0.2 -0.4 -1
- 0.5
(a) C al fc
PROBLEMS
355
a
-0.5 -1
-nF
0.5
F I G U R IE5 . I 3 Comporisonof,Runge'sfunctlon{doshedline)with o tenih-order polynomiolfit to I I points somoledfrom lhe lunction
>> p = polyfrt (x,y,10) ; > > y 1 0 = p o l y v a l (p , x x ) ; >> plot(x,y,'o',xx,y10,xx,yr,'
-'
)
As in Fig. 15.13,the fit has gotteneven worse,particularlyat the endsof the interval! Although theremay be certaincontextswherehigher-orderpolynomialsarenecessary, they are usually to be avoided.In most engineeringand scientific contexts,lower-order polynomials of the type describedin this chaptercan be used effectively to capturethe curvins trendsof datawithout sufferins from oscillations.
PROBTEMS l5.l Giventhe data
Giventhe data
2.5 7
3 65
5r o "f(r)
I 475
3 5.25
5 )9.75
6
36
Calculate/i3.4) usingNewton'sinterpolatingpolynomi- Calculate/(4) usingNewton's interpolatingpolynomialsof alsoforder I through3. Choosethe sequenceofthe points order I through 4. Chooseyour basepoints to attain good for your estimatesto attain the best possibleaccuracy. accuracy.What do your results indicate regarding the order - aL^ r ^_-^-_^ -^l.,nnminl ofthe usedto generatethe data in the table?
356
POLYNOMIAL INTERPOLATION
15.3 RepeatProb. 15.2 using the Lagrangepolynomial of best estimate?Employ this best estimateand inverseinten order I through 3. polation to determinethc conespondingtemperature. 15.,1TableP14.5lists valuesfor dissolvedoxygenconcentration in water as a function of temoeratureand chloride T ' K 204 254 300 350 404 450 concentratlon. Densiry \ 7 0 8 I 3 6 7 I r 3 9 0 . 9 6 1 0 8 5 4 0 / 5 9 (a) Use quadraticand cubic interpolationto determinethe kglm3 oxygen concentrationfor I : 12 'C and c:10 glL. (tl) Use linear interpolationto determinethe oxygen con15.9 Ohm's law statesthat the voltage drop V acrossan centrationfor I : 12 "C and c : 15 glL. ideal resistoris linearly proportionalto the currcnt I flowing (c) Repeat(b) but use quadraticinterpolation. : I R, where R is the resistance. 15.5 Employ inverseinterpolationusing a cubic interpolat- throughthe resisteras in V However, real resistors may not always obey Ohm's law. ing polynomial and bisectionto determinethe value of x that you performed Suppose that some very prcciseexperiments correspondsto /(-t) : 1.6 for the following tabulateddata: to measurethe voltagedrop and correspondingcurent for a rcsistor.The following resultssuggesta curvilinearrelation2,4( ship rather than the straightline representedby Ohm'slaw: .r12 67
f(x) 36
lB
12
09
072
I 5
A . 5 4t 2 9
*"; t
v 15.6 Employ inverseinterpolationto determinethe value of r that coresponds to l (-t) : 0.93 for the following tabulated data:
r01 J@) 0
0.s
23 0B
09
45 o 94lt76
0 9 6 t5 3 8
-2 637
-t 96 5
-05 -245
05 205
t 965
637
To quantify this relationship,a curve must be fit to thedata. Becauseof mcasurcmcnterror, regressionwould typically be the preferred method of curvc litting fbr analyzing such experimentaldata.However,the smoothnessof the relationship, as well as the precisionof the experimentalmethods, suggeststhat interpolationmight be appropriate.Usea fifthorder interpolatingpolynornialto fit the dataand computeV fori : 0.10. 15.10 Besselfunctionsoften arisein advancedengineering analysessuch as the study of electric fields. Here aresome selcctcdvalueslbr the zero-orderBesselfunction ofthe first kind
Note that the values in the table were gencratedwith the function f (x) : x2I 0 + x2). (a) Determinethe correctvalue analytically. (b) Use quadraticinterpolationand the quadraticformula to determinethe value numerically. (c) Use cubic interpolation and bisection to determinethc value numerically. x IB 20 2.2 24 26 15.7 Use the portion of the given steam table for super- , t ' ( r ) 0 . 5 8 l 5 o 5767 0 5560 4 5242 0 4/08 heated water at 200 MPa to tind (a) the conesponding entropy s for a specific volume u of 0. I l8 with linear interpolation, (b) the same coffesponding entropy using qua- Estimate"/1(2. I ) using third-, and fourth-orderinterpolating dratic interpolation,and (c) the volume correspondingto an polynomials.Determinethe percentrelative enor for each casebasedon the true value, which can be determinedwith entropyof 6.45 using inverseinterpolation. MATLAB's builrin tunctionbessel j. 15.11 RepeatExample 15.6but using tirst-, second-,third-, u, m3/kg o to377 0 il 144 4 1 2 5 4 7 and fourth-ordcr interpolating polynomials to predictthe 64147 s, kJl(kg K) 6 5453 6 7664 populationin 2000basedon the most recentdata.Thatis,for the linear predictionusethe datafrom I 980 and I 990,for the quadraticpredictionusethe datafrom I 970, I 980, and I 990, l5.lt The following data for the densityof nitrogengas ver- and so on. Which approachyields the best result? sus temperaturecomesfrom a table that was measuredwith 15.12 The saturationconcentrationof dissolvedoxygenin high precision.Use firsl throughfifth-order polynomialsto water as a function of temperatureand chloride concentraestimatethe densityat a temperatureof 330 K. What is your tion is listedin TablePl5.l2. Use intemolationto estimate
TAB
T, "
o
5 t0 t5 20 25 30
the r
lO o
15. pola
SOIV
the < the r for l
r5.l
in s pres
T, "1 a, Jl
Dett
r5.l
lar a the r
(
Bec has
(
whe dim
PRoBLEMS
I I I f
nfU Pt5. l2 _
I
',
I r: I l; I I I
I
in woteros o functionof iemperoture Dissolved oxygenconcentrotion {"C)ond chloride c o n c e n t r o t i o( g n/ L ) . DissolvedOxygen (mglt) for Temperoture("C)ond Concentrotionof Chloride (g/Ll
I lt,"a
I
ssl
s=OglL
c=l}g/L
146
l?9
t? 3 '3A,
lJt 3?9
theclissolve,l oxygcnlevclfor ?': 18'C with chloride:
l--
I Iosrr-. I
'33u 93! 114
7 46 685
Bt6 756
25 ;0
c=2Og/L
inter15.13 Forthc datain TableP15.12,usepolynomial
673 62A
n
m-----lT
I EU:-',;-*':::.iliH;i:J^"Ti::#T?.,*"H"t; /"/ / i esrimare rhedi:solvcd o\yscn conce-nrrarirrn _ _+ / I ff;:ul.ro fo" |rheeaser,lhel.cch|or.idecollccntrationisequa|tol0g/L.UseJ,#|
rolunre ol'u :upcrlicrte(lsteanris listctl lS.U Thc.specil'ic
TI Forexampre ata I ;;:'.',ll:l*,iffiiil:temrcraturcs F r c u RPEr s . l s I /oo 72.) z*a 7(;a 7Ba I t "t I
r 6613 I u,lt3/lb- a aa-7 a l2t14 a l4060 a li5o9 c-1 ltabulatecI,aportionol'whichisgiveninTablePl5.15.Ifa: 1.6 andh : 14,usea third-orderinterpolatingpolynomiat to I ^ I at : 750 'F compufeo- at it depthI (i rl beiou' thc corner of a rectangu.r I .D.t:tj'Tltt \tre\s,"- under thc c.rncr of a rectangu15'15The lar footing that is sub.iectto a total ]oad of 100 t (metric .venl11t J l I r r a r e a : t r b j c c t e d | o l u n i | t r t' -. t: t' t' l r l"a' 't "J t r"| ' i n 1 c n s i t . r r 7 i r e t ti rr rernsb) r. E r p r e s s ) o u r . t n s $ c r i n ( o n n e s p e r s q u a r e n l c t c r . N ( ) t c L,'.-',-.'.-'-;:'"*"'."" o l B o u s : i n c s q 'esq' u a t i o n : I t h er o l u t i o n inat 4 ts cquatto Inc tolloper area. I
q I z*nJ.2 +-,7+ t r] + n2+2 J ; rI t r r : + n : i l + m 2 n 2 m 2 + n 2 + l L
-si.(#.+#)l Bccau.sc this cquationi.sincorrvenient fo solye nranually,, it hasbeenreformulatedas o- -- t1.f-\nt,n) where.f(rrr. n) is calledthe influenccvalut-.and lr and rr are dirnensionless ratios,with nt : al: and rr : b/; and a and b are defined in Fis. Pl5.l-5. The influcncevalue is then
TABIEPl5.l5
0l o2 03 a4 05 06 a7 OB
tt = 1.2
n=1.4
n=1.6
c 42a26 4.45733 0 08323 0 t063t c 12626 0 14309 0 I 5203 0 I 6843
0 a3aa7 0 a5Bq4 0 . 0 8 5 6r 0 t0941 0 13003 o . 1 4479 016199 0 rz3B9
0 03058 a a59q4 a aB709 0l]t35 a.13241 a.l5a27 0 1 6 5 51 4.17739
358
POLYNOMIAL INTERPOLATION
15.16 You measurethe voltagedrop Vacrossa resistorfor a number of ditl'erentvaluesof current L The resultsare
425 -o.45
475
1. 2 , 5 a70
t5 ]BB
20 60
TABTEP I 5. | 9
"y=O y =2 t=4 Use first- through fourth-orderpolynomial interpolationto t = 6 estimatethe voltagedrop tbr i: 1.15.Interpretyour results. J = 8
Temperotures ("C)ot voriouspoints on o squoreheotedplote.
x=O
x=2
x=4
x=6
10000 8 50 0 2000 s5.00 40.00
90 00 64.49 48.90 38.78 35.00
B0 00 53 . 5 0 3 84 3 30 39 3000
70.04 6000 48 I5 5 00 0 3 5 0 3 4000 2 7 . 0 7 3000 2 5 . 0 0 2000
x=8
| 5. I 7 The currentin a wire is measuredwith greatprecision as a function of time: l-i. 19 Temperaturesare measuredat various pointsona at 0 . 5 0 0 0 heated plate (Table P15.19). Estimate the temperature 0 0 0 0 0 ( a ) . r : 4 , ) ' : 3 . 2 , a n d( b ) r : 4 . 3 .y : 2 ; 7 . 15.20 Use the portion of the given stearntablefor superheated HrO at 200 MPa to (a) tind the conesponding D e t e r m i n ei a l t : 0 . 2 3 . entropys for a specificvolurne u of 0.108 mi/kg with linear l5.llJ The accelerationdue to gravity at an altitudey above interpolation,(b) find the samecorrespondingentropyusrng the surfaceof the earth is given by quadraticinterpolation,and (c) find the volumeconespond. ir-rgto an entropyof 6.6 using inverseinterpolation.
t t
a 0
01250 624
02500 775
03250 4.85
0 30,000 60,000 90 000 120,000 !, m g, m/s2 9 8100 9 7487 9.6879 9 6278 9 568'2 cornputeg ar-y: -55,000 m.
u (m3lkg) s (kJlke.K)
0 14377 6.4147
a.| 144 6.5453
0.r25; 6.7664
..:r,f i,. i'
'...i.i,,,, . ",,*i l'l .
.
Splinesond rrecewrse Interpolotion
CHAPTEROBJECTIVES The primary objectiveof this chapteris to introduceyou to splines.Specificobjectives and topics coveredare . . o . . o '
I6.I
Understandingthat splinesminimize oscillationsby fitting lower-order polynomialsto datain a piecewisefashion. Knowing how to developcode to perform a table lookup. Recognizingwhy cubic polynomialsare preferableto quadraticand higher-order splines. Understandingthe conditionsthat underliea cubic splinefit. Understandingthe differencesbetweennatural,clamped,and not-a-knotend conditions. Knowing how to fit a splineto datawith MATLAB's builrin functions. Understandinghow mr.rltidimensional interpolationis implementedwith MATLAB.
INTRODUCTION TO SPLINEs ln Chap. 15 (n - l)th-order polynomialswere used to interpolatebetweenn data points. For example, for eight points, we can derive a perfect seventh-orderpolynomial. This curve would captureall the meanderings(at leastup to and including seventhderivatives) suggestedby the points. However,thereare caseswhere thesefunctionscan lead to erroneous results becauseof round-off error and oscillations.An alternativeapproachis to apply lower-orderpolynomialsin a piecewisefashionto subsetsof datapoints. Such connectingpolynomialsare calledsplinefunctions. For example, third-order curves employed to connect each pair of data points are calledcubic spl.ines.Thesefunctionscan be constructedso that the connectionsbetween
360
INTERPOLATION AND PIECEWISE SPLINES
FIGUREI6.I interpoloting A visuolrepresentotion of o situotionwhere splinesore superiorto higher-order polynomiols.The funciionto be fit undergoeson obruptincreoseol -r : O. Porls(o) through(c) i n d i c o t et h o tt h e o b r u p tc h o n g ei n d u c e so s c i l l o t i o ni sn i n t e r p o l o i i npgo l y n o m l o l sl n. c o n t r o s t , connections,o lineorsp ine {d) provideso much more becouseit is lirnitedto strcigl-Lt-line occeolobJecoproximotron.
adjacentcubic equationsare visually smooth.On the surface,it would seemthat thethirdYou order approximationof the splineswould be inferior to the seventh-orderexpression. might wonder why a splinewould ever be preferable. Figure 16.1 illustratesa situationwhere a spline performs betterthan a higher-order polynomial.This is the casewhere a function is generallysmoothbut undergoesan abrupt changesomewherealong the region of interest.The stepincreasedepictedin Fig. 16.1is an extremeexampleof such a changeand servesto illustratethe point. Figure 16.la throughc illustrateshow higher-orderpolynomialstendto swingthrough wild oscillations in the vicinity of an abrupt change.In contrast, the spline also connects
1 6 . 2 L I N E AS RP L I N E S
36r
F I G U R EI 6 . 2 t e c h n i q uoef l s i n go s p l i n el o o r o w. -' n[ io' o hc l r u e .l l r o l g - o s e r i eosf p o n i s \ o t i c e T[eorclting h o w o l t e e n dp o i n l sl [ e s p l i n se t r o i g h - eonusi . i s< o le d o ' " o ' u r o l s p i n e
minimum. As such,the splineusuallyprovidesa superiorapproximationof the behaviorof functionsthat have local, abruptchanges. The conceptof the splineoriginatedfrom the drafting techniqueof using a thin, flexible strip (calleda spline) to draw smoothcurvesthrougha setof points.The processis depicted in Fig. I 6.2 for a seriesof five pins (datapoints).In this technique,the drafterplaces paperover a woodenboard and hammersnails or pins into the paper(and board) at the location of the datapoints.A smoothcubic curve resultsfrom interweavingthe strip between the pins. Hence,the name "cubic spline" hasbeenadoptedfor polynomialsof this type. In this chapter,simple linear functionswill first be usedto introducesomebasic conceptsandissuesassociated with splineinterpolation.Then we derivean algorithmfor fitting quadraticsplinesto data.This is followed by materialon the cubic spline,which is the most common and useful version in engineeringand science.Finally, we describeMATLAB's capabilitiesfor piecewiseinterpolationincluding its ability to generatesplines.
r6.2 T I N E A RS P T IN E S The notationusedfbr splinesis displayedin Fig. 16.3.For n datapoints (l : I, 2, . . . , n), there are n - I intervals. Each interval i has its own spline function, s;(x). For linear splines,eachfunction is merely the straightline connectingthe two points at eachend of the interval, which is formulatedas (16.r) si (r) : ar I bi(x - ;ri) where ai is the intercept,which is definedas ai:
Ji
and b; is the slopeof the straightline connectingthe points; Ji+r - Ji , ui--
(16.2)
(16.3)
362
SPTINES AND PIECEWISE INTERPOLATION
J(.tt
attttrr_
F I G U R EI 6 . 3 l \ o ' o ro n u t e dl o d e ' ' , es o l ' ^ e sNotice thot therecre n - I intervolsond n dcio points
where li is shorthandfor l(r;). SubstitutingEqs. (16.1)and (16.2)into Eq. (16.3)gives f..,
Ji (.r ) :
.l ;
_
-r L)----l' rirl-ri
f.
(.r - .ti )
(r6.4)
Theseequationscan be usedto evaluatethe function at any point betweenx1 ard.l,, is by first locating the interval within which the point lies. Then the appropriateequation usedto determinethe function valuewithin the interval.Inspectionof Eq. (16.4)indicates that the linear spline amountsto using Newton'sfirst-orderpolynomial[Eq. (15.5)]to interpolatewithin eachinterval.
E X A M P LI 6E. I
F i r s t - O r d eSr p l i n e s Problem Stotement. Fit the data in Table 16.I with first-ordersulines.Evaluatethe functionotx:5. TABIE l6. l
Dototo be lit with splinefunc]ions.
ix,f, I 2 3 A
30 45 7.4 90
25 I0 2.5 0.5
Solution. The data can be substitutedinto Eq. (16.4) to generatethe linearspline functions.For exarnple,for the secondintervalfrom x : 4.5 to x :7, the functionis sr(x - ) : 1.0 +
2 5-1.0 --::-------:t.r - 4.51 1.0 - 4..5
I6.2 TINEAR SPTINES
363
F I G U R ET 6 . 4 S p l i n ef i t so f o s e lo f f o u rp o i n i s{.o )L i n e osrp l i n e(,b )q u o d r o f iscp l i n eo, n d ( c )c u b i cs p l i n ew, i t h o c u b i ci n f e r p o l o l i pn og l y n o m i o l s op l o t t e d .
The equationsfor the other intervalscan be computed,and the resultingfirst-ordersplines a r ep l o t t e di n F i g . l 6 . 4 c tT h e v a l u ea t x : 5 i s 1 . 3 . 2 . 5- t . 0 -4.5): s-r ( x ) : 1 . 0 + - ( 5 1 . 0- 4 . . 5
l.-l
Visual inspectionof Fig. 16.4aindicatesthat the primary disadvantageof first-order splinesis that they are not smooth.In essence,at the data points where two splinesmeet (called a knot), the slopechangesabruptly.In formal terms,the first derivativeof the function is discontinuousat thesepoints. This deficiency is overcomeby using higher-order polynomial splinesthat ensuresmoothnessat the knots by equatingderivativesat these points,as will be discussedsubsequently. Beforedoing that,the following sectionprovides an applicationwhere linear splinesare useful.
364
ET E R P O L A T I O N S P L I N EASN D P I E C E W I SI N
16.2.1 Toble lookup A table lookup is a common task that is frequentlyencounteredin engineeringandscience computer applications.It is useful tbr performing repeatedinterpolationsfiorrr a tableof independentand dependentvariables.For example,supposethat you would like to setup an M-file that would use linear interpolationto determineair density at a particulartemperaturebasedon the datafrom Table 1-5.1.One way to do this would be to passtlreM-file the temperatureat which you want the interpolationto be performedalong with thetwoadjoining values.A more generalapproachwould be to passin vectorscontainingall thedata and have the M-file detenninethe bracket.This is called a table lookup. variThus, the M-file would perform two tasks.First, it would searchthe independent able vector to find the intervirl containingthe unknorvn.Then it would performthelineat interpolationusingone of the techniques describedin this chapteror in Chap.15. For ordereddata, there are two simple ways to find the interval. The first is calleda sequentialsearch.As the name implies.this methodinvolves comparingthe desiredvalue with each elementof the vector in seqnenceuntil the interval is located.For datain ascendingorder,this can be doneby testingwhetherthe unknownis lessthanthe valuebeing lf so, we know that the unknown falls betweenthis value and the previousone assessed. that we examined.lf not, we move to the next value and repeatthe comparison.Hereisa this objective: simpleM-file that accomplishes functior-i n if
yi
= Tablelook(;<,
y,
rr)
cngrn'X); xx error
<
r(1) xx > x(n) (' Inter1;,olation outside
end 13 seclLrential
range'
)
search
i - r; while (1) if xx .= x(i + 1), break, end i = i + 1; end ? linear .interpolation yr = 1.(i) + (y(i+1) y(i))/(x(i+1)
x(i))*(xx
x(i));
The table's independentvariablesare storedin ascendingorder in the arrayx andthe dependentvariablesstoredin the arrayy. Before searching,an error trap is includedtoen. breakloop s u r e t h a t t h e d e s i r e d v a l u e x x f a l l s w i t h i n t h e r a ntgheeox f' s . A r r r [ ] l s . comparesthe value at which the interpolationis desired,xx, to determinewhetherit isless thanthevalueatthetopoftheinterval,x(i+1).Forcaseswherexxisinthesecondintetval or higher,this will not testtrue at first. In this casethe counter i is incremented byone so that on the next iteration,xx is comparedwith the value at the top of the secondintetval. The loop is repeateduntil the xx is lessthan or equalto the interval'supperbound,in which casethe loop is exited.At this point, the inter-polationcan be performedsimplyas shown. because For situationsfor which thereare lots of data.the sequentialsort is inetTicient it must searchthrough all the precedingpoints to find values. In thesecases,a simple alternativeisthe binan- search.Here is an M-file that performsa binary searchfollowed
365
I 6 . 3 Q U A D R A T IS CP L I N E S
by linear interpolation: yi
f unction n tf
= T,rbleLookBrn
(x,
Lenql-h (x ) ; < x(1) | zx > x(.n) €irror: ( 'InterI,olation outside
\',
xx)
=
zx
range')
end % binary
t;
(1)
rvh i le r.f
search iu
l;
rL
iU
.=
tL/L:'tY((tt rf
< x(iM) rT = rI{:
l,
break, u)
en-l
z);
/
xx
t:lse a:Ilal end % Linear yi 1i
inLerpolati.on ,, i l,
t'1"
/ (x(iL+1)-x(iL)
)*(xx
-
:.(rL)
) ;
The approachis akin to the bisectionmethodfbr root location.Justas in bisection,the index at the rridpoint r1.4is computedas the averageof the filst or "lower" index i L = 1 and the last or "upper" index iu : n. The unknownxx is thencomparedwith the valueof x at the midpoint r (iM) to assesswhetherit is in the lower half of the anay or in the upper half. Dependingon where it lies, either the lower or upperindex is redefinedas being the middle index.The processis repeateduntil the differencebetweenthe upperand the lower index is lessthan or equal to zero.At this point, the lower index lies at the lowel bound of the interval containingx:.:,the loop terminates,ilnd the linear interpolationis pertbrmed. Here is a MATLAB sessionillustratinghow the binary searchfunction can be applied to calculatethe air densityat 350 "C basedon the datafi'ornTable l-5.1.The sequential searchwould be similar. > - r = t 4 0 0 2 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 . 10 0 5 0 o l ; > > d e n s i t v = f 1 . 5 : 1 . 2 9 I . 2 I . Q 9 . 9 4 6 . ' , : r 3 1. ,' l 4 6 . 6 1 1 . 6 1 6 >> TableLoo]
- l5 O l
0.5705 This result can be verifiecl by the hand calculation: os?5-o615
: 0.616 * I (350) 1ffi(3s0 I6.3
- 300): 0.s70s
SPTINES QUADRATIC To ensurethat the nth derivativesare contiDuousat the knots.a splineof at leastn * I order r.nustbe used.Third-orderpolynomialsor cubic splinesthat ensurecontinuousllrst and secondderivativesare most frequently used in practice.Although third and higher
366
SPLINES AND PIECEWISE INTERPOLATION
when usingcubicsplines,they usuallycannotbedetected derivativescanbe discontinuous visuallyand consequently areignored. Becausethe derivationof cubic splinesis somewhatirrvolved,we havedecidedtofirst illustratethe conceptof splineinterpolationusing second-orderpolynomials.These"quadratic splines"havecontinuousfirst derivativesat the knots.Although quadraticsplines are not of practicalin.rportance, they servenicely to demonstratethe generalapproach fordevelopinghi gher-ordersplines. The objectivein quadraticsplinesis to derivea secoud-order polynomialfor eachinterval betweendata points.The polynonial for each interval can be represented generally as s i ( r ) : a i t b t Q - x i ) * c ; 1 x- r i ) 2
(16.5)
wherethe notationis as in Fig. 16.3.For n datapoints (i : 1,2, . . . , n), therearen - 1 (thed's. b's, andc's)to evaluate. intervalsand,consequently, 3(n - l) unknownconstants Therefore.3rrequationsor conditionsarerequiredto evaluatethe unknowns.Thesecanbe developedas fbllows: l.
The function must passthrough irll the points.This is called a continuitycondition.It can be expressedmathematicallyits .fi : ai I b,(.t;- xi) * c;(ri - x;)l which sinrplifiesto ai : Ji
EX (16.6)
Therefore,the constantin eachquadraticmust be equal to the value of the dependent variableat the beginningof the interval.This resultcan be incorporatedinto Eq.(16.5): .r;(r ) :
2.
l, t- lt;\.t - .r;) f ri (.r - -r;)l
Note that becausewe have detenninedone of the coefficients.the nurnberof conditionsto be evaluatedhasnow beenreducedto 2(n - l). The function valuesof adjacentpolynomialsmust be equalat the knots.This condition can be writterrfbr knot i -F I as .filbi\i+r
- t i ) f c i ( r i + r - x i ) 2 : . f i + r * b i + r ( x i + r - - r i + r ) f c ; a 1 ( - r ; -1-1t i + r ) 2 (1 6 . 7 )
This equationcan be simplified nrathematicallyby defining the width of the ith interval as hi :xi+t-xi
Thus,Eq. ( 16.7)simplifiesto .fi I bihl -t c;hl : fi+t
(16.8)
T h i s e q u a t i o cn a nb e w r i t t e nf o r t h e n o d e si,: 1 , . . . . n l . S i n c et h i sa m o u n t s t o n-l c o n d i t i o n si ,t m e a n st h a t t h e r ea r e 2 ( n - 1 ) - ( n - 7 ' ) : n 1 remaining conditions.
I6.3 AUADRATIC SPLINES 3.
367
The first derivativesat the interior noclesmust be equal.This is an importantcondition, becauseit meansthat adjacentsplineswill bejoined smoothly,ratherthanin thejagged fashionthat we saw for the linear splines.Equation( 16.5)can be ditferentiatedto yield s j ( . r 1: b i l 2 c i i . r - x i ) The equivalenceofthe derivativesat an interiornode,i f I can thereforebe written as bi *2ciht:bi+r
4.
116.v.1
Writing this equation for all the interior nodes amounts to ,?- 2 conditions.This meansthat thereis n - | - (n - 2) : I remainingcondition.Unlesswe have some additionalintbrmation regardingthe functionsor their derivatives,we must make an arbitrarychoiceto successfullycomputethe constants.Although thereare a numberof different choicesthat can be made,we selectthe fbllowing condition. Assumethat the secondderivativeis zero at the first point. Becausethe secondderivativeof Eq. (16.5)is 2c;, this conditioncanbe expressed mathematically as ct :0 The visual interpretationof this conditionis that the first two points will be connected by a straightline.
E X A M P L1E6 . 2
Q u q d r o i i cS p l i n e s Problem Stotement. Fit quadraticsplinesto the samedataemployedin Example16.I (Table16.l). Use the resultsto estimatethe valueat,r : 5. Solution. For the presentproblem,we havefourdata pointsand rz: 3 intervals.Therefore, after applying the continuity condition and the zero second-derivative condition,this meansthat2(4-1)-l:5conditionsarerequired.Equation(16.8)iswrittenfbri:1 through3 (with cr : 0) to give .filbrlt1 :7, l ' z * b 2 h 2- l c 2 h i : f t -) Jt lb,h,lcfi\:7., Continuityof derivatives,Eq. (16.9),creirtesan additional3 - I :2 recallthat cr : 0): br:bz b z* 2 c z h z : b t The necessaryfurrctionand interval width valuesare l?,
:4
.\ -
{
ll:
I
'\
J z: 1 . 0
h t : 1 . 0 - 4 . 5: 2 . 5
l.-?s
14:9.9-1.0:2.0
l,:o5
conditions(again,
368
SPLINES AND PIECEWISE INTERPOTATION
These valuescan be substitutedinto the conditionswhich can be expressedin matnx form as
j [Ii'i' ll{l}fl
Theseequationscan be solvedusingMATLAB with the resulrs: br:-l b:. : -1
,._ -
h,-))
r. . ,J . -- - 1
A A,l 6
These results, along with the values for the a's (Eq. 16.6), can be substitutedinto the original quadraticequationsto developthe following quadraticsplinesfor eachinterval: sr(x)-2.5-(x-3) s ; ( . r ): 1 . 0- r - r- - 1 . 5+t 0 . 6 4 ( x - + . 5 ) r . r . 1 ( x:) 2 . 5 + 2 . 2 ( . r- 7 . 0 ) - 1 . 6 ( r - 7 . 0 ) r Becausex : -5lies in the secondinterval,we use.r2to make the prediction, . r 1 ( 5 ): 1 . 0- ( - 5- 4 . . s )+ 0 . 6 . 1 ( -- s 4 . 5 1 2: 9 . 6 6 The total quadraticsplinelit is depictedin Fig. 16.4b.Notice that therearetwo shortcomingstlratdetractfrom the fit: (l) the straightline connectingthe first two pointsand (2) the spline fbr the last interval seemsto swing too high. The cubic splinesin thenext section do not exhibit these shortcomingsand, as a consequence, are bettermethods for splineinterpolation.
16.4 CUBTC SPL|NES As statedat the beginningof the previoussection,cubic splinesare most frequentlyused in practice.The shortcomingsof Iinearand qr"radratic splineshave alreadybeendiscussed. Quarlic ol higher-ordersplinesare not usedbecausethey tend to exhibit the instabilities inherentin higher-orderpolynomials.Cubic splinesareprefered becausethey providethe simplestrepresentation that exhibitsthe desiredappearance of smoothness. Tlre objective in cubic splinesis to derive a third-orderpolynomial for eachinterval betweenknots as representedgenerallyby s , ( , r ) : a i ' l b , ( . r - . r , ) * c i ( . r - , r i ) 2 + r / , ( . r- . r i ) 3
(r6.r0)
T h u s , f o r n d a t ap o i n t s ( i : 1 , 2 . . . . , 1 ) , t h e r ea r e / , - I i n t e r v a l sa n d 4 ( n - l ) u n known coefficientsto evaluate.Consequently,4(,- l) conditionsare requiredfor their evaluation.
I 6 . 4 C U B I CS P L I N E S
369
The tirst conditionsare identicalto thoseusedlbr the quadraticcase.Tlrat is, they are set up so that the functionspassthroughthe pointsand that the first derivativesat the knots are equal. In additronto these,conditionsare developedto ensurethat the secondderivatives at the knots are also equal.This greatlyenhancesthe fit's smoothness. After theseconditionsare developed.two additionalconditionsare requiredto obtain the solution.This is a much nicer outcomethan occurredfor quadraticsplineswhere rve neededto specify a singlecondition.In that case,we had to arbitrarily specily a zero second derivativefor the first interval,hencemaking the resultasymmetric.For cubic splines, we are in the advantageous position of needingtwo additionalconditionsand can, therefore, apply them evenhandedlyat both ends. For cubic splines,these last two conditions can be formulated in severaldifferent ways.A very common approacl.r is to assurnethat the secondderivativesat the first and last knots areequalto zero.The visual interpretationofthese conditionsis that the function becomesa straightline at the endnodes.Specification of suchan endconditionleadsto what is termed a "natural" spline. It is given this name becausethe drafting spline naturally behavesin this fashion(Fig. 16.2). There are a variety of other end conditionsthat can be specified.Two of the more popular are the clampedcondition anclthe not-a-knotconditions.We will describetheseoptions in Section 16.4.2.For the following derivation.we will lirnit ourselvesto natural splines. Once the additionalend conditionsare specitied,we would havethe 4(n - 1) conditions neededto evaluatethe4(n - I ) unknowncoefficients.Whereasit is certainlypossible to developcubic splinesin this fashion,we will presentan altenrativeapproachthat requires the solutionof only n - I equations.Further,the simultaneousequationswill be tridiagonal and hencecan be solved very efficiently.Although the derivationof this approachis less straightforwardthan for quadraticsplines,the gain in efficiencyis well worth the effort.
16.4.1 Derivqtionof CubicSplines As was the case with quadratic splines,the lirst condition is that the spline must pass through all the datapoints. J'i: ai I bi6i-
x i ) * c i ( , r r- . t i ) 2 * d i ( x r- x , ) l
which simplifies to ai : .fi
(16.11)
Therefore,the constantin eachcubic must be equal to the value of the dependentvariable at the beginningofthe interval.This resultcan be incorporated into Eq. (16.10): s ; ( - v ):
-l -,ti) J^i bi(r
* ci(x -
xi)2* di(-r -.r;;3
(t6.12)
Next, we will apply the conditionthat eachof the cubicsmustjoin at the knots.For knot i * 1, this can be representedas J ' i I b i h i - l c y h i - t t ) 1 h l: . f i + r
(16.13)
370
SPLINES AND PIECEWISE INTERPOLATION where hi:xi+t-xr The first derivativesat the interior nodesmust be equal.Equation(16.12)is dift'erentiated to yield sj(x) : bi {2c,(x - rr) * 3tli7 - xi)2
(r6.r4)
The equivalenceof the derivativesat an interior node,I * I can therefore be written as bi l2ciht +3dihl : bi+l
(16.15)
The secondderivativesat the interiot'nodes must alsobe equal.Equation(16.14)can be difTerentiated to yield sj'(-r) : 2ci * 6di(x - xi)
( I6 . 1 6 )
The equivalenceof the second derivativesat an interior node, I f 1 can thereforebe wntten as ci l3dihi : ci+t
(1 6 . 1 7 )
Next, we can solveEq. ( 16.17)fbr d1: t u;:-
ci+l -
ci
(r 6r 8 )
J11;
This can be substituted into Eq. (16.i3) ro give h? Ii I bihi -t lt2ci* -t
c ; - 1) : f i * t
(r6.r9)
E q u a t i o n( 1 6 . 1 8 )c a na l s ob e s u b s r i r u r ei ndt o E q . ( 1 6 . 1 5 )r o g i v e b i + t: b i I h i k i i
ci+r)
(r6.20)
Equation(t6.19) can be solvedfor o, , :
f, ,r-ti h,
hi - i l / - c. ^i
(r6.21)
tc;r1)
The inder of this equationcanbe reducedbl' |; ,
l),-1 :
f, - f'-r lli-1
hi- t .^
--llC;
(16.72)
trC;'l
-,
The index of Eq. (16.20)can alsobe reducedby l: b i : b i - r * f t r r ( c i - r * c ,)
( 16.23)
Equations(16.21)and (16.22)can be sr"rbstituted into Eq. (16.23)and the resultsimplified to yield fffr
I t 1 - 1 c 1 --1t 2 ( h i r - h i ) c i * h ; c ; , r : 3 1 I } - J '
h.;
* rJi - Jr-l -t
( 16.21)
37r
I 6 . 4 C U B I CS P L I N E S
This equationcan be made a little more conciseby recognizingthat the terms on the right-handsidearefinite differences(recallEq. 15.15): / t r i , . r i ]'
f.-f. Jt rl
-
ri-rl
Therefore,Eq. (16.2a) can be written as / r 1 - 1 c i - 1* 2 \ h r - r
3 ( . 1[ - " - i + r , . t i-l
- h)ci I hici+t:
ri "f[.ri.
r])
( 16.2s)
- 2' which Equation (16.25) can be written for the interior knots. i :2.3, '.. 'n I unknown coefficients. results in n - 3 sinrultaneoustridiagonal equations with n cr, c2. ..., (-,r-r.Therefore,r'fwe havetwo additionalconditions,we can solvefor the c's. Once this is done,Eqs.( 16.21)and (16.18)can be usedto determinethe remainingcoefficients.b and d. As statedpreviously,the two additionalend conditionscan be torrnulatedin a number of ways. One common approach,the naturalspline,assumesthat the secondderivativesat the end knots are equal to zero. To see how these can be integratedinto the solution s c h e r n e , t h e s e c o n d d e r i v a t i v e a t t h e f i r s t n o d1e6(.lE 6q ) c. a n b e s e t t o z e r o a s i n s i ' ( , r r ): 0 : 2 c t
* 6 d 1( x 1- . r 1 )
Thus, this condition amountsto settingc1 eQualto zero. The sameevaluationcanbe madeat the last node: s j j - , ( x , ,:)0
:2c,-t
(16.26)
t 6d, tlt,-t
Recalling Eq. (16.17), we can convenientlydefine an extraneousparameterc,,, in which caseEq. (16.26) becomes c u - t * 3 d , , - l t r - 1 : c r r: 0 Thus, to impose a zero secondderivativeat the last node,we set ('/?- 0. The final equationscan now be written in matrix form as [1 I ht
I
II I L
Cl
2 ( h t 1 -h )
hz
h,-t
C2
2(.h,-zI h,,-t1
Cn-l cil
I ,r,l
0 - /[.r2.,rr]) 3(,f[x:,.tz.l 3(flx,,rn rl - flx,-t,x, 0 fhc srrqtem is tridironnal
and
(16.2'7)
qolve
372 .l6.3 EXAMPLE
S P L I N EASN D P I E C E W I SI N ETERPOLATION
NoturolCubic Solines Problem Stotement. Fit cubic splinesto the samedatausedin Examples16.1and16,2 ( T a b l e1 6 . 1 ) U . t i l i z et h er e s u l t st o e s t i m a t e t h ev a l u ea t . r : 5 . Solution. The first stepis to employ Eq. (16.27) to generatetl.resetof simultaneous equations that will be utilized to determinethe c cocfficients:
hz) - l t / l r r.., l o / 1 . , . * , 1 , | ;, 20,,.t 'llI I:l I -: I 3 t / l r r .r r l - / I . r r . x 3I l r o' 2tht-rhtt I
l:l I I
I
The necessary lunctionand intervalwidth valuesare
1t :2'5 fz : 1.0
h 1: ! . J - 3 . 0 : 1 . 5 hz:7.0-1.5:2.5 ht:9.0-1.0:2.0
f--)\
l+ : O.-s Thesecan be substituted to yield
Ir
8
lr-5
2.5
,r e
L
,l{ii}1'l
rt. t
Theseequationscan be solvedusingMATLAB witlr the results: cr :0 ct : -0.166539924
cz:0.831)543726 c+ : 0
E q u a t i o n (s1 6 . 2 1 )a n d( 1 6 . 1 8 )c a nb e u s e dt o c o m p u t et h e b ' s a n dd ' s b1: b2 :
-1.119711863 -1).160456274
h :0.022053232
dt :0.186565212 dz : -0.214144181 dt :0.121156654
These results,along with the valuesfbr the a's lEq. (16.1l)1. can be substituted into Eq. (16.l0) to developthe following cubic splinesfor eachinterval: s r ( , r )- 2 . 5 - 1 . 4 1 9 7 7 1 8 6 3 ( x 3 ) + 0 . 1 8 6 5 6 5 2 7 2 1-r - r . 1 1 s 2 ( x ):
1 . 0- 0 . 1 6 0 4 5 6 2 1 4 (-r 4 . - s )+ 0 . 8 3 9 - 5 + 3 7 2 6-1 r- 1 . - 5 , t r 0.2 Il l-l-l-167t.r -1..5)r
- 7.0)r sj(,r) : 2.5 +0.022053232(x- 7.0) - 0.166539924(.r t 0 . 1 2 7 7 . s 6 6 - rs - l ( 7 . 0 ) ' Tlre threeequationscan then be employedfo computevaluesrvithin eachinterval. For example,the valueat r : 5, which talls within the secondinterval,is calculated as s 2 ( 5 ) : 1 . 0- 0 . t 6 0 4 5 6 2 1 4 ( -54 . - 5 ) + 0 . 8 3 9 5 4 3 1 2 6 (-5 4 . 5 ) 2- 0 . 2 1 4 t 4 4 4 8 7 ( 5- 4 . 5 ) 3 : L102889734. The total cubic splinefit is depictedin Fig. 16.4c.
373
I 6 . 4 C U B I CS P L I N E S
T h e r e s u l t s oE f x a m p l e sl 6 . l t h r o u g hl 6 . 3 a r e s u m m a r i z e d i n F i lg6. . 4 . N o t i c e t h e p r o gressiveimprovementof the flt as we move from linear to quaclraticto cubic splines.We a cubic interpolatingpolynomialou Fig. 16.4c.Although the cubic have also superirnposed spline consistsof a seriesof third-ordercurves,the resultingfit differs fronr that obtained usingthe third-orderpolynomial.This is due to the fact that the naturalsplinerequireszerir secondderivativesat the end knots. whereasthe cubic polvnomial has no such constraint.
16.4.2 End Conditions Althoughits graphicalbasisis appealing,the naturalsplineis only one of severalend conditionsthat can be specifiedfor splines.Two of the mostpopularare .
.
Clunlted End Condition.This option involvesspecifyingthe first derivativesat tlrefirst and lastnodes.This is sometimescalleda "clamped"splinebecauseit is what occurs when you clamp the end of a draftingsplineso that it hasa desiredslope.For example, if zerolirst derivativesarespecilied,the splinewill level off or becomehorizontalat the ends. " Not-a-Knot" End Condition.A third alternativeis to fbrce continuity of the third deknots.Sincethe splinealreadyspecifiesthat rivativeat the secondandthe next-to-last the tunction value and its first and secondderivativesare equal at theseknots. specifying continuousthird derivalivesmeansthat the samecubic functionswill apply to Sincethe first internalknotsno longer eachofthe first and iasttwo adjacentsegments. representthejunction of two different cubic functions,they are no longer true knots. Hence.this caseis refenedto as the "not-a-knot" contlition.It hasthe additionalproperty thatfor firur points,it yieldsthe sameresultasis obtainedusingan ordinarycubic interpolatingpolynomialof the sortdescribedin Chap. 15.
These conditionscan be readily applied by using Eq. tt6.25t fbr the interior knots. i:2,3,...,n2 , a n d u s i n g f i r s t ( l ) a n d l a s t e q u a t(ino-n1s) a s w r i t t e n i n T a b1l e6 . 2 . Figure 16.5showsa comparisonof the threeendconditionsasappliedfo tit the datafrom ?rble 16.l. The clampedcaseis setup so that the derivativesat the endsareequalto zero. As expected,the spline fit for the clarnpedcaselevels off at the ends.In contrast,the natul'aland not-a-knotcasesfollow the trend of the data points nrore closely.Notice how the natural spline tendsto straightenout as would be expectedbecausethe secondderivatives go to zero at the ends.Becauseit hasnonzerosecondderivativesat the ends,the nota-knotexhibitsmore curvature.
TABTE16.2
Thefirstond lostequotionsneededto specifysomecommonlyusedend c o n d i t i o nfso r c u b i cs o l i n e s .
Condition
Firsi ond lost Equotions
Noiuro
r : 1: Q , 6 , , : Q
Cicmped(rvhere .li cnd 7;lore thespeciledfirst dt--rlvotlves vely) ct thefirstond iostnodes,respeci
2 h 1 t 11 ' f t 1 r ' -2 i 7 J , . . . , t r l - 3 . / i h r = 1 c , 1 a 2 h , , 1 t , , , : 3 l , ' ,- 3 . l l - , r , , . x1, ,l
lloto knol
h : t t - { l t t * h : l c z * / z r r ' :: O hn tcn)-0t,:I11, i
2c',:Q
AND PIECEWISE INTERPOLATION SPLINES
374
f(x)
2 t
1
0
I6.5 FIGURE nof-o-knot, ond noturolsplinesfor lhe Comporisonof the clomped (wilh zero firslderivotives), doto from Toble 16 l
I6.5
PIECEWISE INTERPOTATION IN MATTAB MATLAB hasseveralbuilt-in functionsto implementpiecewiseinterpolation.The spline function performscubic splineinterpolationas describedin this chapter.The pchip function implementspiecewisecubic Hermite interpolation.The interpl function canalso implement spline and Hermite interpolation,but can also perform a numberof othertypes of piecewise interpolation. 16.5.1 MATTAB Function: spline Cubic splinescan be easily computedwith the built-in MATLAB function, spli-ne. Ithas the generalsyntax, yy
= spl ine (x,
y,
)<x)
(16.28)
where x and y : vectorscontaining the valuesthat are to be interpolated,andyy = a vector containingthe resultsof the splineinterpolationas evaluatedat the pointsin the vectorxx. By default,splir.re usesthe not-a-knotcondition.However,if y containstwo more valuesthan x has entries.then the first and last value in y areusedas the derivativesatthe end points. Consequently,this option providesthe meansto implement the clamped-end condition. E X A M P L E1 6 . 4
S p l i n e si n M A T L A B Problem Stqtement. Runge'sfunctionis a notorious example of a function that cannot be fit well with polynomials(recallExample r 5.7) : f(x):
I | + 25x2
IN MATTAB I6.5 PIECEWISE INTERPOLATION
375
Use MATLAB to fit nine equally spaceddata points sampledfrom this function in the interval[-1, l].Employ (a) a not-a-knotsplineand (b) a clampedsplinewith end slopes -4. o f f i : 1 a n df i - , : Solution. (a) The nine equally spaceddatapoints can be generatedas in >> x = >> ]1 -
linspace(-I,I,9); (i+25*x."2)
L.,
;
Next, a more finely spacedvectorof valuescan be generaredso tFlatwe can crearca sn?ooai plot of the resultsas generatedwith the spline function: xx
=
>> YY
-
>>
1,1); linspace( (x,Y,xx) ;
sPLine
Recall that I inspace automaticallycreates100 points if the desirednumberof points are not specified.Finally, we cau generatevaluesfor Runge'sfunction itself and display them along with the splinefit and the original data: >> yr >>
plot
= L./ (1+25*xx.^2); (x,y , 'o ' , xx, yy, xx, yr,
'
-
)
As in Fig. 16.6,the not-a-knotspline doesa nice job of following Runge'sfunction without exhibiting wild oscillationsbetweenthe points. (b) The clampedcondition can be implementedby creatinga new vector yc that has the desiredfirst derivativesas its first and last elements.The new vector can then be used to
I6.6 FIGURE Comporisonof Runge'sfunciion(doshedlrne)with o 9-poinlnoto-knotspllnefit generotedwilh MATLAB(solidline).
376
ET E R P O L A T I O N S P L I N EASN D P I E C E W I SI N
0.5 F I G U R EI 6 . 7 (doshedline)wiih o 9 pointclompedend splinefitgeneroted functlon Comporison of Runge's withMATLAB of I ond -4 orespecifred ct thJleltond isolid-ine)Notethotfirstderivoiives r i g h b' o u n d o . i e s ' e s p e c r ; , e l ,
generateand plot the splinefit: >> yc = [1 y 4]; >> yyc = spline(x,yc,xx); >> plot (x,y, 'o' ,xx /yljc,xx,yr,'--')
As in Fig. 16.7,the clampedsplinenow exhibitssomeoscillationsbecauseof the artificial slopesthat we have imposedat the boundaries.In other examples,where we haveknowledgeof the true first derivatives,the clampedsplinetendsto improve the fit.
16.5.2
MATTAB Function: interpl
The built-in function interpl providesa handy meansto implementa numberof different typesof piecewiseone-dimensionalinterpolation.It has the generalsyntax yi
=
interL, 1 (x,
1:, xi,
'rneLhad
t
w h e r e x a n d y - v e c t o r s c o n l a i n i n g v a l u e s t h a t a r e t o b e i n t e r p o l a t ye id: a , ,vectorc0nthe interpolation points taining results of the as evaluatedat the in the vector xj, and 'methocJ' : the desiredmethod.The variousmethodsare .
.
'nearest '-nearest nei-ehbor interpolation. This methodsetsthe valueof an interpolatedpoint to the value of the nearestexisting data point. Thus, the interpolation looks like a seriesof plateaus,which can be thoughtof as zero-orderpolynomials. ' I inear ' -linear interpolation.This methodusesstraightlines to connectthepoints.
EXA
I6.5 PIECEWISE INTERPOLATION IN MATTAB
377
.
'spL.ine'-piecewise cubic spline interpolation.This is identicalto the spline
.
iTItt"Ji
una ' cubic '-piecewise cubicHermiteinrerpolarion.
If the 'method' argunrentis omittecl.the defaultis linearinterpolation. The pchip option (short for "piecewisecubic llermite lnterpolation")merits more discussion.As with cubic splines,pchip usescubic polynomialsto connectdatapoints with continuousfirst derivatives.However,it ditTersfrorn cubic splinesin that the second derivativesarenot necessarilycontinuous.Further the first derivativesat the knots will not be the sarneas for cubic splines.Rather,they are expresslychosenso that the interpolation is "shape preserving."That is, the interpolatedvaluesdo not tend to overshootthe data pointsas can sometimeshappenwith cubic splines. Therefore,therearetrade-offsbetweenthe spline and the pchip options.The results of using spline will generallyappearsmootherbecausethe humaneye can detectdiscontinuitiesin the secondderivative.In addition.it will be moreaccurateif the dataare values of a smoothfunction. On the other hand.pchip has no overshootsand lessoscillation if the dataare not smooth.Thesetrade-ofI\.as well asthoseinvolving the otheroptions.are exploredin the following example. EXAMPLE 16.5
Trode-OffsUsing in; e,''1r l Problem Stotement. You perfom.ra test drive on an automobilewhere you alternately acceleratethe automobileand then hold it at a steadyvelocity. Note that you never decelerate during the experiment.The time seriesof spot nleasurementsof velocity can be tabulatedas
2a 20
4A 2A
56 3B
6B 80
BO BO
B4 r00
96 t00
lo4 t25
I t0 125
Use MATLAB's incerpl fr"rnctionto fit this datawith (a) linear interpolation,(b) nearest neighbor.(c) cubic splinewith not-a-knotend conditions.and (d) piecewisecubic Hermite interpolation. Solution. (a) The datacan be entered,fit with linearinterpolation, and plottedwith the following commands: >> r - [0 20 40 56 68 B0 34 95 t04 110]; >> v = l0 2A 20 38 B0 80 100 10A \25 7251; = finspace ( O, 110 ) ; >> tt = incerpl(t,v,r-t); >> vl >> plot (t,..',' o', tt,vl)
The results(Fig. 16.8c)are not smooth,but do not exhibit any overshoot. (b) The commandsto implementand plot the nearestneighborinterpolationare >> vn - interpl (t,v,tt,'nearest' >> nl ol- (t-,./,'
o' . tt,\-n)
);
378
ET E R P O L A T I O N S P L I N EASN D P I E C E W I SI N
140
't40
tzu
120
100
100
80
80
60
60
40
40
20
20 bu
IUU
120
{a} linear
140
140
120
120
100
100
80
80
60
bU
40
40
20
20 60 ( c )s p l i n e
100
'120
100 60 80 40 (b)nearestneighbor
120
100
120
FIGUREI6.8 Use of severoloptionsof the rnterpt toron ouiomobile.
funciionto performpiecewisepolynomiol nterpolciionon o velocilyiime serles
As in Fig. 16.8b.the resultslook like a seriesof plateaus.This option is neithera smooth nor an accuratedepictionof the underlyingprocess. (c) The commandsto implementthe cubic splineare >> vs >> plot
=
(t, v, tt, interpl ( t , - " - ,' o ' , t t , v s )
'spline'
) ;
Theseresults(Fig. 16.8c)are quite smooth.However,severeovershootoccursat several locations.This makes it appearthat the automobiledeceleratedseveraltimes duringthe
I 6.6 MULTIDIMENSIONAL INTERPOLATION
379
(d) The commandsto implementthe piecewisecubic Hermite interpolationare >>
vh
>> plot
=
interpl(t,\/,tt,'pchip'); (t,V,'o', tt,vh)
For this case,the results(Fig. 16.8d)arephysicallyrealistic.Becauseofits shape-preserving nature,the velocitiesincreasernclnotonicallyand never exhibit deceleration.Although the resultis not as smoothas for the cubic splines,continuityof the f irst derivativesat the knots makesthe transitionsbetweenpointsmore gradualand hencemore realistic.
I 6.6
INTERPOTATION MUTTIDIMENSIONAL The interpolationn.rethods for one-dimensionalproblemscan be extendedto multidimensionerlinterpolation. In this section,we will describethe simplestcaseof two-dimensional interpolationin Cartesiancoordinates.In addition,we will describeMATLAB's capabilities for rnultidimensionalinterpolation.
16.6.1 BilineorInterpolotion 'lnao-dintensional interpolctriondeals with determiningintermediatevalues for functions of two variablesz: .f (xi, ]r). As depictedin Fig. 16.9,we have valuesat four points: yz).Wewanttointerpolatebetweenthesepoints "f(xr,.r'r),,f(rz,.r'r),,f(rr,,r':),andl(.t:, F I G U R EI 6 . 9 G ' o p r ; c oole p c l i o n or {v o - d r r r e r s i oonr lor ln e o ' i n e p o l o ' i o r w l e e o n . l e ' n e o o e , o l- e" {dl i c i r c e )i s e s t i m c t ebdo s e do n f o u rg i v e nv o l u e {so p e nc i r c l e s )
f(xz, yt)
380
S P L I N EASN D P I E C E W I SI N ET E R P O L A T I O N
EX
FIGURE I6.IO Two dimensionolbilineorinlerpolctioncon be implemenledby f irstopplyinq one dir-nensioncl ' e o ^ l ^ ro c l o l i ^ no r o n q' n e . r o i m e n s , o r o d e - ^ r, r n e , o L e co - . r , h e - e , i o l e s c o r t ^ e nb e u s e d t o i n e c r l y i n t e r p o l o t e o l o n g l h e . r ' d i m e n s i o n t o y i e l d t hr ee sf iunl ot oi t - r , , , r , ,
to estimatethe value at an intermediatepoint / (-ri. _vi ) . If we use a linear function,there. sult is a plane connectingthe points as in Fig. 16.9.Suchfunctionsare calledbilinear. A simple approachfor developingthe bilinear tunction is depictedin Fig. 16.10.First, we can hold the v value fixed and apply one-dimensional lineu interpolation in the.rdirection.Usingthe Lagrangeform, the resultat (r;, .r1) is
. f( r i ,r ' r ): 3 f Jt
.fZ
, - r rJ, r )+ S f x2
.rt
t r : ,- v r )
(16.2s)
andat (,r;,rrr)is / ( x i , . r , : :)} . r r y- -rV.2, r : ) It
+ " t --{. ,!r l l ' ( , r r , , v : ) ,t2
16.10)
Thesepointscan then be usedto linearly interpolatealong theI'dimension to yieldthefinat result: -
lr
-
\''
. l { , r ; . . r ' ,:) - / ( x- ; . . r ' 1 )t }:
\'; -
\'r
) - -,f ' l.ri.J:) ,'t): .\'l
116.311
A single equation can be developedby substitutingEqs. (16.29) and (16.30) into Eq. (16.31) to give -.v: r'' -'t2 't '{i-xt '\'i .ft.r,.1-)f l , r , . r r ) f(x.. v,) rl - .\'2 )'r - ")Z X2 - Xt .t,r _ -r': (1634 -{i - rr ),i - lr ^, )'r ... , -ri -t: ]i .rl - rt ): -.r'r 12 -.{l )z _ ).r
I 6.6 MULTIDIMENSIONAL INTERPOLATION E X A M P L1E6 . 6
381
Bilineorlnterooloiion Problem Stotement. Supposeyou have measuredtemperaturesat a number of coordinateson the surfaceof a rectangularheatedplate:
T ( 2 , 1 :)6 0 T(2,6): 55
: 5l .5 T(:9,1) T(9,6):10
Usebilinearinterpolation to estimate thetemperature at.rr: 5.25and1',- 4.8. Solution. Substituting thesevaluesintoEq.(16.32)gives : "/ ( ) . j ) . _ { . 6 )
5.25-9 4 - o. 8 u- 6 2-9 l-6
5.25-2 4 - )./8. -- \6 + 9-2 t-6 s-}s-24R-l
5.25-94.8-l ss + -;; + -;; u-,
u-,
zo : 6r.2113
| 6.6.2 Multidimensionol Interpolotion in MATLAB piecewiseinterpolaMATLAB has two builrin functionsfor two- and three-dimensional tion: interp2 and interp3. As you might expectfrom their names,thesefunctionsoperate in a similar fashionto interpl (Section 16.5.2).For example,a simplerepresentation o f t h es y n t a xo f i n e o 2 i s zi
=
Lt\r-e(p2 lx,
y,
z,
xi,
yi,
'mechod'
)
where x and y - matrices containing the coordinatesof the points at which the values in the matrix z are given, zi : d matrix containingthe resultsof the interpolationas evaluated at the points in the matricesxr and yr, and method: the desiredmethod.Note that the methodsareidenticalto thoseusedby in*'erpi; that is, linear, nearesf , spline, a n dc u b i c . As with interpl, if the,nethod argumentis omitted,the defaultis linearinterpolation. For example,interp2 can be usedto make the sameevaluationas in Example 16.6as >> X= >> y= >>
z-
>>
lnl-
2 9); 1 6l; 6A 51 .5;55 701; erp2(x,y,2,5.25,4.8)
5r .2r43
382
AND PIECEWISE INTERPOLATION SPLINES
. ., H E A T R A N S F E R Bockground. Lakes in the temperatezone can becomethermally stratified duringthe summer.As depictedin Fig. 16.11,warm, buoyantwater near the surfaceoverliescolder, denserbottom water. Such stratification effectively divides the lake vertically into two layers: the epilitnnion and the hypolimnion, separatedby a plane called the thermocline. Thermal stratification has great significancefor environmentalengineersand scientists studying such systems.In particular,the thermoclinegreatly diminishesmixing betweenthe two layers.As a result, decompositionof organic matter can lead to severedepletionof oxygen in the isolated bottom waters. The location of the thermoclinecan be defined as the inflection point of the temperatur€d e p t hc u r v e - t h a t i s . t h e p o i n t a r w h i c h d 2 T / d z 2 : 0 . l t i s a l s o t h e p o i n t a t w h i c ht h e absolutevalue of the first derivative or gradient is a maximum. The temperaturegradient is important in its own right becauseit can be usedin coniunction with Fourier's law to determinethe heatflux acrossthe thermocline: dT
J:-DpC,
(r6.33)
az
where "f : heat flux [call(cm' . s)], cv: an eddy diffusion coefficient (cm'ls), p : density (31 g/cm3),and C: specificheat[: I call(g . C)]. In this case study, natural cubic splines are employed to determinethe thermocline depth and temperaturegradientfor PlatteLake, Michigan (Table 16.3).The latter is also usedto determinethe heatflux for the casewherea = 0.01 cm2/s.
I6.I I FIGURE Temperoiure versus forPlotte depthduring summer Loke, Michigon
r ('c)
Epilimnion Thermocline
Hypolimnion
TABTE16.3 2,6 T,"C
Temperoture versusdepthduringsummerfor PlotteLoke,Michigon.
0 228
2.3 22.8
4.9 228
9 I 2A6
137 t3I
1 83 11.7
22.9 tl.t
27.2 ilt
] 6.2 CASESTUDY
383
" . . , . . : c. o n t i n u e d Solution. As just described,we want to usenaturalsplineend conditionsto performthis analysis.Unfortunately,becauseit usesnot-a-knotend conditions,the built-in MATLAB spl ine function doesnot meetour needs.Further,the sp1 ine functiondoesnot returnthe first and secondderivatives we require for our analysis. However, it is not difficult to develop our own M-file to implementa natural spline and retum the derivatives.Sucha codeis shownin Fig. 16.12.After somepreliminaryerror trapping,we setup and solveEq. (16.27)for the second-ordercoefficients(c). Notice how
FfGUREr6.12 MJile fo deiermineintermediolevoluesond de,"ivofives with o noiurolspline.Note fhot fhe oif f {unclionemployedfor errortroppingis describedin Seclion 19.7.1. -
l \ . / \ ,; . 1 - ;, , t : . ,=:
r r a r - : i p l i n e 1r , y , s;,1 ire
r,'dlii lrl
xx)
r " : i! : ] , r j i f l - - r e n t : . r L 1 ( , r l
spi.irrr,'(x,y,-xril:
uses
. l n . l t s - r - r r ; lc , r b i c
: a)n la.) f r f r d ua t t r:s
j.y, t-he varues oi the ::,r- yector The vecLcf xx.
in
l,
,
rl
j ndepenLlenL
;i
ricpenrlelt
der,vaLiv-:r
::=i:,1: ri aiirr i-iat
ir
tf
c l e ! ) € : n , - . l r t l : ' , ' - -i .- t h L e : :
;{){ r::
ai
anr,ly rrr:st be same L.r!{t-h' ) ; end x n o t s t : - i c " L y . r s c e n c r r : r r ') , e n d
II Il'
l;,-
ar, xr
a:
ii'e:
furr:Lion
slle.ttf ie :
. , ' ar i ; : b r e s . , ' a: : i a h . e s
€ f ilo l.ri e.f \.a l ues
iirsi
r
l...r.
/
i,i.-':';i r:e,J'ialt:es i.t
:pLtne
rtnderlving
-
'.fEL
- 1t(r, I 2 * r l r{ x , -.hi-,.
*
i
Lal-
h.
I
i); L) r
i
hlr:,
i));
i')
fci(j,
il
+
1l-rili
?solve
I')
1,
i,
x,
i-ol c i:oetficier,ts : " S O l . - j e J - c , r -a , b . - . n d
d
i
-
1,
v));
r:c-el-fi,:i+nj-s
ll;
(r lr
+ l, +-lj
t,
x, t,r) - h(x, i) cr{:L)) ,/ 3 / lilx,
/ i i);
*
(2 * c(:)
= c1i
-
1))
(continued)
384
SPLINES AND PIECEWISE INTERPOLATION
coniinued I - 1:m %perforn interpolaLions at desired values - Splinelnrerp(x, lyt'(r),dy{r),d.2(i)l n, a, b, c, d, xx(:.)); end enC frrrlctfon hti = 11*, i) hl'r = x(r + i) - x(i); for
end iun -l n i.C'J fdri, , x , , 1) - y(j)) fcld - (y(i) r' (..r(i) x(j)); enC function lyyy, dyy, d2yl -gp] inelnterp f nr
i
1 ,h
i
(x.
n,
a,
b,
c,
d,
xr )
1
, 0 . 0 0 U 0 0 1 & x r < _ , -x ( i i xi >= x(ri) + i) + 0.000001 yyy-a (1i) +b ( ii) * (xi-x (ri) ) +c (ii) * (x!,x (ir ) ) ^2+o( ii) . . . *(xi x(ri))^3; d y i , = 5 1i I ) + . 1* c ( i , i ) * ( x l - x ( i i ) ) + I * r l ( i i ) * ( x i - x { , i ) ) ^ 2 ; d2y-2*c (ii ) +5*d (ii) * (z.i-x (ii ) ) ;
if
br eak end end end
FIGUREt6.t2 lContinued)
we use two subfunctions, h and fd, to compute the required finite differences.Once Eq. (16.27) is set up, we solve for the c's with back division. A loop is thenemployed t0 generatethe other coefficients(a, b, and d). At this point, we have all we need to generate intermediate values with the cubic equation: f (x) : ai * bt(x - rr) + ct(x * x)2 * di@ - xi)3 We can also determine the lirst and second derivatives by differentiating this equation twice to give '(x) : bi * ci(x - x;) * 3diQ - x;)z .f"{x1 :2ci + 6di, - xi) f
As in Fig. 16.12, these equations can then be implemented in another subfunctign, splineTnterp, to determinethe values and the derivativesat the desiredintermediate values. Here is a script file that usesthe naLspl ine function to generatethe splineandcreats plots of the results: z = i0 2.3 4.9 9.1 13.7 IB.3 22.t) 27.21; r=122.8 22.8 22.8 20.6 13.9 17.7 I1 .l 17.L1; ?i
-
l inqnaeo(z(1\.zi
I6.7 CASESTUDY
"
continued -
natsptine \2,T, zzJ ; (T, z,' a',TT, zz) I ),Pioi ' l, ieqrei.i.ll'ciaca',' l " rcvc!-s /.9r-!v /
,pfor(ai,zzt c 1 z ') ,'reverse'),grid ,!rlot (d't2, zz)
As in Fig. \6.13, the thermoc\ine appearsto be \ocated at a depth of about 11.5 m. We can use root \oeation (zero second denrative) or optimization methods (minimum fust denvative) to refine this estimate.The result is that the thermocline is located at 11.35 m where the gradientis - 1.61'C/m.
FIGURE T6.I3 Plotsof {o) temperoture, depth (ml generotedwith {b} g'odient, ond (c} secondderivotiveuers,,s the cubic sprineprogro.n.The rhermocline islocoted ot the in[lecrionpoint o[ ihe temperoruredeplh curve. ( a )T
(cl dzTtdz2
\bl dr/dz
_L______
10
10
- - - -t-
- - - - - -
10
F_ I
\ t5
15
----F------
-r------
I I I I
t5
20
I
25
30L
10
20
1n
rI I L
I
20
zv
25
25
30 1 -2
30 -0.5
0.5
AND PIECEWISE SPTINES INTERPOLATION
386
cond cubi(
continued The gradientcan be usedto computethe heat flux acrossthe thermoclinewith Eq.(16.33): J:
8 6 ' 4 o- :0t s3 ' 9 . n 2c' aa-l -0.0rt*' * r4 * r-:"1^* (- t6rI) * .^l=- * g..L cmr m / l 0 o c m d s \
The foregoinganalysisdemonstrates how splineinterpolationcanbe usedfor engineeringandscientificproblemsolving.However,it alsois anexampleof numericaldifferfrom differentareascanbeused entiation.As such,it illustrateshow numericalapproaches in tandemfor problemsolving.We will be describingthetopicof numericaldifferentiation in detailin Chap.19,
x
f (x) x f (x't
In ea whic
f
16.5
depi x f (x\
PROBTEMS 16.l Given the data
xl2 f{x1
I
5
5 I
25 7
Fit this data with (a) a cubic spline with natural end conditions, (b) a cubic spline with not-a-knotend conditions,and (c) piecewisecubic Hermite interpolation. 16.2 A reactol is thermaily stratified as in the tbllowing table:
Depih, m Temperoture,'C
0 05 7A 7A
I 55
t5 22
2 25 13 l0
3 l0
to Use a clampedcubic spline tit with zero end derivatives determinethe thermoclinedepth.If k:0.01 cal/ (s . cm . "C) compute the tlux acrossthis interface. 16,3 The following is the built-in humps function that MATLAB uses to demonstrate some of its numerical clpabilities: -0
(x-0.3)2*0.01 (x-0.9)2*0.04
The humpsfunction exhibitsboth flat and steepregionsovet a relatively shon.r range. Here are some valuesthathave beengeneratedat intervalsof 0. I over the rangefrom.r = 0 to l:
r ftt) x
02 03 4.4 05 O 0l 5 . 1 7 6 1 5 4 7 t 45 887 96 500 47 448 19400 06
07
08
0.9
I
Based on these temperatures,the tank can be idealized as f $) ]t 692 12382 t7.846 2t 703 r6.000 two zones separatedby a strong temperaturegradient or thennocliue. The depth of the thermoclinecan be defined as the inflection point of the temperature-depth curve-that Fit this datawith a (a) cubic splinewith not-a-knotendconis, the point at which d2Tldzz:0. At this depth,the heat ditions and (b) piecewisecubic Hermite interpolation. In flux from the surface to the bottom layer can be computed both cases,create a plot comparing the fit with the exact with Fourier's law: humpsfunction. 16.4 Develop a plot of a cubic sptine fit of the following . . d r J:-K-data with (a) naturalend conditionsand (b) not-a-knotend az
Fir tl ditio tions case 16.6 natu cate t6.7 orde 15.4 x
t2
(a) F ditio (b) F slopt tiatir 16.8 and Thes evalr
PROBLEMS
387
conditions. In addition, develop a plot using (c) piecewise cubic Hermiteinterpolation.
mathematicaltables.For example,
lB
.r /(r)
0 0
ro0 0 82436
2aa I 00000
/(x)
600 0 4060l
800 0 t99r5
r000 009r58
Ineach case,colnpareyour plot with the following equation which wasusedto generatethe dala: I"{ , { ) :
x -e
200
16.5Thefollowing data is sarnpledfrom the step function depicted in Fig. 16.I :
/(r)
0
-02 0
02
0.o
I
0lrt
Fitthisdatawith a (a) cubic spline with not-a-knotend con(b) cubic spline with zero-slopeclampedend condiditions, and(c) piecewisecubic Hermite interpolation.[n each trons, createa plot comparing the fit with the step function. case, 16,6Developan M-file to cor.nputea cubic spline 11twith natural end conditions.Test your code by using it to dupliExampleI 6.3. cate 16.7The fbllowing data was generated with the liftlrpolynomial:.f(x): 0.018515- 0.411.14 order * 3.9125xr15.456i+ 27.069x - l4.l: r135679 1000
/(r)
2 172
22 0.556
24 A 5202
26 A 47OB
EstimateJl2.l). (a) using an interpolating polynomialand (b) using cubic splines.Note that the true value is 0.5683. 16.9 The following data defines the sea-levelconcentration of dissolved oxygen for fresh water as a function of tempelature:
T,'C 0 B t6 24 32 40 o,mg/L 1462) IlB43 IBZ0 B4lB 73A5 64)3
-r!r t')rr
-t
2 0 5767
0 5 8t 5 400 "Ir(J) 0 73576
4 22O
5 430
4.912
9.)24
(a)Fit this data with a cubic splinewith not-a-knotend conditions. Createa plot comparing the fit with the tunction. (b)Repeat(a) but useclampedend conditionswherethe end slopes are set at the exact values as determined by differenliating the function. 16,8Besselfunctions often arise in advancedengineering andscientificanalysessuch as the study of electric fields. These functionsare usually not amenableto straightforward evaluation and, therelbre, are olien compiled in standald
Use MATLAB to fit the data with (a) piecewise linear interpolation,(b) a fifth-order polynomial,and (c) a spline.Display the resultsgraphicallyanduseeachapproachto estilnate o(27). Note that the exactresultis 7.986 mg/L. 16.10 (a) Use MATLAB to fit a cubic splineto the followins data:
x
0 2A
v
2 2A
4 12
7 7
l0 6
l2 6
Determinethe value of 1,at "t : 1.5.(b) Repeat(a). but with zero first derivativesat the end knots. l6.ll Runge'sfunctionis writtenas I "
|*25rt
Generatefive equidistantlyspacedvalues of this function over the interval: [, 1]. Fit this data with (a) a fourth-order polynomial, (b) a linear spline, and (c) a cubic spline. Presentyour resultsgraphically. 16.12 Use MAILAB to generateeight points lrom the function / (/) : stn- t from r : 0 to 2n. Fit this data using (a) cubic spline with not-a-knotend conditions,(b) cubic splinewith derivative end conditionsequalto the exactvaluescalculatedwith differentiation,and (c) piecewisecubic hermite interpolation. Developplots of eachfit aswell asplots of the absoluteerror (8, : approximatioll- true) for each.
.: '.
d
t
b
,'i:';,
'ii,;*"'!.1
I
l;'#i$k*tffi#*ri
lntegrotion o nd Diff;rentiotion 5.1 ovERvtEw In high school or during your first year of college, you were introduced to differential and integral calculus.There you learnedtechniquesto obtain analytical er exact derivatives and integrals. Mathematically,the derivuriverepresentsthe rate o[ changeof a clependentvariable willr respectto an independent variable.For example.if we are given a functionXt) that specifiesan object'spositionas a functionof time.differentiationprovidesa meansto determineils velocity.as in: d u(l): -y(l) dt' As in Fig. PT5.1n,the derivativecan be visualizedas the slopeof a function. Infegrationis the inverse of differentiation.Juqt as differentiationusesdifferencesto quantify an instantaneous process,integrationinvolvessumminginstantaneousinformation to give a total result over an interval.Thus. iI-we are providedwith velocity as a function of time, integrationcan be usedto determine the distancelraveled:
: fr'u(t)dr
v(/)
As in Fig. PTs.lb, for functions lying above the abscissa,the integral can be visualized as the areaunder the curve of u(r) from 0 to r. Consequently,just as a clerivativecan be thoughtof as a slope,an integralcan be envisagedas lsummation. Becauseoflhe close relationshipbetweendifferentiation and integmtion, we have opted to devorethis part of lhe book to both processes.Among other
389
390
PART 5 INTEGRATION AND DIFFERENTIATION
0 ) 400
200
0
0 tal
FIGURE PT5.I ond {b) integrotion The confrostbehr;een(o) differentiotion
things,this will provide the opporlunityto highlight their similaritiesand differencesfrom a numericalperspective.ln addition,the materialwill haverelevanceto the next paft of the book where we will cover differentialequations. Although differentiationis taughtbeforeintegrationin calculus.we reversetheir order in the following clrapters.We do this for severalreasons.First, we havealreadyintroduced you to the basicsof numericaldifferenltiationin Chap.4. Second.in part becauseit is much less sensitiveto roundoff errors, integrationrepresentsa more highly developedareaof numericalmethods.Finally, althoughnumericaldifferentiationis not as widely employed, it does have great significancefor the solution of differentialequations.Hence,it makes senseto cover it as the last topic prior to describingdifferentialequationsin Part Six.
5.2
PARTORGANIZATION Chapter 17 is devoted to the most common approachesfor numerical integration-the Newton-Cotes formulas. Theserelationshipsarebasedon replacinga complicatedfunction tabulated or data with a simple polynomial that is easy to integrate.Three of the most widely usedNewton-Cotesformulasarediscussedin detail: the trapezoidalrule, Simpson's 1l3 rule, andSimpson's318rule. All theseformulasare designedfor caseswherethe data to be integratedis evenly spaced.In addition, we also include a discussionof numerical
ORGANIZATION 5.2 PART
39r
integrationof unequally spaceddata. This is a very important topic becausemany realworld applicationsdeal with datathat is in this forrn. All the above materialrelatesto closedintegratiott,where tlre function valuesat the ends of the limits of integrationare known. At the end of Chap. 17, we presentopen integrationfonnulas, wherethe integrationlimits extendbeyondthe rangeof the known data. Although they are not commonly usedfor definite integration.open integrationformulas are presentedhere becausethey are utilized in the solution of ordinary differential equations in Part Six. The formulationscoveredin Chap. 17 can be employedto analyzeboth tabulateddata and equations. Chapter18 dealswith two techniquesthat areexpresslydesignedto integrate equationsand functions:Rombergintegrationand Gaassquadrature.Computeralgorithms are providedfor both of thesemethods.In addition,adaptiveintegrationis discussed. ln Chap. 19, we presentadditionalintbrn.rationon numeric:aldi.fferentiation to supplement the introductorymaterialfrom Chap.4. Topicsincludehigh-accuracl, finite-dffirence formulas, Richardson extrapokttion, and the differentiation of unequally spaceddata. The effect of errorson both numericaldiff'erentiationand intesrationis also discussed.
:' ,
' ;,0;' :
NumericolIntegrotion Formulos
CHAPTER OBJECTIVES The primary objectiveof this chapteris to introduceyou to numericalintegration. Specific objectivesand topicscoveredare .
'
'
. . .
Recognizingthat Newton-Cotesintegrationformulas arebasedon the strategy of replacinga complicatedfunction or tabulateddatawith a polynomial that is easy to integrate. Knowing how to implementthe following singleapplicationNewton-Cotes formulas: Trapezoidalrule Simpson's1/3 rule Simpson's3/8 rule Knowing how to implementthe following compositeNewton-Cotesformulas: Trapezoidal rule Simpson's113 rule Recognizingthat even-segment-odd-point formulaslike Simpson's1/3 rule achievehigher than expectedaccuracy. Knowing how to usethe trapezoidalrule to integrateunequallyspaceddata. Understandingthe differencebetweenopen and closedintegrationformulas.
YOU'VE GOT A PROBTEM h ftf l\
ecall that the velocity of a free-falling bungeejumper as a function of time canbe computed as
( ,f8a'),) uu\: .14 ,unn V ,'.i \''l n
rrrrr
I Z.I INTRODUCTION ANDBACKGROUND
393
Supposethat we would like to know the vertical distancez the jumper has fallen after a certaintime r. This distancecan be evaluatedby integration: fl
:(r):
(1j.2)
/ t'(ndr .to
S u b s t i t u t i nE g q . 1 1 7 . 1i)n t o E q . ( 1 7 . 2 )g i v e s
z(t):I,',8""n(,l*,)0,
( I 7.-r.1
Thr.rs,integrationprovidesthe meansto deterrrrinethe distancefiom the velocity.Calculus can be usedto solveEq. (17.3)for
,E')l
f '"[.."'n(
(t7.4)
Although a closedform solutioncan be developedfor this case,thereare other functions that cannot be integratedanalytically.Furlher. supposethat there was some way to measurethejumper's velocityat varioustimesduringthe lall. Thesevelocitiesalongwith their associatedtimes could be assembledas a table of discretevalues.ln this situation,it would also be possibleto integratethe discretedatato determinethe distance.In both these instances, numericalintegrationmethodsareavailirbleto obtainsolutions.Chaptersl7 and l8 will introducevou to someof thesemethods.
I7.I
INTRODUCTION AND BACKGROUND 17.l.l
Whof ls Integrotion?
According to the dictionarydefinition, to integratemeans"to bring together,as parts,into a whole;to unite;to indicatethe total amount.. . ." Mathemafically, definiteinteglationis representedby
t:
a^ l,,of {.r)
(r 7 . 5 )
which standsfor the integralof the function f (x) with respectto the independentvariable .r. evaluatedbetweenthe limits x : a to ,r : b. As suggestedby the dictionarydefinition,the "meaning"of Eq. (17.5) is the roral value. or summation,of /("r) dx overthe rangex : a to b. In fact, the symbol / is actually a stylizedcapital S that is intendedto signify the closeconnectionbetweenintegration and summation. Figure 17.1 representsa graphicalmanifestationof the concept.For functions lying above the x axis, the integral expressedby Eq. (17.5) correspondsto the area under the curve of l (x) betweenr : a andb. Numerical integrationis sometimesreferredto as quadrature.This is an archaicterrn that originally meantthe constructionof a squarehaving the sameareaas somecurvilinear figure. Today, the term quadratureis generallytaken to be synonymouswith numerical definite integratior.r.
394
NUMERICAL INTEGRATION FORMULAS
I
.{ F I G U R EI 7 . I Grophicolrepresentotion of theintegroo1 l ( x ) b e t w e e nt h e l i m i t sr : equivoenl lo theoreounderlhecurve.
t t l o D T h e i n t e g r oi ls
nonl
17.1.2 Integrotionin Engineeringond Science Integrationhas so many engineeringand scientific applicationsthat you were requiredto take integral calculusin your first year at college.Many specificexamplesof suchapplicationscould be given in all fields of engineeringand science.A number of examples reIate directly to the idea of the integralas the areaunder a curve. Figure 17.2depictsa few caseswhere integrationis usedtbr this purpose. Other common applicationsrelateto the analogybetweenintegrationand summation. For example,a common applicationis to determinethe meanof a continuousfunction.Recall that the meanof discreteof n discretedatapointscan be calculatedby tEq. (i3.2)1.
i,,
14s2n- 5r t17.6) n where -yiare individual measurements. The determinationof the meanof discretepointsis depictedin Fig. 17.3a. In contrast,supposethat ,r is a continuousfunction of an independentvariable.r,as depictedinFig. 17.3b.For this case,thereare an infinite numberof valuesbetweena and b. Just as Eq. (17.6) can be appliedto detenninethe mean of the discretereadings, you might also be interestedin computingthe mean or averageof the continuousfunction .l : / (x) for the interval fron a to b. Integrationis usedfor this purpose,as specifiedby Mean:
l'! f t.utax ,"-
FIG Exo nee kno
(11.1)
D-A
This formula has hundredsof engineeringand scientific applications.For example,it is used to calculatethe center of gravity of irregular objectsin mechanicaland civil engineeringand to determinethe root-mean-square currentin electricalengineering.
I Z.I INTRODUCTION ANDBACKGROUND
395
i _l -l
(b)
(c)
I7.2 TIGURE (o)A surveyor of how integrotion is usedto evoluole oreosin engineering Exomples cnd scientific opplicotions. might to knowtheoreoof o fied boundedbv o meonderinq need ond hryoroods.{b)A hvdroloqist miqhtneedto streom k n o w i h e c r o s s s e col iroenooo fo r i v e ri c )A s t r u c t u reonl g i ' , e e r m i gnhete d i o d e t e r m i n e t h ef in, ce ei d J e t o o nonuniform wind blowingogo nsfthesideof c skyscroper.
4 (a)
r : "f(-r)
(b)
FIGUREI7.3 An illustrotion of themeonfor(o)discreie ond(b)continuous dotc
396
INTEGRATION NUMERICAL FORMUTAS
Integralsare also employedby engineersand scientiststo evaluatethe totalamount0r quantity ofa given physicalvariable.The integralmay be evaluatedover a line, anNea,sr a volume.For example.the total massof chemicalcontainedin a reactoris girenasth product of the concentrationof chemicaland the reactorvolume, or Mass : concrentration x volume where concentrationhas ur.ritsof mass per volume. However, supposethat concentration varies from location to location within the reactor.In this case,it is necessaryto sumthe productsof local concentrationsc; and correspondingelementalvolumes A Vi:
: Mass I.,on whererr is the numberof discretevolumes.For the continuouscase,wherec(x, y, t) isa known function andx, y, and : are independentvariablesdesignatingpositionin Cartesian coordinates,integrationcan be usedfor the samepurpose: Mass:
III,,-
t , .z ) d r d v d z
ttt
Mirss: lllrtvtdV JJJ v
which is referredto as a volume integrctL.Notice the strong analogybetweensummation and integration. Similarexamplescould be givenin otherfieldsof engineering and science. Forexample, the total rate of energy transferacrossa plane where the flux (in caloriesper square centimeterper second)is a lunctionof positionis givenby ??
Flux:llnu*ae " "r.
which is referredto as an areal integral. whereA : areo. Thesearejust a few of the applicationsof integrationthat you might taceregukuly in the pursuit of your profession.When the functionsto be analyzedare simple,you will normally chooseto evaluatethem analytically.However, it is often difficult or impossible whenthe functionis complicated, as is typicallythe casein morerealisticexamples. Inaddition, the underlyingfunction is often unknownand definedonly by measurement atdisvalues crete points. For both thesecases,you must have the ability to obtain approximate for integralsusing numericaltechniquesas describednext.
17.2
NEWTON-COTES FORMULAS The Newton-Cotes.formulasare the most common numerical integrationschemes, They are basedon the strategyof replacingn complicatedfunction or tabuiateddatawitha polynomial that is easyto integrate: 7b
7b
I:lf(r\dr=1f..(,r\tl.x
FORMULAS I 2.2 NEWTON-COTES
397
where ll, (,r) : a polynomial of'the form a,, rtr"I la,tx" .fr(..r):a0+alx +"'+
(17.9)
wheren is the order of the polynomial.For example,inFig. 17.4a, a first-orderpolynomial (a straightline) is usedas an approximation.In Fig. 17.4b,a parabolais employedfor the samepurpose. The integralcan also be approximatedusing a seriesof polynomialsappliedpiecewise to the function or data over segmentsof constantlength. For example,in Fig. 17.5,three
FIGURE I7.4 ineond(b)o porobolo bythecreounder(o)c stroight Theopproximction of on integrol "f(;r)
.f (tt)
FIGURE I7.5 three byfheoreounder stroightline segmenis Theopproximotion of on integro l(r)
398
FORMULAS NUMERICAL INTEGRATION
FIGUREI7.6 Thedifference behween ondib)openinlegrotion formulos {o)closed
straight-linesegmerltsare usedto approximatethe integral.Higher-orderpolynomialscan be utilized for the samepurpose. Closedand openforms of tlre Newton-Cotesformulas al'eavailable.Theclosedlorns are those where the data points at the beginning and end of the limits of integration ac known (Fig. 17.6a).The openfornrs have integrationlimits that extendbeyondtherangc m of the data (Fig. 17.6b).This chapteremphasizesthe closedforms. However,material open Newton-Cotesformulas is briefly introducedin Section17.7.
RUIE 17.3 THETRAPEZOIDAT The trapezoidalrul.eis the first of the Newton-Cotesclosedintegrationformulas. It con+ spondsto the casewherethe polynomialin Eq. (17.8)is first-order:
t:
I .f (b) - .f (a) (.r, , ")]d, D-0
l""lt^ 1-
T h e r e s u l to f t h e i n t e g r a t i oins l:(b
kl) + .[(b) - r r ) f' 2
(17.10
which is calledthe trapezoidalrule. Geometrically,the trapezoidalrule is equivalentto approximatingthe areaof fu trapezoidunderthe straightline connecting I O1 andJ @) inFig. l7 .7. Recallfromgeor etry that the formula for computingthe areaof a trapezoidis the heighttimestheaverage of the bases.In onrcase,the conceptis the sarriebut the trapezoidis on its side.Therefog the integralestimatecan be representedas / - width x ,verageheight
EXAM
17.3 THETRAPEZOIDAL RULE
399
FIGURE I7.7 Grophicol depicrion of thetropezoidol rule
I : (b - rr) x averageheight
(r7.r3)
where, fbr the trapezoidalrule, the averageheight is the averageof the function valuesat the end points,or l.f (o) + J @)112. All the Newton-cotes closed lormulas can be expressed in the generalfbrmat of Eq' ( 17'13)' That is, they difl'er only with respectto the formulationof the averageheight.
| 7.3.1 Error of the TropezoidolRule when we employ the integral under a straight-rine segmentto approximatethe integral undera curve,we obviouslycan incur .oo, that rnay be substantial (Fig. l 7.g). An esti_ mafe forthe local truncationenorof a "n singleapplicationof the trapezoidalrule is I
E,:--;f"tE)tl,
-ul.
t!
(17.11)
wheref lies somewherein the intervalfrom a to b. Equation(:i.14)indicatesthat if the tunction being integratedis linear,the trapezoidalrule wiil be exactbecausethe secondderivative of a straightline is zero. otherwise. for functions with second-and higher-order derivatives(i.e., with curvaturc),someerror can occur. 'lZ.l EXAMPLE
S i n g l eA p p l i c o r i o no f r h e T r o p e z o i d o lR u l e ProblemStqiement. Use Eq. (Ij.lD ro numerically integrate l ( r) : 0.2 _t 25x _ 200x2* 615x3_ 900ra f 40015 from a : 0 to D : 0'8. Note that the exactvalue of the integralcan be determinedanalytically to be 1.640533.
400
FORMULAS NUMERICAL INTEGRATION
lnteqralestimate
0.8 F I G U R EI 7 . 8 ruleto opproximote Grcphicoldepiction of theuseof o single-opplicotion of thetropezoidol i h e i n t e q rool l ( . r ) - - A . 2 + 2 5 x - 2 o a ^ - r * 6 7 5 r r - 9 O O - r + * 4 O O r ' t r o m ; t - : 0 t o 0 . 8 .
S o l u t i o n . T h e f u n c t i o nv a l u e sl ( 0 ) : 0 . 2 E q .( l 7 . t I ) t o y i e l d 1:{0.8-n,
a n d . f ( 0 . 8 ) : 0 . 2 3 2 c a nb e s u b s t i t u t e d i n b
0 . 1+ 0 . 2 3 1 :0.17:8 ,
which represents an effor of E, : 1.640533- 0.1128: 1.461133,whichconespondsh a percentrelatil,eerror of st :89.5c/c. The reasonfor this largeerror is evidentfron|b g r a p h i c adl e p i c t i o n i n F i g .l T . 8 . N o t i c e t h a t t h e a r e a u n d e r t h e s t r a i g h t l i n e n e g l e c t s a s i g nificant portion of the integrallying abovethe line. hr actual situations,we would have no for-eknowledge of the true value.Therefoq serord an approxirnateerror estimateis required.To obtain this estimate,the function's twicc derivativeover the interval can be computedby differentiatingthe originalfunction to give f" (x) : -400 + 4,050.t- 10.80012t 8,000x3 The averagevalueof the secorrdderivativecan be computedas lEq. (17.7)l
'' '^'f
/,.\
_
/1'* t-+oo * 4.050x - 10,800:rr* 8,00013)r/:r
0.8-o
which can be substituted into Eq. (17.14)to yield I E,,: - |(-60)(0.8)3 - 2.56 | ./,
RULE 17.3 THETRAPEZOIDAT
401
doesexist, which is of the sameorderof magrritudeand sign as the trueerror.A discrepancy' however,becauseof the fact that tbr an interval of this size,the averagesecondderivative is not necessarilyan accurateapproximationof l"(6). Thus, we denotethat the error is by using the notation E,,, ratherthan exactby using E,. approxin.rate
17.3.2 The CompositeTrqpezoidolRule One way to improve the accuracyof the trapezoidalrule is to divide the integrationinterval liom a to b into a numberof segmentserndapply the ntethodto eachsegment(Fig. 17.9). The areasof individual segmentscan then be addedto yield the integralfbr the entire inor muhiple-application,integration terval. Tlre resulting equationsare called compo,site, .fitrmulas. Figure 17.9 shows the generalfbrmat and nomenclaturewe will use to characterize compositeintegrals.Therearen * I equallyspacedbasepoints(ro,,rr, x2,..., -r,,).Consequently,there are ,l segmentsof equalwidth: la-
b-a
(l7.l-s)
-
t7
If a and b are designatedas ,rs and 1,,. respectively,the total integral can be representedas
r : L''.f(x)ru +1,,,' f ( t ) d x + . . ' +l . ' ' f , r r u , F I G U R EI 7 . 9 trcoezoidcrl rue Comoosite
, n :
b-ct ---il-
402
FORMULAS NUMERICAL INTEGRATION
Substitutingthe trapezoidalrule for eachintegralyields
,
, I(rn\* /trr) 22
-
, " [ ( r r ) . 1/ ( x : ]
t t -
or, grouplng terms: nl
,T
nt
I : : l / ' ( x o+ ) 2) .rl -L
i-l
I
I
I ( r i ) l - J ( x , , )|I J
or, usingEq. ( 17.l5) to expressF.q.(17.17) in the generalform of Eq. ( 17.l3): n-l
f (rn) r 2 - / r\ r-\ ' - I '/ t . 1 ,) r
r
r ( . r / i)
j-1
t:(b-a) \.....--
2rt
widlh
Avcragc hcight
Becausethe summationof the coefficientsof /(:r) in the numeratordividedby 2il is equal to to I , the averageheightrepresentsa weightedaverageof the function values.According Eq. ( 17.l8), the interior points are given twice the weight of the two end points/(.ro) and .f (x,). An error fbr the compositetrapezoidalrule can be obtainedby summingtheindividual errorsfor eachsegmentto give ,' lD
-
.1 tll'
E,: _,:;:: tln'
il -
) f- /,,re,
{r7re)
where /" (6i) t, ,n" ,".ond derivativeat a point {; locatedin segmenti. This resultcanbe simplified by estimatingthe mean or averagevalue of the secondderivativefor theentire interval as t1
\-
1'
f ' l r \l \. i ' , r
J
n
ThereforeL, .l'" (€,) = n f " and Eq. (.11.19) can be rewritten as ,l tD -
.1
ot'
-.
L , , - - - I
| 1,n'
Note Thus, if the number of segmentsis doubled,the truncationerror will be quarlered. t h a t E q .( 1 1. 2 1 )i s a n a p p r o x i m a t e e r r o r b e c a u s et hoef a p p r o x i m a t e n a t u r eEoqf . ( 1 7 , 2 0 ) . E X A M P L EI 2 . 2
C o m p o s i t eA p p l i c o t i o no f t h e T r o p e z o i d o lR u l e Problem Stotement. Use the two-segmenttrapezoidalrule to estimatethe integralof -t * f (x) : 0.2 * 25x 200x2 675r.3 900x4 * 400x5 from a : 0 to D : 0.8. Employ F,q.(17.21) to estimatethe error.Recallthat theexactvalue of the intesralis 1.640533.
.I2.3
403
RULE THETRAPEZOIDAL
Solution. Forn : 2(h :0.4): /(0) : 0.2 1:0.8
f ( 0 . 4 ): 2 . 4 5 6
, f ( 0 . 8 ): 0 . 2 3 2
0 . 2 + 2 ( 2 . 4 s+60) . 2 3 2 4
:1.0688
E , : 1 . 6 4 0 5 3 3- 1 . 0 6 8:80 . 5 1 1 7 3
€t :34.97c
0.gr [ ,"n : - - 1 - ( - 6 0 ) : 9 . 6 4 t2(2\t where -60 is the averagesecondderivativedeterminedpreviouslyin Example 17.1.
rl o d
The resultsof the previousexample,along with three-through ten-segmentapplications of the trapezoidalrule, are surnmarizedin Table l7.l. Notice how the error decreases as the numberof segmentsincreases. However,alsonoticethat the rateof decreaseis gradual. This is becausethe error is inverselyrelatedto the squareof r [Eq. (1,1.21)).Therefore, doubling the number of segmentsquartersthe error. In subsequentsectionswe develop higher-orderfbrmulasthat iue more accurateand that convergemorequickly on the true integralas the segmentsareincreased.However,belbreinvestigatingtheseformulas,we will first discusshow MATLAB can be usedto implementthe trapezoidalrule.
| 7.3.3
MATTAB M-file: trap
A simple algorithm to implement the compositetrapezoidalrule can be written as in Fig. 17.l 0. The function to be integratedis passedinto the M-file along with the limits of integrationand the numberof segments.A loop is then employedto generatethe integral f o l l o w i n gE q . ( 1 7 . 1 8 ) .
TABLE| 7. I
hopezoidolruleto Resulis for thecomposite estimotethe integrolof /(.r'l :0.2 + 25.\200-t2+ 675x3- 9O0r++ 4OO,rs from - r : O t o 0 . 8 .T h ee x o c tv o l u ei s 1 . 6 4 0 5 3 3 . e, ("/"1
2 3 4 5 6 7 B I t0
0.4 a 2667 0.2 0 t6 0 .r 3 3 3 0l143 0t 0.0889 008
06BB 3695 484B 5399 5703 5B87 6008 609 I 6t50
349 t6.5 95 6l 4.3 32 2.4 1.9 t6
404
NUMERICAL INTEGRATION FORMULAS
function I = trap(func,a,b,n,varargin) % trap: composi.te tt:apezoidal rule quadraLure 'Z I - trap(f unc,a,b,n,pi,p2,... ): % composite trapezoidal rule % input: func = name of function ? Lo be integrated Z a, b = inteqration limits ? n - number of seg,ments (defaulL = 100) pl ,p2, . .. - additional parameters used by func ? Z output: I=integralestimate % if nargin<3,error('at least 3 input arguments required'),end if -'(b>a),error('upper bound must be greater than lower,),end r f narqin-4 risempLylnt,n-100;end x = a; h = (f - a)/n; S-func (q,vdrarg i -,{:' ) ; fori=1;n-1 X = X + h; s = s + 2*func(x,vararg:in{:}); end s - s + func(b,varargin{:}); I = (b - a) * s/(2*n);
F I G U R IE7 . I O
M-f le -o imp enr-cnl'he conpo:iie rope./ooo 'J e.
An applicationof the M-file can be developedto determinethe distancefallenby the free-falling bungeejurnper in the first 3 s by evaluatingthe integralof Eq. ( 17.3),For this exarnple,assumethe following parametervalues:g : 9.81 mlsz,m : 68.1kg, and ca :0.25 kg/m. Note that the exact value of the integralcan be computedwith Eq.(17.4) a s4 1 . 9 4 8 0 5 . The function to be integratedcan be developedas an M-file or with an anonymous function, >> v=@(t) sqrt (9.81*68.L/0.25) *tanh(sqrt (9.81*A.25i68.1)*t) C ( t ) s q r i ( 9 . 3 1 * 6 8 . 1 , 't ). 2 5 ) * t a n h ( s q r t ( 9 . B 1* A . 2 5 / 6 8 . 1 ) * t ) First, let's evaluatethe integralwith a crudefive-segmentapproximation: formaL long >;' tt ap (v, 0, 3, 5 ) o, ..6oo2o
:,ar" )-1>5
IZ.4 SIMPSON'S RUTES
405
As would be expected.this resulthasa relativelyhigh true enor of 18.67o.To obtaina more accurateresult,we can use a very fine approximationbasedon 10,000segments: > > t . r a p ( r r ,O , 3 , 1 0 t 1 0 0 ) 4!.948049999r1528 which is very closeto the true value.
17.4 SIMPSON'SRULES Aside from applyin-qthe trapezoidalrule with finer segmentation,anotherway to obtain a more accurateestimateof an integral is to use higher-orderpolynomials to connectthe p o i n t s . F o r e x a m p l e ,tihf e r e i s a n e x t r a p o i n t m i d w a y b e t w ef (ean) a n d / ( b ) , t h e t h r e e points can be connectedwith a parabola(Fig. 17.11a).lf there are two points equally spacedbetween.f (a) and f (b), the tbur points can be connectedwith a third-orderpolynomial (Fig. l7.l1b). The formulasthat resultfrom takingthe integralsunderthesepolynonrialsare called Sintpson'srules.
17.4.1 Simpson'sl/3 Rule Simpson'sI /3 rule corespondsto the casewherethe polynomialin Eq. ( 17.8)is secondorder: (x-xr)(;-rz) (x - xo)1.r- rz)
,: I:,'l
. / (rtt) ( x 1 y- x y ) ( x s - x 2 )
+
+ (xr -
x e ) ( x 1-
:
l(xr)
(,r - xo) ("t - :rr)
4 rr',rla, I
(x2 - xe)(.r2 - Y t )
F I G U R EI 7 . I I ( o )G r o p h l c odl e p i c l i oonf S i m p s o n1s/ 3 r u i e l:t c o n s i sol sf t o k i n gt h eo r e ou n d eor p c r o b o l o c o n n e c i i nt g h r e ep o i n t s(.b )G r c p h i c od e p i c f i oonf S i m p s o n3' s/ 6 r u l e l: t c o n s i soi sf t o k i n gt h e ^rA^,
n/l^r
1(/.)
' g 'I O J r p O t n ' 5 . 1 - - , O nC O ' n e c l
^-^ . ^ r i
f(x)
406
NUMERTCAL tNIEGRAT|ONFORMULAS wherea andb arc designated&s-:16 and 12, respectively.The result of the intesration is lt -
/ :
-r 1 [/(xo) 4.f(x) + J-(.y)l
where,forthiscase,/z -(!srems
(n
a ) / 2 . T h i s e q u a t i o n i s k n o wa ns S i n t p s o n , s I/3rule.T'e
fromthe r r w rfact 4 v r that r r 1 4 r //r t is l s atvided u t v l ( j e o by Dy 3 l n Eq. r7.22). J in l2t. Si Simpson'sl/3 rulecan
:?::,,^t1]_ also be expressed usingthe formatof Eq. (17.l3): .l (:to)+ 4f (r) * .f (xzl -
I-(b-a
f)
wherea :
xo,b:.r2,
'fld.rr : the point midway betweena andb,which is rs glven si by
that,accordingto Eq. (17.23),the middlepoiint is weightedby two,"otice
::,:!11and rhe rwo thirds end points by one_sixih. It can be shown that a single_segment applicati ication of Sin npson'sl/3 rulehasa trunca. tion enor of I E ,' :
-
90
h 5 r/ ' f r t '. \l t'
or,becauseh:(b-a)12: -
E ,' - - '
( b - u-t"s
f t 1 , t b /,
2990 "\
where f lies somewherein the interval from a to b. Thus, Simpson's1/3 ruleismoreaccuratethan the trapezoidalrule' However, comparisonwith Eq. (17.] )indicatesthatit is nore accuratethan expected'Rather than being proportionalto the third denvative, the etror is proportional to the fourth derivative. consequently,Simpson,sl/3 rule is thirdorder accurateeven though it is basedon only threepoints. In other words,it yieldsexact resultsfor cubic porynomialseven thoughit is derivedfrom a parabora! E X A M P L E1 7 . 3
S i n g l eA p p l i c o r i o no f S i m p s o n , sl / 3 R u l e ProblemSiotement. Use Eq. (17.23) ro inregrate f ( . x ' ) : 0 . 2 * 2 5 " u - 2 0 0 1 2* 6 i 5 x 3 _ 9 0 0 x a *400rs frorn.l 0 b : 0'8' Employ Es. o7.24) to esrimare rhe enor. Recall that theexacrin!9 regratis 1.640-533. S o l u t i o n .n : 2 l h : J'(0) : 0.2
1 :0.8
0.4); f ( 0 . 4 ): 2 . 4 5 6
/(0.8) : 0.232
0.2+4(2.456)+0.232 = 1.367467
- 1.36746j: 0.2730667 E, : 1.640533
€t:16.67o
407
I 2 . 4 S I M P S O N 'R SULES
which is approximatelyfive times more accuratethan for a singleapplicationof the trapezoiclalrule (Example17.I ). The approximate errorcan be estimatedas oRs E ", , : - l t - 2 4 0 0 ) 1880
: 0.17-10667
where -2400 is the averagefourth derivativelbr the interval.As was the casein Example 17.1,the error is approximate(E,,) becausethe averagefourth derivativeis generally not an exactestimateof /'+rt{ ). Hower,'er, becausethis casedealswith a fifth-olderpolynomial, the result matchesexactly.
17.4.2 The CompositeSimpson/sl/3 Rule Just as with the trapezoidalrule. Simpson'srule can be improvedby dividing the integration interval into a number of segmentsof equal width (Fig. I7 .12).The total integralcan be representedas
r:
[^,,'f{*)ar+ l,'.f
(x)ttx+. . ' +
o, 1."' .rU,
( 17.2-s)
FIGURE 17.12 lim^.^n'< 1/3 rule.The relotiverveightsore depictedcbove the funclionvclues Not-^thoi the methodcon be employedonly if the numberof segmenlsis even.
fnrrnn
408
NUMERICAL INTEGRATION FORMUTAS
SubstitutingSimpson'sI /3 rule fbr eachintegralyields
I (xo)+
I :2lt
+..'+2h
66 (r,, z) * 4J'Q,, r) + l (r,,) J 6
or, groupingtermsand usingE q .( 1 7 . 1 - 5 ) : n
l(,ro)+4I
I
n2
fQi)+2 L
j:2,4.6
i- 1.3.5
J:(b-tt)
fG)+f(r,,)
3n
Notice that, as illustratedin Fig. 17.12,an even numberof segmentsmustbe utilized to implement the method. In addition, the coefficients"4" and "2" in F,q.(17.26)might seempeculialat first glance.However.they follow natul'allyfrom Simpson'sl/3 rule.As illustratedin Fig. 17.12,the odd pointsrepresent the rniddleterm tbreach applicationand hencecarry the weight of four from Eq. (11.23).The even points are commonto adjacent applicationsand henceare countedtwice. An error estimatefor the compositeSimpson'srule is obtainedin the samefashionas tor the trapezoidalrule by summing the individual errors for the segnlentsandaveraging the derivativeto vield rh -
t ,", : - " ,
'' ,' 1 5 ! 'I ' o '
\1j.21)
l80na
where f EXAMPLE I2.4
(l)
is the averasetourthderivativetbr the interval.
C o m p o s i t eS i m p s o n ' sI / 3 R u l e ProblemSiotement. UseEq. (17.26\ with n : 4 to estimatethe integralof f (.x) : 0.2 * 25r - 200x2-| 675;;3- 900x4* 400x5 from a : 0 to b : 0.8. Employ Eq. (11.27) to estinlatethe error. Recall that theexact integralis 1.6.10533. Solution.n :4(h:
.f(o'): o.z ( I ) . 1 ): 2 . 4 5 6 l (0.8): 0.232
0.2):
f \0.2): 1.288 . f( 0 . 6:)3 . 4 6 4
From Eq. (11.26): / -nrl
Et :
0.2+ 4( 1.288 + 3.464)+ 2(2.456) + 0.232 : 1.623467 I2
1 . 6 4 0 5 3 -- l 1 . 6 2 3 4 6 :10 . 0 1 7 0 6 1
€t :
I.04Vo
RULES IZ.4 SIMPSON'S The estimatederror 1Eq.17.27';is (o'8)5 E', , : - . _ t-2100) : ().U17067 l t t 0 ( 4) + which is exact(aswas alsothe casefor Example17.3).
As in Example 17.4, the compositeversionof Simpson's1/3 rule is consideredsuperior to the trapezoidalrule fbr most applications.However,as mentionedpreviously,it is Iimitedto caseswherethe valuesareequispaced. Further,it is limited to situationswhere thereare an evennumberof segments and an odd numberof points.Consequently, as discussedin Section 17.4.3,an odd-segment-even-point formula known as Simpson's3/8 rule can be usedin conjunctionwith the I /3 rule to permit evaluationo1'bothevenand odd numbersof equispaced segments.
17.4.3 Simpson's378 Rule In a similar mannerto the derivationof the trapezoidaland Simpson'sl/3 rule, a fhirdorder Lagrangepolynomial can be fit to four points and integratedto yield f :
3h -t3f(x) +.f(.fr)l o If(.rr1)*-]l(.rr)
where/r:(b- a)/3.ThisequationisknownasSimpsons3I8rulebecause/rismultiplied by 3l8.It is the third Newton-Cotes closedintegrationformula.The 3/8 rule can alsobe
expressed in thefbrmof Eq.(17.13): i:\U-UrT
- f( x r , ) * 3 / ( r r ) * 3 . f ( . x )+ . / ( - r : )
(11.28)
Thus, the two interior pointsare given weightsof three-eighths, whereasthe end pointsare weightedwith one-eighth.Simpson's3/8 rule hasan emorof E, : -]/,'f 6t/
't'te t
or, becauseh : (l) - a) 13:
, , : - ' b . , . i L l ' . / 'r' ' r { 64rJ0
(1.29)
Becausethe denor.nina.tor of Eq. (11.29)is largerthanfor Eq. (17.24),the 3/8 rule is somewhat more accuratethanthe 1/3 rule. Simpson'sl/3 rule is usuailytlre rnethodof pref-erence becauseit attainsthird-order accuracywith three points ratherthan tlre tour points requiredtbr the 3/8 version.However,the 3/8 rule has utility when the nunrberof segmentsis odd. For instance,in Example 17.4 we usedSimpson'srule to integratethe function for four segments.Supposethat you desiredan estimatefor five segments.One option would be to usea compositeversion of the trapezoidalrule as was done in Example 11.2. Thts may not be advisable,however, becauseof the large truncationerror associatedwith this method.An alternativewould be to applv Simpson'sl/3 rule to the first two seqments Simoson's3/8 nrle to the last
4ro
NUMERICAL INTEGRATION FORMULAS
0
0.2
0.4
# 1/3 rule
0.6
0.8
3/8 rule
FIGURE 17.13 jntondem lllusirotion of howSimpsons 1/3cnd 3/Brules conbeopplied 1ohondle muriple
o p p l i c o l i o nw s i t h o d d n u m b e r so f j n t e r v o l s .
three(Fig. ll .13).In thisway,wecouldobtainanestimate with third-order accuracy across theentireinterval.
EXAMPLE 12.5
S i m p s o n3' s7 8 R u l e ProblemStqtement.(a) UseSimpson's 3/8 ruleto integrare
TABTEI
-l400x5 ./(r) : 0.2r 25x- 20012* 675x'r- 900;r+ froma :0 to b :0.8. (b) Useit in conjunction with Simpson's l/3 ruleto integratethe samefunctiontbr five segnlents. Soluiron. (c) A singleapplication of Simpson's 3/8 rule requires four equally spaced points: /(0):0.2 :3.4811'71 /(0.,5333)
f (0.266j):t.432j24 ,f (0.8):0.232
UsingEq.(17.28): + 3(t .432724 + 3.487117) + 0.232 l,: t ) .^l J-0.2 :1.51970
Segmentr (n)
HIGHER-ORDER NEWTON.COTES FORMULAS
4tl
(b) Thedataneededfor a five-segment appI ication(/z: 0.16) is .f (0) : 0.2 ./ (0.16): | .296919 f ( 0 . 3 2 1: 1 . 7 4 3 3 9 3 l ( 0 . 4 8 ): 3 . 1 8 6 0 1 5 f Q.A+1: 3.181929 ,f (l.Stl;:0.232 Theintegralfor thefirst two segnrents is obtainedusingSimpson's1/3 rule: I :0.32
0.2+ 4(.1.2969t9)+ t.143393 :0.3803237 0
For the last threesegments,the 3/8 rule can be usedto obtain 1:0.48
1 . 1 4 3 3 9+3 3 ( 3 . 1 8 6 0 1+5 3 . r 8 r 9 2 9 )+ 0 . 2 3 2 :1.264154 g
The total integralis computedby summingthe two results: : 1.645017 1 : 0.3803237 + 1.264754
17.5 HIGHER-ORDER FORMUTAS NEWTON.COTES As noted previously,the trapezoidalrule and both of Simpson'srules are menbers of a family of integratingequationsknown as the Newton-Cotesclosedintegrationformulas. Some of the formulas are summarizedin Table 11.2. along with their truncation-eror estimates. Notice that, as was the casewith Simpson'sI l3 and 3/8 rules,the five- and six-point formulas have the sameorder error. This generalcharacteristicholds for the higher-point fornrulasand leadsto the resultthat the even-segment-odd-point fonnulas(e.g.,l/3 rule preference. and Boole's rule) are usually the methodsof
IABLE 17,2 Newton-Cotes closedintegrotion formulos. Theformulosore presented in theformotof Eq. {l Z.I 3) the overogeheightis opporent.Thestepsizeis so thoi theweightingof thedoto pointsto estimote givenbyh:(b-a)ln. Poinls
Formulo
Nome Trcpezoidol rule
., \t)
./(ro)+l(-rr) n, )
S i m p s o r '1: 3 ' r - ^
., . , f ( r o ) + 4 . f ( - r r )+ f ( - r : ) \p - r,)-
S m p s o n3s/ 8 r u e . ," . . , 8 5
Bccles rr,le
\ 1 ,- u ) \b-0)
Truncolion Error -ll /I2lhtl"tI) - l 1, ' 9 O 1 t r1:' r ' t , ,
/ { . \ , , r - ' . / { . r 1 r -3 t t . r ' , - 7 r . t . r
7 . 16 0 ) + 3 2 1 ( x t ) + l 2 l ( x l + 3 2 . (1r t ) + 7 J \ t 1 )
- 1 3 l 8 0 ) 1 5f 1 1 ) ( 6 ) Q 4 5 ) l i r . ; r1( '6r 1
182 90 I 9 l (-rx) + Z5.l(.rr) + 501(.r:)+ 501(,{r)+ 75l (xa)+ I 9 l (r5) -127 5 / I 2,a9dnt .1rt':16, 288
412
N U M E R I C AI LN T E G R A T I OFNO R M U T A S
However,it must alsobe stressedthat, in engineeringand sciencepractice,thehigherorder (i.e., greaterthan four-point) lbrrnulasare not commonly used.Sirnpson'srulesare sufficientfor most applications.Accuracycan be improvedby usingthe compositeversion. Furthermore,when the function is known and high accuracyis required,methodssuchas Rombergintegrationor Gaussquadrature,describedin Chap. 18, offer viableandattractive altematives.
17.6 INTEGRATION WITH UNEQUALSEGMENTS To this point, all lbrrrrulasfor numericalintegrationlrave been basedon equispaced data points. In practice,there are ntany situittionswherethis assurnptiondoesnot holdandwe r.nustdeal with unequal-sizedseqments.For example,experimentallyderiveddatais often of this type. For thesecases,one method is to apply the trapezoidalrule to eachsegment and sum the results: I -ht
.f(.ril+f@r)
lhz
f (u) + f (r.) * " . r h ,.l, G, t) * .1(x,,)
whereft; : the width of segmenti. Note that this was the sameapproachusedfor thecompositetrapezoidalrule.The only differencebetweenEqs.( 17.16)and( 17.30)is thattheft's in the formel are constant. E X A M P L EI 2 . 6
T r o p e z o i d o lR u l ew i t h l ' l 1 s ^ " ^ l s a a m a n l c Problem Stotemeni. The informationin Table 17.3was generated usingthesamepolynomialemployedin Example17.1.Use Eq. ( 17.30)to determinethe integralfor thisdata. Recallthat the correctansweris 1.640533. TABTE 17.3
Doto for "f(.r) : 0.2 + 25x - 200x2I 67513- 9O0ra+ 400"t', with uneguollyspocedvoluesof :r.
f (-x) 000 0t2 o22 o32 036 040
0 200000 1 .ic9729 t 345241 ) 7,13393 2 074943 2.456040
f {x1 044 o5a 064 o70 080
S o l u t i o n . A p p l y i n gE q . ( 1 7 . 3 0 y) i e l d s
I :0.12
1.309129 0.2+ r.309129 + 1.305241 + 0.10
2 . 1 6 1+ 0 2 1 2 +...+Ot0:_::'lj:_:::=t.-s9-1801 which represents an absoluteperceutrelativeerrorof €t :2.8Vc.
2 842985 3 547297 3 rBr929 2 363000 0 232AAA
413
12.6 INTEGRATION WITHUNEOUALSEGMENTS
17.6.1
MATLAB M-file: trapuneq
A simple algorithm to implement the trapezoidalrule for unequally spaceddata can be written as in Fig. 17.14.Two vectors,x andy, holding the independentand dependentvariablesarepassedinto the M-file. Two error trapsareincludedto ensurethat (a) the two vectors are of the samelength and (b) the x's are in ascendingorder.rA loop is employedto generatethe integral.Notice that we havemodified the subscriptsfrom thoseof Eq. (17.30) to accountfor the fact that MATLAB doesnot allow z.erosubscriptsin anays. An applicationof the M-file can be developedfor the sameproblemthat was solvedin E x a m p l e1 7 . 6 : >> x - l0 .I2 .22 .32 >> v - 4.2+25*x-200*x. > > l -r a p L l n e q ( x , y )
:
(o
.Bl; .36 .4.44.54 .64.1 ' 2 + 5 15 * x . ^ 3 - 9 0 0 " x . ^ 4 + 4 0 0 * x . " 5 ;
t!
which is identicalto the resultobtainedin Example17.6.
FIGURE 17.14 thetropezoidol rulelorunequolly spoced dofo. M{ileio lmplement function o. o
i r _ nJ r . u
f \y.
= trapuneq(x,y) J,--4)a
L
.-r-.i,. >lJdLFu
L r a p s T u ', v. (: ,d r_ l
, t rt - L o
dlt:alt_ftte YJ
T = trapuneq (x, y) : Z rule to determine the integral ? Applies the trapezoidal (x, y) where x and y must be of t,he * for n daLa points ascending % same lengrth and x must be monotonically % input: x = ./ector of independent var:rables % y = vector of dependenE variai:les ? % output: 2 I=inteqralestimate least 2 input arguments resuired'),end if nargin<2,error('at not monotonically ascendrnq' ),end if any(diff (x)<0),error('x n = lengrth(x); if lengch (y) -=n, error ( 'x and y must be same length' ) ; end s = 0; for k = 1;n-1 +y\k+\\) /2; s : s + (x(k+1)-x(k))-(y(k) end T
' The
-
c '
r : l u n c t i o ni s J e s e r i h eiJn S c c t i o r1r 9 . 7I.
414
FORMULAS NUMERICAL INTEGRATION
17.6.2
MATTAB Functions: trapz
Qrd
cumcrapz
MATLAB has a built-in function that evaluatesintegralsfor datain the samefashion asthc M-file we just presented in Fig. i7.14. It hasthe generalsyntax z
-
t rapz
(x,
l')
wherethe two vectors,x &od-r,.hold the independentanddependentvariables, respectively. Here is a simple MATLAB sessionthat uses this function to integratethe datafrom T a b l e1 7 . 3 : >> x = t0 .I2 .22 .32 .3,h .4 .44 .54 .64 .7 .81; > > y = 0 . 2 + 2 5 * x - 2 0 0 " x . " ' 2 + 6 15 * x . ^ 3 - 9 0 0 * x . ^ 4 + 4 0 0 * x .^ 5 ; >> trapz (x,y)
1.5948 In addition, MAILAB has another function, cumr rapz. that computes the cumulativc inte_ural. A sirnple representationof its syntax is c u m t r a p z ( x. ,
z -
y.)
where the two vectors, x and y, hold the independent and dependent variables,respectively, a n d z : a v e c t o r w h o s e e l e n r e n t sz ( k ) h o l d t h e i n t e g r a lf r o r n x ( 1 ) t o x ( k ) .
EXAMPLE l7.7
UsingNumericolIntegrotion to ComputeDistonce fromVelociiy ProblemStotement. As described at the beginningof this chapter, a niceapplication of integrationis to computethe distancez(r)of an objectbasedon its velocityu(t)asin (recallEq. 17.2): I u \ t ld t
:trl:
J(\
Supposethat we had measurements of velocityat a seriesof discreteunequallyspaced tirno during free fall. Use Eq. (17.2) to syntheticallygeneratesuch inforrnationfor a 70-kg jumper with a drag coefficient of 0.275 kg/m. Incorporate some randomerrorby rounding the velocitiesto the nearestinteger.Then usecum1, rapz to determinethedistance fallen and comparethe resultsto the analyticalsolution(Eq. 17.a).In addition,developa plotofth analyticalar.rdcornputeddistancesalong with velocity on the samegraph. Solution. Someunequallyspacedtimesandroundedvelocitiescanbe generated as '. lornot shu'r (J ;';5=[Q 1 L.4 2 3 4.3 5 5.1 B); > > 9 - 9 . 8 L ; m = 7 0 ; c c t = O. 2 ' 1 5; > > v = r o u n d ( s q r t ( g * m , / c d )* t a n h ( s q r t ( g * c d / m ) * t ) ) ; The distances can then be comouted as /
J^Ltopt(
o
5
9.6
,vJ
1-9.2
4L1
80.7
I44.45
173.85
23r.t
I2.6 INTEGRATION WITHUNEQUAL SEGMENTS
the
)ly. om
4t5
Thus, after 8 seconds,thejumper hasfallen 231.7m. This result is reasonablycloseto the analyticalsolution(Eq. 11.4):
A graph of the numericaland analyticalsolutionsalong with both the exact and rounded velocitiescan be generatedwith the following commands: (t (1) , t (length(r) ) ) ; za=n/ cd*1og (cosh (sqrt (qr*cd/m) *t,a) ) ; >> ploL (La, za,t, z,' a' ) >>
ra=linspace
>> >>
title('Distance t-1me') versus ('t (s) ') ,ylabel ('x (m) ') 'numerical Legend ( 'analyticaL' , t
>> xlabel >>
rve
,ly,
As in Fig. 17.15,the numericaland analyticalresultsmatchfairly well.
FIGURE 17.15 Ploiof distonce timeThelinewoscompufed versus withtheonolyticol rne solufion, wnereos p o i ' r ' sw e e d e l e ' r i r e d ^ l n e c o l , , , r- [ .- h e u r
on in
Distanceversustime
O LCS
kg ng nd he
-. , fu^clio..
a nalytical numerical
416
NUMERICAL INTEGRATION FORMULAS
TABTE| 7.4
Newton-Cotes open integrotion formu/os. Theformulosore presenied in the formotof Eq.{17.l3) so thottheweightingof the doto pointsto estimote theoverogeheightis opporent.Thesiepsize i s g i v e nb y h : ( b - u ) l n .
Segmenls (n\
Poinls
2
I
3
2
Nome
Formulq
M dpolf t rn-.1hod
( b - a ) . f( . r t ) . . - . . f ( x t )+ l ( - r 2 . 1 \D
a)-
Truncotion Enor
| /3V13 f"G) t,314)h3 f"G)
2 J ( r t ) - . 1( r : ) I I I ( r r ) (l4rJ5ll5.;iar161 3 I l . / ( . t)r + . 1( f r ) * . f ( , r r )* 1l . l ( - r 1 ) (1)(6) (95/l44lr5.f d) lD 2A i l . l ( - t r -) l : 1 1 ( f r +) 2 b l ( x t )- 1 4 .(f x r ,*) I 1 . 1 ' 1 . r r 1 (b-u) l+t,l14A)h1 f\6)Gl 2A (b
a\
17.7 OPENMETHODS Recalltiorn Fig. 11.6b that openintegrationformulashavelimits that extendbeyond the rangeof the data.Table 17.4summarizesthe Newton-CotesopeninteT4rotion.fbnnulas.The fbrmulasare expressedin the tbmr of Eq. ( 17.I 3 ) so that the weightingfactorsareevident. As with the closed versions,successivepairs of the formulas have the same-order enor. The even-segment-odd-point fbrmulas are usuallythe methodsof pref'erence because they require f'ewerpoints to attainthe sameaccuracyas the odd-segment-even-point fbrmulas. The openformulasarenot ofien usedfor definiteintegration.However,theyhaveutility for analyzingimproperintegrals.In addition,they will haverelevanceto our discussion of methodsfor solving ordinary differentialequationsin Chaps.20 and 21.
17.8 MULTIPLE INTEGRATS Multiple integralsare widely used in engineeringand science.For example,a general equatiorlto compute the averageof a two-dimensionalfunction can be written as lrecall E q .( t 1 . 1 ) l
l . ' '( I i l t r , 1 ' ) t 1 x ) (d-c)(b-a)
(r7.3 r)
The numeratoris called a double integntl. The techniquesdiscussedin this chapter(and Chap. l8) can be leadily employedto evaluatemultiple integrals.A simpleexarnplewould be to take the doubleintegralof a functionover a rectangular area(Fig. 17.16). Recallfrom calculusthat suchintegralscan be computedas iteratedintegrals:
a,)at: L' (1,'/(r' r)or) L' (1,';(.r,,v)
(17.32)
Thus, the integral in one of the dimensionsis evaluatedfirst. The result of this firstintegration is integratedin the seconddimension.Equation(11.32) statesthat the orderofin-
EXA
I 2.8 MULTIPLE INTEGRATS
417
FIGURE17.16 Double infegrol ostheoreounder thefunction surfoce
A numericaldouble integralwould be basedon the sameidea. First, methodssuchas the cornpositetrapezoidalor Simpson'srule would be applied in the first climensionwith eachvaltteof the seconddimensionheld constant.Then the methodwould be appliedto integratethe seconddimension.The approachis illustratedin the following example.
EXAMPLE 12.8 U s i n g D o u b l e I n t e g r o lt o D e i e r m i n eA v e r o g eT e m p e r o t u r e Problem Stotement. Supposethat the temperatureof a rectangularheateclplate is describedby the following function: I ( t . - r , ): 2 x ) * 2 r - x 2 - 2 t ' 2+ 1 2 If the plate is 8 m long (;r dimension)and 6 m wide (,y dimension),computethe average temperature. Solution. First,let us merelyusetwo-segment applications of thetrapezoiclal rulein each dirnension.The temperatures at the necessary,rand y valuesare depictedin Fig. 17.17. Note that a simple averageof thesevaluesis 47.33.The function can also be evaluatedanalyticallyto yield a resultof 58.66667. To make the same evaluationnumerically,the trapezoidalrule is first implemented along the ,r climensionfor each-r'value. Thesevaluesare then integratedalong the I' dimensionto give the final result of 2688. Dividing this by the areayields the averagetemperature a s2 6 8 8 / ( 6 x 8 ) : 5 6 . Now we can applya single-segment Simpson'sl/3 rule in thesamefashion.This results in an integralof 28 l6 andan averageof 58.66667,which is exact.Why doesthisoccur?Recall thatSimpson's| /3 ruleyieldedperfectresultsfor cubicpolynomials.Sincethehighest-
4t8
NUMERICAL INTEGRATION FORMULAS
Iftiit U
40
54
70
72
64
48
( u - u^ ). 0 + 2 { 4 0 ) + 4 8 + 2 5 6 4
24
( 8 - 0 ) 5 4 + 2 | \ 7 0 ]+. 5 4 4-+
496
8_u7?:3gej4+
448
X
I
(6-0) 256+?(4961+448
FIGURE 17.17 Nurnericol evoluotion of c doubleintegrol usingthetwo-segmenl rule tropezoidol For higher-ordelalgebraicfunctionsas well as transcendental functions,it wouldbe necessaryto use compositeapplicationsto attain accul'ateintegralestimates. In addition, Chap. l8 introducestechniquesthat are more efficient thanthe Newton-Cotes formulasfr evaluatingintegralsof given f-unctions. Theseoften providea superiormeansto implement the numericalintegrationsfor multiple integrals.
17.8.1
MATTAB Functions: dblquad ond triplequad
MATLAB has functions to implernentboth double (dnlquad) and triple (rriptequad) integration.A simple representationof the syntaxtbr dbtquad is q = d b l q u a d ( f u n, x n t i n , x m a : . ,y m i n , y m a x , t o l ) where q is the double integralof the function fun over the rangesfrom xmrn to xmaxad to ymax.If rol is not specified,a defaulttoleranceof I x 10-6is used. ,vm,zn Here is an exampleof how this function can be usedto computethe doubleintegral evaluatedin Example17.7: >> q - dblquad(@(x,y) 2*x*y+2xx x. "2-2*y.''2+72,A,8,0,6) 2816
CASESTUDY
419
Bockground. The calculation of work is an important component of many areasof engineering and science.The general formula is Work:force
x distance
When you were introducedto this conceptin high schoolphysics,sirnpleapplicationswere presentedusing forces that remained constant throughout the displacement.For example, if a force of 10 N was used to pull a block a distanceof 5 m, the work would be calculated as50J(ljoule:lN.m). Although such a simple computationis useful for introducing the concept,realistic problem settingsareusually morecomplex.For example,supposethat the force variesduring the courseofthe calculation.In suchcases,the work equationis reexpressed as
w:
F(x)dx 1.,,'
( 17.33)
where W: work (J), xo and x, : the initial and final positions (m), respectively,and F(x) : a force that varies as a function of position (N). ff f(-r) is easyto integrate,Eq. (17.33) can be evaluatedanalytically. However, in a realistic problem setting,the force might not be expressedin such a manner.In fact, when analyzing measureddata, the force might be available only in tabular form. For such cases,numerical integration is the only viable option for the evaluation. Further complexity is introduced if the angle between the force and the direction of movement also varies as a function of position (Fig. 17.18).The work equationcan be modified further to account for this effect. as in
w:1""
F(,r) cos[9(x)] dx
(r7.34)
Again, if F(-r) and e @) arc simplefunctions,W 07.34) might be solvedanalytically.However, as in Fig. 17.18,it is more likely that the functionalrelationshipis complicated.For this sifuation, numerical methodsprovide the only alternativefor determining the integral. Suppose that you have to perform the computation for the situation depicted in Fig. 17.18.Although the figure showsthe continuousvaluesfor F(x) and d(x), assumethat, becauseof experimentalconstraints,you are providedwith only discretemeasurements at x : 5-m intervals(Table 17.5).Use single- and multiple-applicationversionsof the trapezoidal rule and Simpson's1/3 and 3/8 rules to computework for this data. Solution. The resultsof the analysisare summarizedin Table 17.6.A percentrelative error r/ was computed in referenceto a true value of the integral of 129.52 that was estimatedon the basisof valuestakenfrom Fig. 17.18at 1-m intervals. The results are interesting becausethe most accurateoutcome occurs for the simple two-segment trapezoidal rule. More refined estimates using more segments,as well as Simpson'srules, yield lessaccurateresults. The reason for this apparently counterintuitive result is that the coarse spacing of the points is not adequateto capturethe variations of the forces and angles.This is particularly
420
FORMULAS NUMERICAL INTEGRATION
continued
0 ,{, m
FIGURE17.18 forceoctingono block.Forlhiscosetheongle,oswellosihe Thecoseof o vorioble mognitude, of theforcevories.
TABLE t 7.5
Doto for forceF("r)ond ongle 0(x) os o functionof position-r. F(r), N
0 5 t0 t5 20 25 30
0.0 9.0 r3.0 l4.o 105 12.0 5.0
0, rod 050 1.40 0.75 090
r30 I .48 r.50
F(x) cos d
0.0000 1.5297 9 . 5r 2 0 8.7025 2.8082 1. 0 8 8 1 0.3532
I2.9 CASESTUDY
421
continued TABIE 17.6
Estimoies of work colculotedusingthe tropezoidolruleond Simpson's rules.Thepercentrelotiveerror€, os computedin referenceto o truevolueof the integrol1129.52Po)thot wos estimoted on the bosisof voluesot l-m intervols.
Technique Trnnpzni.lnl
Segmenls
r,'lo
Simpson!1/3 rule Simpson's 3/B rule
I 2 3 6 2 6 3
o/o
Work
e[
5 3t 1 3 3r 9 124.98 r19.09 175.82 I l 7. 1 3 r39.93
95.9 2.84
35 r 8.05 35 75 L57 804
5
5
F|GURE17.19 A continuous plotof F(x)cos[d(r)]versus posilion withtheseven discrete pointsusedto develop ihe numericolintegroiionestimotesin Toble 17.6 Notice how the use of sevenpointsto chorccterizethisconfinuously voryingfunctionmissestwo peoksot x :2.5 ond 12.5 m.
evident in Fig. 17.19, where we have plotted the continuous curve for the product of F(.r) and cos [0(x)]. Notice how the useof sevenpointsto characterizethe continuouslyvarying function missesthe two peaks at r :2.5 and 12.5 m. The omission of thesetwo points effectively limits the accuracyof the numerical integration estimatesin Table 17.6. The fact that the two-segment trapezoidal rule yields the most accurateresult is due to the chance positioning of the points for this particular problem (Fig. 17.20). The conclusion to be drawn from Fig. 17.20 is that an adequatenumber of measurements must be made to accurately compute integrals. For the present case, if data were ^.'^ir4rrr6 af E"/t {\ onc tA(1 5\l -
?
422
NUMERICAL INTEGRATION FORMULAS
(a) anal (c) coml applicat | /3 rule the num cent rela 17.4 Ev
continued
1a
I tt
a. 6
(a) analy (c) coml applicat and (f) E through 17.5 Th f (x)
,t, ft
FfGURE17.20 Grophicol depiction of whvthetwo-seqment lropezoidol rulevieldso ooodestimote ofthe forthisporiictrlor integrol cose.Bychoice,theuseof trrotro[ezoids h"oppens to leodtoon even bolonce between posifiveond negotive errors.
can be r spacedd rO f(x)
determine an improved integral estimate.For example, using the MAILAB trapz tion, we could compute >> x=t0 >> y=[0
2.5 5 r0 L2.5 15 20 25 301; 3.9007- L.5291 9.5t"20 11.3940 8.1025 2.8087 ...
1.0881 0.35371; >> trapz (x, y )
2
Evaluate lytical rn of the tri attain thr cent rela 17.6 Ev el
I J-t J
r32 .6458
Including the two additional points yields an improved integral estimateof I (e,:2.l6Eo). Thus, the inclusion of the additionaldata incorporatesthe peaksthat missed previously and, as a consequence,lead to better results.
PROBtEMS 17.1 DeriveEq. ( 17.4)by integratingEq. (17.3). 17.2 Evalnte the lbllowing integral: r4 I
JO
tt - e-2'1tlx
applicationof Simpson'sl/3 rule, (e) composite Simpson\ 1/3 rule with n : 4, and (f) Simpson's3/8 rule.Foreachof the numerical estimates(b) through (f), determinetheprcent relativeerror basedon (a). 17.3 Evaluatethe following integral: rn /2
(a) anal with n = 1/3 rule
423
PROBLEMS
(a)analytically, (b) singlc applicationof the trapezoidalrule, 17.7 Evaluatethe triplc integral (c)conrposite tlapezoidal rule with n :2 iutd 4, (d) single n) n) rl application of Sirnpson'sl/3 rule, (e) compositeSimpson's 1 . r 3- 3 r ' : ) d t d t ' d : 1/3rulewith n : 4. and (f) Simpson's3/8 rule. For eacho1' thenumericalestimates(b) through (f), determinethe per(a) analytically.and (b) using single applicationsof cerrt relltirc erlor hasedon 1a.1. Simpson'sl/3 rule. For (b), computethe percentrelative 17.,1 Evaluatethe lirllowins inte-eral: efror. 17.8 Detcrmine the distancetravelcd tionr the follouinc + 2xs)ri-r velocitydata:
I .J,, J^
(a)analytically, (b) singlc applicationof thc trapezoidalrule, (c)compositetrapezoidalrule with n : 2 antl .1,(d) single l/3 rule, (e) Simpson's3/8 rulc. application of Sirr.rpson's and(f) Boole'srule. For cach of the numericalestimates (b) (f), determinethc percentrelativeerror basedon (a). through 17.5The firnction ,f(-r): 2e
r 5'
canbe useclto generatethe following table of unequally spaced data:
tl2 u56
325 55
45 7
6 B5
7 B
B 6
B5 7
93
t0 5
(a) Use the tlapezoidalrule. (b) Fit the data with a cubic equationusine polynomial regression.Integratethe cLrbicequation1odeterminethe distance. I 7.9 Walcr exertspressureon thc upstreamface of a dam as by shown in Fig. P17.9.The prcssurecan be characterized
pQ): pg(D- z) 0o where p(;) - pressurein pascals(or N/ml) exertedat an 0s308 08131 elevationi mctersabovethe rescrvoirbottorn;p : densityol to be a constant water, which fbr thi.sproblem is a.ssunred Evaluate the inlc-eralfrom rr :0 to b :0.6 using (a) ana_ 103kg/rn3;.9 : acceleration due to -eravityt9.81m/s2):antl rule,and (c) a combinltion D : elevation(in m) ofthe watersurfaccabovethe reservoir lytical means,(b) the trapezoidal ofthetrapezoidaland Simpsou'srules whereverpossibleto bottom.Accordingto Eq. (P17.9),pressureincreaseslinearly aurinthehighestaccuracy.For (b) and (c), computethe per- with dcpth,as depictedin Fig. Pl7.9rz.Ornittingatmospheric centrelativeerror'. prcssure(becauseit works againstboth sidesof the dam facc 17.6Evaluatethe double intesral and essentiallycancelsout). the total force ./, can be dctermined by multiplying pressuretimes the areaof the dam tace (as shown in Fig. P17.9b).Becauseboth pressurcand area vary with elevation,thc total force is obtainedby evaluating (a)analytically,(b) using thc composite trapezoidalrule r I) withir : 2. and (c) using singleapplicationsof Simpson's f, : I psw(z)(D-i,)(1. JO l/3 rule.For (b) and (c). compute the percentrelativeeror.
425 O OO5 015 03-5 f ( x ) 2 1 8 5 5 - 51 5 9 7 0 1 3 7 4 6 l l 8 3 l r
A.{5
OU
40 zu
200 190 175 160
(b)
FIGURE P17.9 foce of o Woterexerting pressure on lheupslrecm
dom { o J: ; d e r i e w s L o * r n gf o ' t e . ^ r r e o s , n g
424
INTEGRATION FORMULAS NUMERICAL
where u(l): width of the dam face (m) at elevation.: (Fig. Pl7.9b). The line of action can also be obtainedby evaluating IoDpgzu:(z)(D z.)tJz, p,su(:)(D ;) t1:
ft
71.
Use Simpson'srule to compute J, andcl. 17.10 The tbrce on a sailboatmastcan bc reprcscntcdby the following function:
/ - \ Jr.-r:200(.,.),:s'a \/T'l
where: : the elevationabovethe deck and H : thc height of the mast.The total force F exertedon the mast can be dctermined by integratingthis function over the height of the mast:
O:J,,I k.) dz The linc of action can also be deterrninedby integratron: r
rH J0 :.It:ta: -u
J,,'.fr:\ 'l:
(a) Use the compositetrapezoidalrule to compute F rnd d for the casewhcre 11 : 30 (n : 6). (b) Repeat(a), but use the compositeSimpson'sl/3 rule. l7.ll A wind fbrce distributedagainstthe side of a skyscraperis measuredas Height /, m Force, F(/), N/m
0 0
Height l, m Force, F(/), N/m
I50 3r00
30 340
If M,,iszero and.r : I l, calculateM using(a) analyticalintcgration,(b) multiple-applicationtrapezoidalrulc, and(c) multiplc-applicationSimpson'srules.For (b) and (c) usel-m lncrements. 17.13 The total mass of a variable density rod is givenbi-
60 l2aa
90 t2a t600 27aa
IB0 2)A 240 3200 3500 3800
,r:
pG)A,.(r)dx
I
Jt) p(,r) : dcnsity,A,(,r): cross-sectional whererz : filoSS, area, -r : distancealongthe rod and L : the total lengthoftherod, The following datahasbeenmeasuredfor a l0-m lengthrod. Determinethe massin grams to the best possibleaccurac),
x,m a 2 4 l0 3 6 B p,glcm34ao 395 389 3Bo 3.60 34t 33l A,, cm2 loo lo3 l06 I lo )2a I33 lsc 17,14 A transportationengineeringstudy requiresthatyou determinethe numbcr of cars that passthroughan intersection traveling during morning rush hour. You standat the sideol the road and count the numberof carsthatpassevery 4 minutesat severaltimes as tabulatedbelow. Use thebest numericalmethodto determine(a) the total numberof can that passbetwecn7:30 and 9:15, and (b) the rateofcan going throughthe interscctionpcr minute. (Hint: Be careful withunits.) Time (hr) 73A Rote (cors per 4 min) I B
7 45
B:OO B:15
24
l4
24
B : 4 5 9 :i 5 21
9
17.15 Determinetheaveragevaluefor thedatain Fig.PI 7.15. Perfbrm thc intcgral necdedtbr thc avcragcin theordershown by the following equation: r, f r,-
l- I ll J,.
I
/(\.r),./.rl,1r
L/", .l Compute the net fbrce and thc line of action due to this diswind. tributed 17.16 Integrationprovides a meansto computehow much 17.12 An I l-m beam is subjectedto a load, and the shear massentersor leavesa reactorover a specifiedtimeperiod. force follows the equation a si n V(-r):5+0.2512 where V is the shearforce, and,r is length in distancealong the beam.We know that V : dM ldx, and M is the bending moment. Integrationyields the relationship rI
M:M.,+
I Vttt' .l,t
ft)
M:
I
et tlr
Jt where I, and t2 - the initial and final times, respectively, This formula makesintuitive senseif you rccall thc analogy betweenintegrationand summation.Thus, the integralreprcscntsthe summationof the product of flow times concenqivo
Use numerica data listed bel t, mln Q, m3/mi;r c, fng/Jn"
17.17 The c Putedas
A,: I ./0
where R : t and,r : dist averagetlov
o:
1
w h e r eU : a numerica data:
)rfi I;1, ffi
U, mls -
PROBTEMS
425
-u
-8
-4
-3
-z
1
4
7
I
10
0
12
F I G U R EP I 7 . I 5
Usenumericalinte_uration to evaluatethis equationfbr the datalistedbclow: l, min Q, m3/min c, mg/m3
0 4 t0
t0 2a 30 4B 52 50 3_s .-55 52
35 4A 45 4.6 43 43 4A 37 32
50 5.0 34
17,17The cross-sectional area of a channelcan bc computedas
J0
whercB : the total channelwidth (rn), H : rhedepth(m), andr' : distanceliom the bank (m ). In a similarfashion,thc l1ou,'0 (mr/s) can be contputcclas average rl] I U{.r.lrrlr,r.)z/r. Jo
whereU : watervelocity'(m/s).Use theserelationships and a nurnericalmethodto determineA,.and B fbr the fbllorving data:
J,m H, ftr U, m/s
n)/1
05 003
r3 006
(-:
,t'(,+,da, wherc Z : the total depth (ni). Determinethe averageconcenlrationbasedon thc followins data:
O 4 z,n B 12 )6 A , l a 6 m 2 9 8 1 7 5 5 t 0 5 t 1 . 9 6 3 5A 3 5 2 7 O O O O O c,g/m3 )a'2 85 7.4 52 4l
rll
A,= I H(r')r1r.
0 :
l7.lll The averageconcentfationofa substance-(g/rn3)in a lake u,hercthe arcaA.1m:) varieswith depth l(m) can be computcdby integration:
56 1.25 l7 005 412
l 0Ir
a.25 oa2
17.19 As wasdonein Section17.9.cletermine the work perfbrmed ifa constantforce of I N appliedat an anglegresults in the iollowing displacernents. Use the curnLrcipzfunction 1odetcrminethe cunrulativcwork andplot the resultversusg. r,h 0,rad
0 0
I 30
2.7 60
38 90
37 l2O
3 150
1.4 lB0
if*
.,
NumericolIntegrotion of Functions
CHAPTER OBJECTIVES The primarv objectiveof this chapteris to introduceyou to numericalmethodsfor integratinggiven firnctions.Specificobjectivesand topicscovereclare ' . o
I 8.I
Understandinghow Richardsonextrapolationprovidesa meansto createa more accurateintegralestimateby combiningtwo lessaccurateestimates. Understandinghow Gaussquadratureprovidessuperiorintegralestimates by picking optimal abscissasat which to evaluatethe function. Knowing how to useMATLAB's built-infunctionsq,-r.:dand r-1uadlto integrate functions.
INTRODUCTION In Chap. 17, we noted that functionsto be integratednumericallywill typicallybeoftno lbrms: a table of valuesor a tunction. The form of the datahas an importantinfluenceon the approachesthat can be usedto evaluatethe integral.For tabulatedinformation, youare limited by the numberof points that are given. In contrast.if the fuuctionis available, yut can generateas many valuesof.l (r) as are requiredto attainacceptable accuracy. At tacevalue.thecompositeSirnpson'sl/3 rulemight seemto be a reasonable toolfa such problems.Although it is certainly adequatefor many problems,therearemoreefficient methodsthat are available.This chapteris devotedto three suchtechniques. Bolt capitalize on the ability to generatefunetion values tcl develop efflcient schemes fr numericalintegration. The flrst technique is based on Richardson extr(tpolatiorz,which is a method fa combining two nunrericalintegral estimatesto obtain a thild, more accllratevalue. Tlc computationalalgorithm lbr implementingRichardsonextrapolationin a highlyefficied rnanneris called Rombergintegration.This techniquecan be usedto generate anintegnl esrimatewifhin a nresnecified eror
ROMBERG INTEGRATION The secondmethod is called Gcuss qundrnfure.Recall that, in Chap. 17, values of /(x) for the Newton-Cotesformulas were determinedat specifiedvaluesof ,r. For example, if we usedthe trapezoidalrule to determinean integral,we were constrainedto takethe weightedaverageof /(x) at the endsof the interval.Gauss-quadrature formulasemploy,r valuesthat arepositionedbetweenthe integrationlimits in sucha mannerthat a much more accurateintegralestimateresults. The third approachis called adaptivequadrature.This techniquesappliescomposite Simpson's I /3 rule to subintervalsof the integrationrangein a way that allows error estimates to be computed.These error estimatesare then used to determinewhether more refined estimatesare required for a subinterval.In this way, more refined segmentation is only usedwhereit is necessary. Two built-in MATLAB functionsthat useadaptivequadratureare illustrated.
18.2 ROMBERGINTEGRATION Rombergintegrationis one techniquethat is designedto attainefficientnumericalintegrals of functions.It is quite similar to the techniquesdiscussedin Chap. 17 in the sensethat it is basedon successiveapplicationof the trapezoidalrule. However,throughmathematical manipulations,superiorresultsare attainedfor lesseffort.
18.2.1 RichordsonExtropolotion Techniquesare availableto improve the resultsof numericalintegrationon the basisof the integral estimatesthemselves.Generally calledRichardsonextrapolallon,thesemethods use two estimatesof an integralto computea third, rnore accurateapproximation. The estimateand the error associatedwith the compositetrapezoidalrule can be representedgenerallyas
r:t(h)+E(h) where 1 : the exact value of the integral, I (h) : the approximationfrom an n-segment applicationof the trapezoidalrule with stepsizeh : (b - a) I n, andE (h) : the truncation error. If we make two separateestimatesusing stepsizesof h1 andh2 andhave exact values for the error: I (hr)-t E(ht) : I (h) -f E(hz)
(1 8 . 1 )
Now recall that the error of the compositetrapezoidalrule can be representedapproxlm a t e l yb y E q . ( 1 7 . 2 1 )[ w i t h n : ( b - a ) l h ) : b-u E-__htf" 12
.
(18.2)
lf rtis assumedthat f" is constantregarcl\ess of stcp sizc,Eq. (18.2) canbc uscdto
E(hz)
_ - hh! i
(1 8 . 3 )
This calculationhas the importanteffect of removing the term 1" from the computation. In so doing, we have made it possibleto utilize the information embodiedby Eq. (18.2)
428
N U M E R I C AI L N T E G R A T I OONF F U N C T I O N S
without prior knowledge of the function's secondderivative.To do this, we reanange E q . ( 1 8 . 3 )t o g i v e / h, \:
E ( h t t= E ( h : t l ;
\n2/
I
which can be substituted into Eq. (18.1): /h,\2 I(h1tLEthrtl , | -I(h2tlE(htl \tt2 /
which can be solvedfor p E\ (" /1r/ ' ) :
Itlttl- I\hzl 1-(htlh)2
Thus, we have developedan estimateof the truncationelror in tenns of the integral estimatesand their step sizes.This estimatecan then be substitutedinto I : I(hz) _t E]tz) to yield an improvedestimateof the integral: I : l(hz) + -
- 1 - t - -- .
\tt1/h2)'
|
ll ttt:l -
ttttrtl
fl8.4)
It can be shown (Ralston and Rabinowitz, 1978) that the error of this estimate is we havecombinedtwo trapezoidal of O(h2) to yielda newesO(h4). Thr-rs, ruleestimates timateof O(h\. For the specialcasewherethe intervalis halved(h2: hr12),thisequation becomes
' :
4l
a'(")
-
(185)
lt(nt;
EXA
Extropolotion E X A M P1 LB E . I Richordson ProblemStotement.UseRichardson to evaluate extrapolation theintegral of /(.r)= : : 0.2t 25x 200x2t 675x-r 900r+*400x5 tioma 0 to b 0.8. Solution. Singleandcomposite rulecanbeused toevaluapplications of thetrapezoidal atetheintegral: Segments
h
I 2 4
0B a4 02
Integrol
o t72B I 0688 t 4B4B
E
89.5% 349% 5.5%
Richardsonextrapolationcan be usedto combinetheseresultsto obtainimprovedestimates of the intesral. For example.the estimatesfor one ancltwo sesmentscan be combined
429
INTEGRATION I 8.2 ROMBERG
to yield
t :lt.oazsr- lro.rzz8):1.36146i 3'
3' : 0.2130(t1 (st : 76.6c/r:), is Et : 1.640533 1.361461 improvedintegral Theerrorofthe which it was based. estirnates upon whichis superiorto the for two andfour segments canbecombinedto give In thesamemanner,theestimates 41 I : - ( 1 . 4 8 4 8- ); ( 1 . 0 6 8 8 ) : 1 . 6 2 3 4 6 1 -53
an errorof E, : which represents
1.640533 1.623461:0.011061 (et :
l.ja/c).
Equation(18.4)providesa way to combinetwo applicationsof the trapezoidalrule with error O (h2) to computea third estimatewith error O (ha1. This approachis a subsetof a more generalmethodfor combiningintegralsto obtainirnprovedestimates.For instance, in ExampleI 8. l, we computedtwo improvedintegralsof O (ha) on thebasisof threetrapeThesetwo improvedintegralscan,in turn, be combinedto yield an zoidalrule estimates. estimates evenbettervaluewith Oth6l. For the specialcasewherethe originaltrapezoidal fbr is halving the step size, the equation used O(li6) accuracy of are basedon successive I :
r
16
(r 8 . 6 )
-lt I-5
-1,,, 1.5
where f,, and 11 arc the more and less accurateestimates,respectively.Similarly, two O(ft6) resultscan be combineclto computean integralthat is O(/r8)using [ :
,l8.2 :XAMPLE
64
-1,,,
63"'
-
I
-ll
61
'
( 18 . 7 )
Higher-OrderCorrections ProblemStotement. In Example18.1,we usedRichardson extrapolation to computetwo integralestimatesof Othl). Utilize Eq. (18.6)to combinetheseestimatesto computean i n t e g r awl i t h O t l r 6 t . Solution. The two integralestimates of O(h1) obtainedin Examplel8.l were 1.361461 and 1.623461.Thesevaluescan be substitutedinto Eq. ( I 8.6) to yield
r : ! l . en 4 fl ) - !t.ru ro u r): r .640533 t5' t5 which is the exactvalueof the intesral.
18.2.2 The Romberg Infegrotion Algorirhm Notice that the coefficientsin eachof the extrapolation equations[Eqs.(18.5),(18.6),and (18.7)l add up to l. Thus. thev reoresentweichtinefactorsthat. as accuracvincreases.
430
INTEGRATION OF FUNCTIONS NUMERICAL placerelatively greaterweight on the superiorintegralestimate.Theseformulationscanbe expressedin a generalfbrm that is well suitedfor computerimplementation: I..
-
4r tlj*r.r 1-11.r<,1 4i-l - I
where 17-.1.1-1 and I1.r r : the more and less accurateintegrals,respectively,andllt = the improved integral.The index ft signifiesthe level of the integration,whereft : I cotrespondsto the original trapezoidalrule estimates,k:2 conespondsto the O(ft4)e$imates,ft : 3 to the O (ho).and so forth. The indexj is usedto distinguishbetweenthemore U + l) and the less (l) accurateestimates.For example,for ft : 2 and j : 1, Eq.(18.8) becomes
tr.t:!4
J
which is equivalentto Eq. ( I 8.-5). The generalform representedby Eq. ( 18.8)is attributedto Romberg,andits systematic applicationto evaluateintegralsis known as Rombergintegratiut. Figure18.1isa graphical depiction of the sequenceof integral estimatesgeneratedusing this approach. rule Each matrix correspondsto a singleiteration.The first column containsthetrapezoidal ($ep application evaluationsthat are designated1,.r, where : I is for a single-segment "r sizeisb-a),j:2isforatwo-segmentapplication[stepsizeis(b-d)12],j=3isfor a four-segmentapplication[stepsize is (b - a) l4], andso forth. The othercolumnsofthe better matrix are generatedby systematicallyapplying Eq. (18.8) to obtain successively estimatesof the integral. For example, the first iteration (Fig. 18.1a) involves computing the one-andtwos e g m e n t t r a p e z o i d a l r u l e e s t i m a t e1sa( 1n1d 1 21 ) . E q u a t i o n ( 1 8 . 8 ) i s t h e n u s e d t o c o m p u t e the elementIt.z: 1.361461 , which hasan errorof O(h41.
FIGURE I8.I G ' o p n i c o d e p i q f i e no f t l . es e q , e ' c e o f i n e g r o Je c i m o l e 5g e n e ' o l e dr s i n gR o r n [ 6 1i9n i e g r o t i o n (o) Firstiterotion.ib) Second iferotion.(c)Third iterotion.
o(h2)
o(h4)
o@8)
o(h6)
t'r ?:163333-1s67467 0j72800
1.367467---------?
F 1.640533
,r, l:13?333-1623467 0.172800 1.068800
1.367467 1.623467--..*
," 1:333333---163s467
1.640533----* 1.640533
'
1.640533
I 8.2 ROMBERG INTEGRATION
th
8.8) t:
:orstiore i.8)
m;a )h. lle ep br le er
)re
Now, we must check to determinewhetherthis result is adequatefbr our needs.As in othe-rapproxirnate rnethclds in thi.sbook,a lernrinrrtion. e15lpppin!.criterionis requiredto assessthe accuracyof the results.One methodthat can be employedfor the presentpurposesis
1,,,r:lu;;-lxroo"/.
(18.e)
wheret,, : an estinrate of the percentrelativeerror.Thus,as was donepreviouslyin other iterativeprocesses. we conrparethe new estimatewith a previousvalue.For Eq. (18.9),the previousvalue is the most accllrateestimatefiom the previouslevel of integration(i.e., the fr - I level of integrationwith j :2). When the changebetweenthe old auclnew values as represented by r:,,is below a prespecified error criterion0,. the computationis terminated.For Fig. 18.1a,this evaluationindicatesthe following percentchangeover the colrrseof the first iteration;
lp.,':l
I r.36i461 - | .068800| lx1001 :21.8q t..167-167 I I
The objcctof the seconditeration(Fig. 18.10)is to obtairrthe O(h6l estirnate-Ir..r. To do this, a four-segmenttrapezoidalrule estimate,1:.r : 1.4848,is determined.Then it is combinedwith 11I usingEq. (18.8)to generateI:t : |.623461. The resultis, in turn, cornbinedwith /1,1to yielcl1r : : 1.640-533. Equationi18.9) can be appliedto determine that this result representsa changeof 1.}c/r.when comparedwith the previousresult 12.2. The third iteration(Fig. l8.lc) continuesthe processin tlresamefashion.In this case, an eight-segment trapezoidalestimateis addedto the first column.and then Eq. (18.8)is appliedto computesuccessively more accurateintegralsalong the lower diagonal.Afier only three iterations,becausewe are evaluatinga filih-order polynomial,the resr-rlt ( l t t : 1 . 6 4 0 - 5 3 i3s)e x a c t . Romber-e integlationis more efficientthan the trapezoidalrule and Sinrpson'srules. For example,fbr determination of the integralas shownin Fig. 18.l, Sirnpson'sl/3 rulc' would requireabouta 2lS-segment applicationin doubleprccisionto yield an estimateof the integralto sevensignificantdi-sits:1.640533.ln contrast,Rombergintegrationprodr-rces the sameresult baseclon cornbiningone-,two-, fbur-. and eight-segnrent trapezoidal rules-that is, with only l5 functionevaluations! Figure18.2presents an M-file lbr Rombergintegration. By usingloops.this algorithm implementsthe methodin an eficient manner.Note that the function usesanotherfunction (recallFig. 17.10).Here is trrep to implementthe compositetl'apezoidal rule evalutrtions a MAILAB sessior.r showinghow it canbe usedto determinetheintegralof thepolynomial fiom Examole18.l: >> >>
f =(i lx) 0.2+25*x rombcrg(f,0,0.8)
1.5405
2 0i]*t'-+b75*x
432
INTEGRATION NUMERICAL OF FUNCTIONS
function le, ea, i terl'romberg ( func,a,b,es,maxjr,varargin) quadrature ? romberg: Romberg integration q = romberg(func,a,b,es,maxit.,pI,p2,...) % : Romhero inteoration. % % input: ? func - name of function Co be integrated % a, b = int.egration lirnits (default = 0.000001?) % es = desired relative error (default. = 30) ? maxit - maximum allowable it.erations pl,p2,... = additional parameters used by func % % output: q-integralestimate % (%) Z ea = approximate relative error = number of iterations ? iter if narqin. l,errot ('dL leasu : inpuc argumenrs requireo'),end rt nargin<4 isempty (es ) , es-0 .000001; end i I nargin-5tis-mpfy(maxlr. ), maxit=50;end r
-
1.
= trap(func,a,b,n,varargin{: I(1,1) }) ; = 0; iter while iter<maxit. = iter+l; iter n = 2^iter; I (iter+1, 1) - trap(func,a,b,n,varargin{ : } ) ; for k = 2:iter+l j = 2+iter-k; I (j , k ) = (4 ^ (k - 1 ) * I (j + 1 , k - 1 ) - r (j , k - 1 ) | / ( 4 ^ (k - 1 ) - 1 ) ; end -T(2,iter) ea = abs( (I(1,iter+l) ) /I(1,iter+l) ) "100; if ea<-es, break; end end q = I(1,iter+l);
FIGUREI8.2 MJileto implement Romberg integrotion.
I8.3
GAUSSQUADRATURE In Chap. 17, we employedthe Newton-Cotesequations.A characteristicof theseformulas (with the exceptionof the specialcaseof unequallyspaceddata)was that the integralestimate was basedon evenly spacedfunction values.Consequently,the location of thebase points usedin theseequationswas predeterminedor fixed. For example,as depictedin Fig. 18.3a,the trapezoidalrule is basedon takingthearea underthe straightline connectingthe function valuesat the endsofthe integrationinterval. The formula that is usedto computethis areais = (h -
+ f (u)
I8.3 GAUSS AUADRATURE
433
"f("r)
(b)
F I G U R EI 8 . 3 of thelropezoidol ruleos fheoreounderthestroight {o)Grophicoldepiction lineloininofixed end points{b)An improved integrol eslimofe obtoinedby tokingtheoreounderthestroight line possingthroughtwo intermediote points,By positioning thesepointswisely,thepositive ond negcliveerrorsore bellerbolonced,ond on improved integrol estimote resulls
whereaandb:thelimitsofintegrationandb-a:thewidthoftheintegrationinterval. Becausethe trapezoidalrule must pass through the end points, there are casessuch as Fig. 18.3awhere the fbrmula resultsin a large error. Now, supposethat the constraintof tixed basepointswas removedand we were free to evaluatethe areaunder a straightline joining any two points on the curve. By positioning thesepointswisely, we could definea straightline that woLlldbllancc thc positil'c tnd ncgativeerrors.Hence,asin Fig. 18.3b,we would arriveat an improvedestimateof the integral. Gttussquadratureis the name for a classof techniquesto implementsuch a strategy. The particular Gauss quadratureformulas describedin this section are called GaussLegendreformulas.Before describingthe approach,we will show how numericalintegration formulas suchas the trapezoidalrule can be derivedusingthe methodof undetermined coefficients.This methodwill then be employedto developthe Gauss-Legendre formulas.
18.3.1 Method of UndeterminedCoefficients In Chap. 17,we derivedthe trapezoidalrule by integratinga linearinterpolatingpolynomial and by geometricalreasoning.The methodof undetermined
434
N U M E R I C AI L N T E G R A T I OONF F U N C T I O N S
FIGURE I8.4 Twointegrols thctshould beevoluoted exocfly by l h a ' r 6 p " r o , 6 o lr u l ^ .{ o )o (b)c stroight line
Lonsio"rro'ro
To illustratethe approach, Eq. (18.10)is expressed as I = c g J @ )+ c t f ( b )
(r8.il)
where the c's : constants.Now realizethat the trapezoidalrule shouldyield exactresults when the function being integratedis a constantor a straightline. Two simple equations that representthesecasesare J : I and ) : -r (Fig. 18.4).Thus, the following equalities shouldhold:
:l
{i;l-t;
r(h-u\ 12 I
ldx
J -(b-u)12
and b-a
2
o, *,,+ : I',0,,_,",'j,,
I8.3 GAUSS QUADRATURE
435
or, evaluatingthe integra.ls, co*ct:b-a and -cu
b-a
b-o
2
2
n
Theseare two equationswith two unklrownsthat can be solvedfor b-a L() -
Ll
-
^
z
which, when substituted back into Eq. (18.11),gives . I:
b-a^ - J(It+ zz
b-u^. ^ J(bl
which is equivalentto the trapezoidalrule.
| 8.3.2 Derivqtion of the Two-PointGouss-legendreFormulo Just as was the casefor the previousderivationof the trapezoidalrule, the object of Gauss quadratureis to detenninethe coefficientsof an equationof the lbrm I = cof 6il + clf {.rt)
rrg.rz)
wherethe c's : the unknowncoefficients.However,in contrastto the trapezoidalrule that usedfixed end points a and b, the function arguments16 and r1 &rerot fixed at the end points, but are unknowns (Fig. 18.5).Thus, we now have a total of four unknowns that must be evaluated,andconsequently, we requirefour conditionsto determinethem exactly. Just as for the trapezoidalrule, we cirn obtain two of theseconditionsby assuming that Eq. ( I 8. 12)fits the integralof a constantiurda linearfunctionexactly.Then,to anive at the other two conditions,we merely extendthis reasoningby assumingthat it also fits the i n t e g r aol f a p a r l b o l i c( ) : x 2 ) a n da c u b i c( , y: x r ) l - u n c t i o nt s. yd o i n gt h i s .w e d e t e r m i n e
F I G U R EI 8 . 5 Grophicoldepiction of theunknown voriobles.rs ond r1 for integrotion by Goussquodrolure
436
N U M E R I C AI L N T E G R A T I OONF F U N C T I O N S all four unknownsand in the bargainderivea lineartwo-pointintegrationformulathatis exactfor cubics.The fbur equationsto be solvedare fl
crtltt:
I
(r8.13)
ldr:2
J -l
r i r . \ r r- f ( t . r t :
fl | Jl
(18.14)
.t d,r :0
cut; t.1*i :
rt 2 / 1 2 r l r : =3 lt
c , r x +i l c l x .f :
/
(r8.15)
pl
(r8.r6)
.rr,/x:o
Jt
E q u a t i o n s( 1 8 . 1 3 )t h r o u g h( 1 8 . 1 6 )c a n b e s o l v e ds i n r u l t a n e o u sfloy r t h e l b u r u n k n o w n s . F i r s t . s o l v e E q . ( 1 8 . 1 a ) f oarncdr s u b s t i t u t e t h e r e s u l t i n t o E q . ( 1 8 . 1 6 ) , w h i c h c a n be solvedfor
EX
-.r2) _- - 2^ |
Sincex6 and 11 cannotbe equal,this rneansthat.re - -.r1. Substitutingthis resultinto E q . ( 1 8 . 1 4 )y i e l d sc s : c r . C o n s e q u e n tfl iyo m E q . ( 1 8 . 1 3 )i t t b l l o w st h a t ca :61
:
I
Substitutingtheseresultsinto Eq. (18.15)gives , f r r:
I
--
:
-0.5173503...
/a
VJ
.r':]:0.5773503... V3 fbrmulais Therefbre, thetrvo-pointGauss-Legendre
t:r(i)*/f+) " \,./3/ '\'/3/
(r8.r7)
at result that the simple addition of the functionvalues Thus, we arrive at the_interesting .\ : - 1/ J3 andl/V3 yields an integrirlestimatethat is third-orderaccurate. Noticethattheintegrationlirnitsin Eqs.( 18.13)through( I8.I6) arefrom - I to l. This was done to simplify the mathematjcsand to make the fbmrulation as generalaspossible. A simple changeof variable can be used to translateother limits of integrationintothis form. This is accomplishedby assumingthat a new variablex,7is relatedto the original variable-r in a linear fashion.as in -r:al
(18.r8)
+azxd
to-r,/: If tlre lower limit,.l : r.r,corresponds E q . ( 1 8 . 1 8r)o y i e l d
-1, thesevaluescan be substituted into
a : at + a2el) Similarly, the upperlimit, r : b, correspondsto x4 : 1, to give
(18.19)
437
QUADRATURE 18.3GAUSS E q u a t i o n(s1 8 . 1 9 )a n d( I 8 . 2 0 tc a nb e s o l v e ds i m u l t a n e o u sfloyr ot :
blcr 2
antl
a::
b-a
(18.21)
2
into Eq. (18.18)to yield which can be substituted (b + rr) 'l (b - a)r,1 2
rl8.llr
This equationcan be differentiatedto give b-u dx - _ dr.l 2'
(r 8 . 2 3 )
in theeqtration fbr,r andd,r, respectively, Equarigns(18.22)and ( 18.23)can be substituted without integration interval the transform to be integrated.Thesesubstitutionseff'ectively is how this done in illustrates example changingthe value of the integral.The fbllowing practice.
E X A M PIL8E. 3 Two-PointGouss-LegendreFormulo ProblemStotement. Use Eq. (18.17)to evaluatethe integralof ,f(r) : 0.2 * 25r 200.rr+ 675-rr 900"ra+ 400.15 betweenthe limits r : 0 to 0.8. The exactvalueof the integralis 1.640533. Solution. Before integratingthe function, we must perform a changeof variableso that rz : 0 andb : 0.8 into Eqs.(18.22) the limits arefiom - I to +1. To do this,we substitute a n d( 1 8 . 2 3t)o y i e l d x : 0.4* 0.4x7
and
dx :0.4dxd
into the originalequationto yield Both of thesecan be substituted r 0.8
|I Jt) :
n2 " " - 200:12 + 6j5x3 - 900x4+ 400x5)dx ' " - +' -25r
I
- 200(0.4 10.4r7)3 + 675(0.4 * 0.4x.1;2 [0.2+ 25(0.4* 0.4x,1)
J_l
- 900(0.4* 0.4-t.1)a + 400(0.4+ 0.4xi5l0.4dra Therefore,the right-hand side is in the form that is suitablefor evaluationusing Gauss quailrature.The transtormedfunction can be evaluatedatxT : - | I \4 as 0.5167ttl and at iu:1/J1 asl.305S3T.Therefore,theintegralaccordingtoEq.(18.17)is0.516741* a percentrelativeeror of -11.17c.This resultis 1.305837: 1.822578,which r-epresents applicationof the trapezoidalrule or a single a in to four-segment magnitude comparable Tlris latterresultis to be expectedbecause 113 3/8 rules. and ol'Simpson's application accurate. However, becauseof the clever choice of rules are also third-order Sirnpson's quadrafure on the basis of only two function this accuracy poinfs, attains Gauss base evaluarions.
436
N U M E R I C AI L N T E G R A T I OONF F U N C T I O N S
all fbur unknownsand in the bargainderive a linear two-point integrationlbrmulathatis exactfor cubics.The fbur equationsto be solvedare
c a* c 1 :
l',rrtx:2 pt
(ir-ruT(l,rl :
| JI
.rJx:0
pt
. i , t , j' , 1 r i
:
)
/ *l,1t:= -l J-, rl
c r r x i*]c 1 x f :
|
(18.16)
,f ax:O
J "-1
for the fourunEquations(18.13) through (18.16) can be solved sir.nultaneously knownsF . i r s t ,s o l v eE q . ( 1 8 . 1 4 )f o r c r a n ds u b s t i t u tteh er e s u l ti n t oE q . ( 1 8 . 1 6 )w, h i c h c a n be solvedfor ^- to --
EXA
'-1l
Since xe and x1 cannot be equal, this r.neansthat.re - --.r1. Substitutingthis resultinto from Eq. ( | 8.l3) it follows that Eq. ( 18.14)yieldsc1;: cr . Consequently c r t: c t - l Substitutingtheseresultsinto Eq. ( 18. t 5) gives . \ ' r:) xt:
I - 0..5773503... ^. : v')
I /::0.-5773503... v --)
Therefore,the two-point Gauss-Legendre formula is /.t\
t : f l " \ . ' / 3^/l + f l
r
l\
(18.17)
-l
\'/3/
result that the simple addition of the functionvalues at Thus, we affive at the_interesting r : - | l./3 and I /V3 yields an integralestimatethat is third-orderaccurate. Notice that the integrationlimits in Eqs.( 18.I 3) through( 18.16) arefrom - 1 to 1.This was done to simplify the mathematicsand to make the formulation as generalaspossible. A simple changeof variablecan be usedto translateother limits of integrationintothis form. Tlris is accomplished by assumingthat a new variable.r7 is relatedto theoriginal variablex in a lineartashion.as ir.r " x: a t I a z x a If the lower limit, x : d, corresponds to xt: E q . 11 8 . 1 8t)o y i e l d
(18.18) -7, thesevaluescan be substituted into (18.19)
0:a1*a:(-l) Similarly,the upperlimit, .t : b, corresponds to x,7:
1. to give
437
QUADRATURE 18.3GAUSS for Equations(18.19)and (18.20)can be solvedsimultaneously at :
b'ta -;:_
b-a (tZ: -;
and
(i8.21)
into Eq. ( I 8.I 8) to yield which can be substituted (b*a)l(b-tt)x,r .t
(18.22)
----
:
2
This equationcan be diffbrentiatedto give dx :
b-n d.t,t 2'
(18.2.1)
in theequation for-r-anddx, respectively, Equations\18.22)and (18.23)canbe sLrbstituted to be integrated.Thesesubstitutionseffectively transformthe integrationinterval without changingthe value of the integral.The following exampleillustrateshow this is done in practice.
E X A M P1L8E. 3 Two-PointGouss-LegendreFormulo ProblemSlotement. Use Eq. (18.17)to evaluatethe integralof .f (.x) :0.2 * 25.r - 200x2* 67513 900.14+ 400x5 betweenthe limits x : 0 to 0.8. The exactvalueof the integralis 1.640533. Solution. Before integratingthe function, we must perform a changeof variableso that thelimitsaretiom-lto+l.Todothis,wesubstitutea:0andb:0.8intoEqs.(18.22) and ( I 8.231to yield r : 0.4 * 0.4,r,i
and
th :O.4elxd
Both of thesecan be substitutedinto the otiginal equationto yield n 0.u
,I Jo
rc.2 * 400x5)r/r ' " ' - '*- ' '25.r- 20012+ 6i5r3 - 900,11 nl
:
I
-lLv'!'-"\
n.2+25rc.4i_0.4x,;- 200(0.4t0.4x,)2+675Q.4*0.4-r.7)l
J _l
- 900(0.4* 0.1x,)a + 400(0.4+ 0.4.r41510.4dx,1 Therefore,the right-hand side is in the form that is suitablefor evaluationusing Gauss quaclrature. The transformedfunction can be evaluatedat x.1: - | I J1 as 0.5I 674 I and at l c c o r d i n gt o E q . ( 1 8 . 1 7 .i )s 0 . 5 1 6 7 4 1 + , { , / : I l " E a s 1 . 3 0 5 8 3 7T. h e r e f b r et,h e i n t e g r a a a percentrelativeerrorof -lI.l7o. This resultis 1.305837: 1.822518,which represents compa'ablein magnitudeto a four-segmentapplicationof the trapezoidalrule or a single applicationof Simpson'sl/3 and 3/8 rules.This latterresultis to be expectedbecause Simpson'srules are also third-orderaccurate.However, becauseof the clever choice of base points, Gauss quadratureattains this accuracyon the basis of only two function evaluations.
438
NUMERICAL INTEGRATION OF FUNCTIONS
TABTE| 8.t
Weightingfoctorsond functionorguments usedin Gouss-Legendre formulos. Weighting Foclors
Points I
co:2
2
co:
I l
Function Argumenls ; 1 6: 0 . 0
-
-ro:
=
-tl :
I lJ3
r11: -nl3/5 ;rt : 0.0
c6:{lB-"tr6ltso c j: ( 1 8 + ' t r d t s a
x o: * , / 5 2 5+ 7 0 \ / 3 0 1 3 5 r r : - v 5 2 5- 7 O \ / 3 0 1 3 5 x2: y'\15 -7sJfr135
X2:
/JO
c.r:{tB-,trotza q1:1322- t3.na)PO] c 1: 1 3 2 2 + ) 3 y r y 1p)A j c2: 128/225 q : 1 3 2 2 + ) 3 ! r y op A O c a: 1 3 2 2 - l 3 ! r y d / 9 O A rx:0171324492379174 t ' t : O 3 6 0 7 6 15 7 3 0 4 8 1 3 9 c z: 0 . 4 6 7 9 1 3 9 3 4 5 7 2 6 9 1 c z: 0 . 4 6 7 9 31 9 3 4 5 7 2 6I9 c t : 0 . 3 6 4 7 6 1 5 2 3 0 4I83 l r - :: 0 1 7 1 3 2 4 4 9 2 3 7 9 1 7 A
1t2){{) 7t1)G)
l/VJ
co : 5/9 (t : B/9 cz -* 5/Q
c::lu+V.,Ul
Truncolion Error
=./to'({)
\/ J/)
'. : Jili +7aJi6t35 -
ro:-t/245+14\/7Al2l ,y : -li4514Jfi121 .r,:: 0.0
= 1t8){{)
-
/{to)16,
,.:n/1[i-14ffip1 u: '/ltii1tJfrlzt
ro : -a 9324695142031 52 1{tzr16, -0.66 ,rr : I 249386466265 , r z: - 0 2 3 8 6 1 98l 6 0 8 3I 9 Z x::O2386l9l86083l97 -ra= 0 661209386466265 xs:0.9324695142A3152
I 8.3.3 Higher-PointFormulos Beyond the two-point formula describedin the previoussection,higher-pointversions can be developedin the generalform I = c s f ( x 0 )+ c l l ( x r ) + . . . + cu-rf(xn-t) (18.24) wherer?: the numberof points.Valuesfor c's andx's for up to and includingthe six-point fonnula are summarizedin Table 18.I .
E X A M P LI 8E. 4 T h r e e - P o i nGt o u s s - L e g e n d rFeo r m u l o Problem Siotement. Use the three-pointformula from Table 18.1to estimatetheintegral for the samefunctionas in Example18.3. Solution. According to Table 18.1, the three-pointformula is 1 : 0.5555556f(-0.114s967) + 0.88888891'(0) + 0.555s556 f (0.7'74s961) which is equalto / : 0.28t3013 + 0.8132444+ 0.4859876: 1.640533 which is exact.
439
I 8.4 ADAPTIVE QUADRATURE
BecauseGaussquadraturerequirestunctionevaluationsat nonuniformlyspacedpoints within the integrationinterval,it is not appropriatefor caseswherethe functionis unknown. problemsthatdealwith tabulated data.However.wlrere Thus,it is not suitedforengineering This is particularlytrue the function is known, its efficiency can be a decidedadvantage. performed. when numerousintegralevaluationsmust be
I8.4
ADAPTIVEQUADRATURE Althoughthe compositeSimpson'sI /3 rule can certainlybe usedto estimatethe integral thatit usesequallyspacedpoints.This constraint of givenfunctions,it hasthedisadvantage doesnot take into accountthat somefunctionshaveregionsof relativelyabruptchanges where more refined spacingmight be required.Hence,tcl achievea desiredaccuracy,the fine spacingmust be appliedeverywhereeven though it is only neededfor the regionsof sharpchange.Adaptive quadraturemethodsremedythis situationby automaticallyadjusting the stepsize so that small stepsare takenin regionsof sharpvariationsand largersteps are taken where the function changesgradually. the corrrposite Simpson'sl/3 rule to are basedon applyin-u Most of thesetechniques subintervalsin a mannersimilal to how the trapezoidalrule was usedin Rombergintegration. That is, the 1/3 rule is appliedat two levelsof reflnementand the differencebetween thesetwo levelsis usedto estimatethe truncationerror.If the truncationerror is acceptable, no fi,rrtherrefinementis required and the integral estimatefbr the subintervalis deemed acceptable.If the error estimateis too large,the stepsizeis r-efinedand the processrepeated levels. until the enor falls to acceptable MMLAB includes two built-in functions to implerrrentadaptivequadrature:cluad how they can be applied. and quad1.The following sectiondescribes 18.4.1
MATTAB Functions: quad ond quadl
MATLAB has two functions.botlr basedon algorithrnsdevelopedby Canderand Gautschi (2000), for irnplementingadaptivequadrature: . .
quad. This function usesadaptiveSimpsonquadrature.lt may be more efficient fbr low accuraciesor nonsmoothfunctions. quadl. This function useswhat is calledLobotto quadrature.It may be more efficient for high accuraciesand smoothfunctions.
The following function syntax for the quacl function is the same for the qr-Ladl function: g
= qu.ad(fun,
a,
b,
taf ,
t.race,
pl,
p2,
.)
where fun is the functionto be integrated.a and b: the integrationbounds,rol : the desiredabsoluteerror tolerance(default: l0 6). rra.-e is a variablethat when set to a nonzerovalue causesadditionalcomputationaldetail to be displayed,and pt, p2, are parametersthat yor-rwant to passto f rrn.It shouldbe noted that array operators. * . . / and .^ shouldbe usedin the definitionof fun. ln addition.passemptymatricesfbr rol or tt ace to use the def-aultvalues.
440
NUMERICAL INTEGRATION OF FUNCTIONS
,l8.5 AdoptiveQuodroiure EXAMPLE ProblemStotement. Usequadto integrate thefollowingfunction: (, - q)2+ 0.01
-s
(x - r)2 + 0.04
belweenthelimitsr:0tol.NotethatforQ:0.3,r:0.9,ands:6,thisisthebuiltin humps function that MATLAB usesto demonstratesome of its numericalcapabilities. The humps t-unctionexhibits both flat and steepregions over a relatively short.r range. Hence,it is uselulfor demonstrating and testin-s routineslike quad and quadl. Notethat the t.r,-rmps functioncan be integratedanalyticallybetweenthe given limits to yield anexact integralof 29.85832539549861. Solution. First, let's evaluatethe integralin the simplestway possible,usingthebuilt-in version of humps along with the defaulttolerance: >> format lonEt > > q u a d ( @ h u m p s ,0 , 1 )
29.85832612842164 Thirs. the .solution i.scorrect to .seven-si-clnificantdigits. Next, we can sojve the sarneproblem, but using a looser tolerance and passingq, r, and .r as parameters. First, we can develop an M-file for the function: functi-on y Y - 1./((x
= m"yhumps(x,q,r,s) cl).^Z + O.01) + l./
((x
r).^2+0.04)
-
si
Then. we can integrate it with an error tolerance of l0-a as in , ,1, le-4, > > q u a d ( @ m y h u m p sO
11,0.3, 0.9, 6)
29.85812133214492 Notice that because we used a larger tolerance, the result is now only accurate to five significant digits. However, although it would not be apparent liorn a single application, fewer function evaluations were made and. hence, the computation executes f'aster.
Bockground. Becauseit resultsin efficient energytransmission,the currentin anAC circuit is often in the form of a sine wave: i : ipeatsin(arr) where i : the current (A : c/s), lo.ur: the peak current (A), ar : the angularfrequency (radians/s)and / : time (s). The angularfrequencyis relatedto the period r(s) by a =zrlT.
I8.5 CASESTUDY
441
continued The power generatedis relatedto the magnitude of the current. Integration can be used to determine the averagecurrent over one cycle: , :
I fr. i.",r. lpearsln(tr;/)dt - -!:= (- cos(2n) + cos(Q)):0 T Ju
Despite the fact that the averageis zero, such a cuffent is capable of generating power. Therefore, an alternative to the averagecuffent must be derived. To do this, electrical engineersand scientistsdetermine the root mean squarecurrent i*, (A), which is calculatedas
il"ssin?(att)dt
ipea.k
/;
(r8.2s)
Thus, as the nameimplies, the rms current is the squareroot of the mean of the squared current' Because1/ \/2 : 0.70707, l'.., is equal to about j}vo of the peak current for our assumed sinusoidal wave form. This quantity has meaning becauseit is directly related to the averagepower absorbed by an element in an AC circuit. To understandthis, recall thatJoule's lai statesthat the instantaneouspower absorbed by a circuit element is equal to the product of the voltage acrossit and the current throush it: P:iV
(18.26)
whereP: the power (W: J/s),and y: voltage(V: J/C).For a resistor,Ohm,slqw states that the voltage is directly proportional to the currenr: V:iR
(18.27)
whereR : the resistance(Q : V/A: P:izR
J . s/Cz).SubstitutingEq (18.21)into (18.26)gives (18.28)
The averagepowercanbe determined by integrating Eq. (lg.2g) overa period with the result: r
p E -_ ;. 2 mcr\
Thus, the AC circuit generatesthe equivalent power as a DC circuit with a consrant current of 1*r. Now, although the simple sinusoid is widely employed, it is by no means the only waveform that is used.For some of theseforms, such as triangular or squzuewaves, the l*" can be evaluated analytically with closed-form integration. However, some wavefbrms must be analyzedwith numerical integration methods. In this case study, we will calculate the root-mean-squarecurrent of a non-sinusoidal wave form. We will use both the Newton-Cotes formulas from Chap. 17 as well as the -__--^^^L^^ )^^^;l^^A
i-
t}|ic
nhanfar
442
N U M E R I C AI L N T E G R A T I OONF F U N C T I O N S
continued Theintesralthatmustbe evaluated is
Solution.
,k,:
Io''
dr eoe-tsin2nt)2
(18.29)
For comparative purposes,the exact value of this integral to fifteen signilicant digitsis 15.41260804810169. Integral estimatesfor various applications of the trapezoidal rule and Simpson's1/3 rule are listed in Table 18.2.Notice that Simpson'srule is more accuratethan the trapezoidal rule. The value for the integralto sevensignificantdigits is obtainedusing a 128-segment trapezoidalrule or a 32-segmentSimpson'srule. The M-file we developed in Fig. 18.2 can be used to evaluate the integralwith Rombergintegration: >> format longi (10*exp(-t) >> i2-@(t)
.*sin(2*pi*t)
| .^2;
15 .4r'2608A4288977 1 . 4 8 0 0 5 8 7 8 7 3 2 6 94 6 e - 0 0 8 5 Thus, with the default stopping criterion of es : 1 x 10-6. we obtain a result that is correct to over nine significant figures in five iterations. We can obtain an even better resultif we impose a rnore stringent stopping criterion:
r5.4L260804810169 0 i ra-
-
l
Gaussquadraturecan also be used to make the sameestimate.First, a changein variable is performedby applying Eqs. (18.22)and (18.23)to yield 1lI t:-*-t.r 4 4"
dt:-dt,t " 4
Theserelationshipscan be substitutedinto Eq. (18.29)to yield t,1",:
lt -) zs+ozs',)sinzt(0.25 + 0.25q)l'0.25 dt / [l0e-,o -
J-t
(18.30)
I8.5 CASESTUDY iir' rh-T:'-
"
continued
TABTE18.2
Voluesfor the iniegrolcolculoted usingNewton-Cotes formulos.
Iechnique
Segmenls
I 2 4 B )b 32 o4 I?B
Tropezoidol rule
S i m p s o n1s/ 3 r u l e
TABTE18.3 Poinfs
2 3 ,4
5 6
2 4 B t6 32
Inlegrol
e,l"/ol
0.0 r00 0000 1s.163266493 ) 6178 15.401429095 0.A725 l 5 4 11 9 5 8 3 6 0 4 . 2 2x 1 O - 3 15.412568151 259x1O-a 15412605565 I6lxlO-s 15.412607893 lOl x 10-6 1 54 1 2 6 0 8 0 3 8 6 . 2 8x l O - b 20217688657 t 5 .480816629 r5415468il5 I5 412771415 l5 412618037
3\ 1763 4.4426 0.0r86 I 06x 1O-3 6 48 x l0-5
Results of usingvorious-point Goussquodroture formulosto opproximote the integrol. Estimote
t L9978243 15.65755A2 r5 4058023 I 5. 4 1 2 6 3 9 1 154126109
e, l"/"1 22.1 1.59 4 . 4 2x l 0 - 2 2 . 0 1x l 0 - a L B 2x l 0 - 5
Forthe two-point Gauss-Legendre formula, this function is evaluatedat t,t : -1 I J3 and, 1/V3. with the resultsbeing 7.684096and 4.313728,respectively.These values can be substitutedinto Eq. (18.17)to yield an integralestimateof 11.99182,which representsan error of €r : 22.1c/o. The three-pointformula is (Table 18.1) 1:0.5555556(1.237449)+0.8888889(15.16321)+0.5555556(2.684915):15.65755 which has €t: l.6Vo.The resultsof using the higher-pointforrnulasare summarizedin Table 18.3. Finally, the integral can be evaluatedwith the built-in MATLAB function quad and quadL: >> irms2-quad(i2, 0, . 5) i. rms:2 ,-, \ t r , . 4 I 2 , b 0 84C9 13 4 5 0 9
444
N U M E R I C AI N L T E G R A T I OONF F U N C T I O N S
18.4 Tt
continued >> irirs2=quad1 (i:,
erf(c
0, . 5)
irms2 = 1 5 . . 11 : 6 0 8 0 4 8 C 9 r 957 Both these results are very accurate, with quadt being a little better. We can now compute the i,n by merely taking the square root of the integral. For ex" arnple, using the result computed with quadl, we get >> i rrrs=sqrt- (irrns2 )
Use the formula error for termine
r8.5 Th followir 1;:
irms = 3.92588945948554 This result could then be employed to guide other aspects of the design and operationof the circuit such as power dissipation computations. As we did for the simple sinusoid in Eq. (18.25), an interesting calculation involves comparing this result with the peak current. Recognizing that this is an optimization prob lem, we can readily employ the fminbnd tunction to determine this value. Becausewe are looking for a maximum, we evaluate the negative of the function:
where ; of the n (a) Rom two-poir quad fu Iti.6 Th
I*^
ForT = 0 . 2 2 4 B 1 9 4 0 3 1,9 3 2 L inax = - 7 . B B 6 B 5 3 8 l3 9 3 2 5 A
t (r) i(I)
A maximum cuffent of 7.88685 A occurs at t : 0.2249 s. Hence, for this particular wave form, the root-mean-square value is about 49.87o of the maximum.
Evalual ance ol formulr 18.7 T represe
PROBTEMS l8.l Use Rombergintegrationto evaluate f:
/
1\-
1: I {:,- -),/r r,/ Jt \
GaussquadraturetbrmLrla,and (d) MATLAB quadfunction:
| :
,"8
I
- ().0547.rr -r 0.s6-t6.rr - -1.1562.i:
,to
] 6 . 2 9 1 7 xI 2 d . r to an accuracvof s. : 0.5a/..Your resultsshouldbe presented 18.3 Evaluatethe tbllowing integralwith (a) Romberg intein the tbnnat of Fig. 18.1.Use the analyticalsolutionof the eration (tr :0.5%), (b) the two-point Gaussquadrature integralto deterrninethepercentrelativeerrorof theresultobformula. and (c) MATLAB quad and quadl functions: tainedr.vithRombergintegration.Checkthate, is lessthane.. 73 18.2 Evaluate the following integral (a) analytically, t: I xe'd.t rk\ Dnmhprcinreorarion(F" :0.5%). (c) the three_point Ji
where the pip meter. using ( two-p( quad'l 18.8 l riod oi M
PROBLEMS
445
18.4Thereis no closedfcrrnrsolution fbr the error functiol
whereM = nlass(mg), rr : the initial time (min), r: : rhe llnal tirne (rnin), Q1r) : flow rare (m3/min), and c(r.) : a rd = :f (mg/rn3). The following functional repreconcentration sySqal ,.-' ,/,r | Jn Jt sentations define the temporal variations in flow and Usethe (a) two-point and (b) rhree-pointGauss-Legendre croncentration: formulas to estimateeri( 1.5).Determinethe percentrelarive QQ):9+,icos21o.4r1 enorforeach casebasedon the true value.which canbe dec ( t ) - 5 e - ( )' 5 /* Z e t t ' 1 s r rvith MATLAB's built-in tirnction +r:f . termined 18.5Theforce on a sailboatmastcan be representedby the Determine the mass transported between 1r : 2 and following function: l: : 8 min rvith (a) Romberg integrationto a toleranceof rH 0.1Vc: and(b) the MMLAB qu,:il function. ,, n F = I too{ ----, ),, d.18.9 Evaluatethe doubleinteglal \ / t--,/ Jo where;: the elevationabovethe deck and 11: the heighr of themast.Compute F fbr the case where I1 : -10using (a)Rombergintegrationto a toleranceof e. : 0.5olc.(b) the tu,o-point Gauss-Legendreformula. and (c) rhe MNLAB q:r;dI'unction. 18,6Theroot-mean-square cunent cln tre cornputedas
'rrr: I\nu, E /
(.r2 - 3-r,2* -r,r'3)r1.rr1-r,
(a) analyticallylnd (b) usinr the MATLAB dl.t cr,.r,:cr function. Usehetp to understandhor.vto implementthe tunction. 18.10 Computework as describedin Sec. j7.9, but usethe tbllowing equationsfor F(_r)and d1.r): F(.r):1.6x-0.045.12 d(t) : -0.000-5-5rr* 0.0123;12 * 0. l3.r
Forf : l. supposethat i (r) is definedas -
l_, l,'
r\
t ( r ): l o d - /' s i n ( Z r 7J
[or0
l ( t ): 0
forTl2
The force is in new,tonsanclthe angle is in radians.perfbrm the integrationf}onr -r : 0 to 30 m. l8.ll Perfbrnrthe samecomputarion as in Sec.18.5,but fbr the curlent as specifiedby
i(i1:5r'ls'sin2zr
n,=L,'
Qft)c(t) dt
Compute the averagevoltage over / : multiple-segment Simpson's1/3 rule.
0 to 60 using the
446
N T E G R A T I OONF F U N C T I O N S N U M E R I C AI L
where center
0.02 40.0 0.05 0 . 1 0 43.0 0 . 1 5 52.0 0 . 2 0 60.0 0.25 55.0
Rupture
where using result 18.18 stretcl tor:
M o d u l u so j t o u g hn e s s
F,IQ x, fn F, l( rrfi
(al
F I G U R EP I 8 . I 6 ( o )A r o du n d eor x i o lo o d i n go n d ( b )t h er e s u l i nsgt r e s s s l r coui nr v ew, h e r e i sd l m e n s i o n l e s s i n c hi l 0 r l b / i n ' 7 o) ,n ds t r o i n s t r e si s i n k i p sp e rs q u o r e
18.1 the vr U
u U
The areaunderthe curve from zero stressout to thepoint0f It ruptureis calledthe modulusof toughnessof the material. providesa measureof the energyper unit volumerequired to lfr ol causethe materialto rupture.As such,it is representative Vttt:; I ittllt LJo the material'sability to withstand an impact load.Usenufol If C: l0 s farad,use the fbllowing currentdata to develop mericalintegrationto computethe modulusof toughness curve seenin Fig. Pl 8. I 6b. the stress-strain a plot of voltageversustime: Itt.l7 If the velocity distributionof a fluid flowing through pipe is known (Fig. Pl8.l7), the flow rate Q (thatis,the a 04 06 t'3 0 a2 volume of water passingthroughthe pipe per unit time)can 03683 038t9 0.2282 i, Io-3 A o2 be computedby Q : J u tl A, where u is the velocity,andA ol is the pipe's cross-sectionalarea.(To graspthe meaning ) 12 t,s 0B bethis relationshipphysically,recall the close connection i, Io-3 A 0 0486 0.0082 0 )441 tween summation and integration.) For a circular pipe, A: r12 anddA:2rr dr. Therefore. Itl.l5 The work doneon an object is equalto the fbrce times rl the distancemoved in the direction of the force. The velocO -- I u(2rr)dr ity of an object in the dircction of a fbrce is given by JIJ llt.l{ If a capacitorinitially holds no charge,the voltage acrossit as a function of time can be computedas
u:4t
0
u:16*(4-r)t
4
I Al
where r is in m/s. Determinethe work if a constantforce of 200 N is appliedfor all r. llt.16 A rod subjectto an axial load (Fig. Pl8.i6n) will be curvein Fi P I 8. I 60. deformed.as shownin the stress-strain
I
FIGURE PI8.I7
r
PROBLEMS
447
r is the radial distancemeasuredoutward from the where ol thepipe.If the velocilydistributionis given by center /
r,=lIl-]l \
,
/ -.. \ r , - r r l n { "-' ' - ) - s ,
\l'6
\.tno qt /
ltt,/
(in this case,3 cm), conrpute0 where r,,is the total ladir"rs the multiple-applicationtrapezoidalrule. Discussthe using results. 18,18Usingthe following data,calculatethe work done by a spling that has a springconstantof ft : 300 N/nr slretchrng to;r: 0.35m:
0 0
F ,l 0 3 .N rrf,
4 t 0 3 .N rrf,
18.20 The upward velocity of a rocket can be computedby the fclllt>win-e tomula:
0 063 020
00r 005 O C)82 025
0 028 o l0
0 046 0 15
0ll 0.30
0 13 035
18.19Evaluatethe vertical dislancetraveledby a rocket if theverticalvelocityis given by r':lll---\1
0
r ,= 1 1 0 0 - 5 r
l0 < r < 20
r'=-50I+2(t-20))
2O
whererr: upwardvelocity,rr : velocityat which fuel is expelled relativeto the rocket,rno- initial massof the rocket al time / :0, q : fuel consumption rate.andg : downward accelerationof gravity (assumedconslant: 9.8 m/sr).If I : 1800m/s. n,, : 160.000kg. and q : 2500 kg/s,determine how high the rocket will f1y in 30 s. 18.21 ]'he not'rraldistributionis defineclas . / t - rt :
t,^ --E:e-\-t2 \/ L]T
(a) Use MATLAB to integratethis function fiom.r : - I to I and frorn -2 to 2. (b) Use MATLAB to determinethe inflection points of this function. 18,22 Use Rornbergintegrationto evaluate
^)/ - €'sln-\ I
-d,
Jo l +.\to an accuracyof e, : 0.57c.Your results should be presentedin the form of Fig. I 8. L
NumericolDifferentiotion
CHAPTEROBJECTIVES The primary objectiveof this chapteris to introduceyou to numericaldiftbrentiation. Specific objectivesand topicscoveredare ' ' ' ' ' '
Understandingthe applicationof hish-accuracynumericaldifferentiation lbrmulas for equispaceddata. Knowing hclwto evaluatederivativestbr unequallyspaceddata. Understandinghow Richardsonextrapolationis appliedfbr numerical differentiation. Recognizingthe sensitivityof numericaldifferentiationto dataerror. Knowing how to evaluatederivativesin MATLAB with the dif f and qlradient functions. Knowing how to generatecol.rtourplots and vector fields with MATLAB.
YOU'VE GOT A PROBTE'I/I velocity of a free-talling bungeejurnper as a function of time canbe
R::*,*::TJ u ( r)
At the beginningof Chap. 17, we usedcalculusto integratethis equationto determine the vertical distancez the jumper has fallen after a time /.
z(t.)
(19.2)
I9.I INTRODUCTION ANDBACKGROUND
449
Now supposethat you were given the reverseproblem.That is, you were askedto determine velocity basedon the jumper's positionas a function of time. Becauseit is the inverseof intergration. differentiationcould be usedto make the determination: dz0) , at
r'(l):
119.3)
Eq. (19.2)into Eq. (19.3)anddifferentiating wouldbring us backto Eq. ( 19.1). Substituting Beyond velocity, you might also be askedto computethe jumper's acceleration.To do this, we could either take the first derivativeof velocity, or the secondderivative of displacement: du(t\ .t\i/:-----:-:
dt
d 2 z t tl
dt2
(19.4\
In eithercase.the resultwould be a(t) :S
,".ftt (
#,)
(1 9 . 5 )
Although a closed-formsolutioncan be developedfor this case,thereare other functions that may be difficult or irnpossibleto differentiateanalytically.Further,supposethat there was some way to measurethe jumper's position at various times during the fall. Thesedistancesalong with their associatedtimes could be assembledas a table of discrete values.In this situation,it would be usefulto differentiatethe discretedatato determinethe velocity and the acceleration.In both theseinstances,numericaldifTerentiationmethods are availableto obtain solutions.This chapterwill introducevou to someof thesemethods.
I9.I
INTRODUCTION AND BACKGROUND | 9. | . I Whor ls Differentiorion? Calculusis the mathematicsof change.Becauseengineersand scientistsmust continuously deal with systemsand processesthat clrange.calculusis an essentialtool of our profession. Standingat the heartof calculusis the mathematicalconceptof differentiation. According to the dictionary definition, to dffirentiate means"to mark off by differences;distinguish;. . . to perceivethe differencein or between."Mathematically,the derivettive, which servesasthe fundamentalvehicle for diff-erentiation,representsthe rate of change of a dependentvariablewith respectto an independentvariable.As depictedin Fig. 19.l, the mathematicaldefinitionof the derivativebeginswith a differenceapproximation:
f (xi + Ar) - /(.t;) Ar
A.y A.r
(1 9 . 6 )
where -y and /(:r) are alternativerepresentatives for the dependentvariable and x is the independentvariable.If A:r is allowedto approachzero,asoccursin moving from Fig. 19.1a to c, the differencebecomesa derivative: -' dr
-
lim
{r+0
.f (xi + Ar) - /("r;) A.r
(t9.7)
450
NUMERICAL DIFFERENTIATION
.l(.r-,+ Ar; f(x)
r,
.r, + ar AI
(b)
(c)
F I G U R EI 9 . I Thegrophicoldefrnition zeroin goingfrom(o)to (c),thedifference opprorimolion of o derivotive: os Ax opprooches becomes o derivotive. whereh ldx [which can erlsobe designatedoS.l/ o. .f '(.ri) ]' is the first derivativeof .r'with r e s p e c t t o - r e v a l u a t e d a t r , . A s s e e n i n t h e v i s u a l d e p i c tFi oi gn.ol 9f . l c , t h e d e r i v a t i v e i s the slopeof the tangentto the curve at,rr. The secondderivativereDresents the derivativeof the first derivative.
#:*(#)
( 19.8)
Thus, the secondderivativetells us how fast the slopeis changing.It is commonlyrefened to as the cun,ature,becausea high value for the secondderivativemeanshigh curvature. Finally.partialderivativesareusedfor lunctionsthat dependon morethanonevariable. Partialderivativescan be thoughtof as taking the derivativeof the functionat a pointwith all but one variableheld constant.For example,given a function/ that dependson both.r and y, the partial derivativeof/with respectto.{ at an arbitrarypoint (x, 1,)is detinedas
./(r * A.r,t') - ./(;r.r')
af 3;r
A,r+o
Ax
( 19.9)
Similarly, the partial derivativeof./with respectto y is definedas
a" f- l i m 3y
Ar-o
/ ( ; r , - r ' * A . r ' )- / ( x , l ' ) Ay
(19.r0)
To get an intuitive graspof partial derivatives,recognizethat a function that depends on two variablesis a surfaceratherthan a curve.Supposeyou aremountainclimbingandhave accessto a function.f that yields elevationas a function of longitude(the east-west oriented 'The fbrm dt'ldx was devisedby Leibnitz, whereasy' is attributedto Lagrange.Note that Newton usedthe so-calleddot notation: i. Todav.the dot notationis usuallv usedfbr time derivatives.
I9.I INTRODUCTION ANDBACKGROUND
451
.r axis) and latitude(the north-southoriented-r'axis).If you stopat a particularpoint (,rn,-r'e), the slopeto the eastwould be 0f(xu,y,) I 0.r,andthe slopeto the noth would be 8/(xo,y6)/3"r. 19.1.2
Differentiotion
in Engineering
ond Science
The differentiationof a function has so rnany engineeringand scientific applicationsthat you wererequiredto takedifferentialcalculusin your first yearat college.Many specificexamplesof suchapplicationscould be given in all fields of engineeringand science.Differentiationis commonplacein engineeringand sciencebecauseso much of our work involves characterizingtlrechangesof variablesin both time and space.In fact,many of the laws and other generalizationsthat figure so prominentlyin our work are basedon the predictable ways in which changemanifestsitself in the physicalworld. A prime exampleis Newton's secondlaw, which is not couchedin termsof thepositionof an objectbut ratherin its change with respectto time. Aside from suchtemporalexamples,numerouslaws involving the spatialbehaviorof variablesare expressedin terms of derivatives.Among the most colnmon of thesearc the cortstitutive/au's that define how potentialsor gradientsinfluencephysicalprocesses. For example,Fourier's law oJ heat conductionquantifiesthe observationthat heatflows from regions of high to low temperature.For the one-dimensionalcase,this can be expressed mathematically as Q:
- K, d T ,' ax
( 1 9rr. )
where q (.r) : heat flux (Wm2), k : coefficientof thermalconductivitytW(m . K)1, f : temperature(K), andx : distance(m). Thus,the derivative, or gradient,providesa measure of theintensityofthespatial temperaturechange,whichdrivesthetransferofheat(Fig. 19.2).
F I G U R EI 9 . 2 Grophlcoldepiciionof o temperoture grooient.Becouse heolmoves"downhill" fromhighlo low lemperoture, ihuflo* in {o)is fromleftiorlght However, dueto theorientotion of Corteiion theslopeis negotive coordinoies, for thiscose.Thus,o negolive grodientleodsto c positive flow.Thisis iheoriginof lhe"minus low of XeotcJnduction. Thereverse signin Fourier's coseis depiciedin {b),wherethepositrve grodlenlleodslo o negotive heotflow fromrightto left
Directionof heatflow
452
DIFFERENTIATION NUMERICAL TABTE| 9. I Low Fourier's lor,v
Theone-dimensionol formsof someconstitutive lowscommonly usedin e n g i n e e r i nogn d s c i e n c e . Equotion q : - K *,
d7'
dc
lcw Fick's
a.r ,dh
D'Arcy'sow
dT
Ohm'slow
I -
Newion's viscosily low
| -t^
flooke'sow
o :
_^
d v _ d.r
du , ax
EL-! L
PhysicolAreo !eot conduclion
Grodient -[:mperoture
Flux
Proportionolily
Heotfiux
Thermoi Conductivily
Mossdiffusion
Cor,e,rorio1
Mc.. Iu
Drru:iu]
Flowthrough porousmedicr
Heod
For'vflux
ic Hydrou Conductivi\
C u r r e n ft l u x
Elecfricol ConductivilY
flow Currenl Fluids
Velocity
Sheor Sfress
Dynomic Viscosily
Eosticily
Deformction
Siress
Young s Modulus
Sirnilar laws prcvide workablemodelsin nranyotherareasof engineeringandscience, elecincluding the modeling of lluid dynurmics,masstransf'er.chernicalreactionkinetics. isan tricity, and solid mechanics(Table19.1).The ability to accuratelyestimatederivatives importantfacet of our capabilityto work effectivelyin theseareas. Beyond direct engineeringand scientificapplications,numericaldifferentiation isalso importantin a variety of generalmathernaticalcontextsincluding otherareasof numerjcal methods.For example,recall that in Chap. 6 the secantmethod was basedon a finitedifferenceapproximationof the derivative.ln addition,probablythe mostimportant appliWehave cation of numericaldifferentiationinvolvesthe solutionof differentialequations. alreadyseenan examplein the fbrm of Euler's methodin Chap. 1. In Chap.22,wewillinvestigatehow numerical differentiationprovides the basis for solving boundary-value problemsof ordinary differentialequations. These arejust a few of the applicationsof differentiation that you might faceregularly in the pursuitof your profession.When the functionsto be analyzedaresimple,youwillnormally chooseto evaluatethem analytically.However,it is oftendifficult or impossible when the function is complicated.In addition,the underlyingfunction is often unknownanddefined only by measurement at discretepoints.For both thesecases,you rnusthavetheability next. to obtain approximatevaluesfor derivatives,usingnumericaltechniquesasdescribed
r9.2 HIGH.ACCURACY DIFFERENTIATION FORMUTAS We have already introduced the notion of numerical differentiation in Chap.4. Recalltha we employedTaylor seriesexpansionsto derivefinite-differenceapproximations of derivatives.In Chap.4. we developedforward,backward,andcentereddifferenceapproximations of tirst and higher derivatives.Remernberthat,at best.theseestimateshaderrorsthatwere O(h2)-that is, their enors were proportionalto the squareof the stepsize.Thislevelof
EXA
I 9.2 HIGH-ACCURACY DIFFERENTIATION FORMULAS
derivationof thesefbrnrulas.We wrll now illustratehow high-accuracyfinite-differenceformulascan be generatedby includingadditionaltermsfrom the Taylor seriesexpansion. For example,the forward Taylor seriesexpansioncan be written as [recall Eq. (a.13)] f"!:t) ^z *... f (ri+t) : f (xi)+ .f'(x)h *
(1e.12)
which can be solvedfor
f'(r): {!tP
-
o(h2)
+r+
(19.13)
In Chap. 4, we truncatedthis result by excludingthe second-and higher-derivativetenns and were thus left with a fbrward-differenceformula:
f'(,,): tSt#
+ o(h)
(r9.r4)
In contrastto this approach,we now retain the second-derivative term by substituting the following forward-differenceapproximationof the secondderivative[recallEq. (4.27)]:
.f,,Gi):W+o&)
(19.1-5)
i n t o E q . ( 1 9 . 1 3 )t o y i e l d J l x i + z t- 2 f ( r ; - 1 ) + . / t x ; ) . h+Otlr) ;;; zn'
( l s .1 6 )
or, by collectingterms:
.f'(.x):-ry+o(h2)
(r9.r7)
term hasimprovedthe accuracyto O\h21. Notice that incl-rsionof the second-derivative Similar irnprovedversionscan be developedfor the backward and centeredformulas as well as for the approximationsof higher-orderderivatives.The formulasare summarizedin Fig. 19.-lthroughFig. 19.5along with the lower-orderversionsfrom Chap.4. The following exampleillustratesthe utility of theseformulasfor estimatingderivatives. H i g h - A c c u r o c yD i f f e r e n t i o t i o F normulos
E X A M P TlE9 . l
Problem Stotement. Recall that in Exarnnle4.4 we estimatedthe derivativeof t.
I
f ( r ) : - 0 . 1 , r *- 0 . 1 5 x ' - 0 . 5 , r ' 0 . 2 5 x* 1 . 2 at -r : 0.5 using finite-differencesand a step sizeof h: 0.25. The resultsare summarized in the following table. Note that the errors are based on the true value of /'(0 5) : -0'9125' Centered
Forword
o(h)
o(h\
o(h)
-4 714
-a 934
Bockword
Estimote
a/a
Renearfhis comnutation-hut emolov
-t 155 a^
qa/
formulasfrom Fis. 19.3throueh
454
NUMERICAL DIFFERENTIATION
Enor
F i r s lD e r i v o l i v e f ( xi + t t f l x , l -
|
0(h\
\xil:
-f (r,*21+ 4f (xi+) - 3f (x) 2h
oftll
Second Derivotive
-f (,rr*J - 2f (x,*,) * f (.r,)
o(h)
h2
--f (xin:)+ 4.f (xi+) - 5"f (.t,*,)+ 2f (x,)
o(h2)
h2
ThirdDerivotive (ti*z) * 3/(;,*') - /(.r,) (x,)= f @i*) 3f
;1"'
0(h)
h3
-31 (1.,+1) -l 14f (x,n.1- 24f $i+2)+ iB.f(r.+r)- 5,f (r,)
o(h2)
ln'
FourthDerivoiive
f (x,*) - 4f (t,+r)* 6/('r;*r) - 4l'6i+) + f (ri) f"" (x,)=
o(h)
h1
f "" (r,) =
- 2 . f ( x , * ) + l l / ( x , + + )- 2 4 f ( . x i * ) * 2 6 7 1 . r n 1 * 1 4 . f( x , * , )+ 3 / ( . r , )
0T'?)
Ita
FIGURE I9.3 Forwordf inite-difference formulos:hryoversionsore presentedfor eoch derivotive. Thelotier version incorporclesmore termsof the icyior seriesexponsionond rs,consequenlly, moreoccurole.
Solution. The data neededfor this exampleare xi-2:0
f(ti-):1.2
xi,r :0.25 .ri : 0.5
f(xi-t): 1.1035156 J'Gi) :0.925
ri+r :0'75
, f ( x i + r ): 0 . 6 3 6 3 2 8 1
li+l
:
I
f ( . r i + ): 0 . 2
Theforwarddifference of accuracyO (h2)is computed as(Fig. 19.3) / ( 0 . 5 ):
-0.2 + 4(0.636328 r) - 3(0.9t5) - 0 . 8 5 9 3 7 5 : € r: 5 . 8 2 V t 2(0j,
Thebackward difference of accuracy O \h2I iscornputecl as(Fig.19.4) / t t . l . -:s )
l ( 0 . 9 2 5-) 4 ( 1 1. 0 3 15 5 6+) 1 . 2 : -0.878125 €,:3.llVt 2(025)
Thecentered difference of accuracy Otha;is computed as(Fig.19.5) - 0 . 2 + 8 ( 0 . 6 3 6 3 2 8- 18) ( 1 . 1 0 3 s 1 s+6 1 ) .2
19.3 RICHARDSON EXTRAPOLATION
Firsl Derlvotive +rr\ J
\
f
i'
Error frY J '4i
-
\
t,
o(h)
n
' ( - t , ) 3J(.\i)- 4f (r,-J + J(.\i 1) :
o(hr)
Lt1
Second Derivolive
.f (r) - 2f (,r, ,) * f (x,'r) f"(xi)=
o(h)
h2
.f"(r') :
2f (x) - 5/[r'-r) * 4f (x,-r)- -f (.r, ,)
o(h?)
h7
Ihird Derivoiive 3,f (xir,) + 3f (xi,.2) - l (r;-:)
o(h)
hl
f"'( r',)=
5.f(il - 18f(x,_)+ 24f (.t,_r)- 14f (.r,_)-t 3f (.r,_,)
o(.h'1)
2h3
Fourth Derivotive f lt,t - 4f rr,_,t+ 6f (r,_r)- 4.f (xi_t)* /(r:,_r) lr' 3"f(.r;)- I 4 f (x,-) + 26 f @-.) - 24f (ti ) + I I /(.{,,4) - 2f(x,-s) 1""(t) =
o(h) o(h2)
FIGUREI9.4 formulos.h.rzo versionscre r.,resenled for eoch derivotive.Thelotier Bockwordfinlte-diiference moreoccurote versionincorpoTotes more fermsof ihe Toylorserresexponsionond is, corrsequently,
As expected,the errors for the forward and backwarddifferencesare considerably more accuratethan the resultsffom Exanrple4.4. However,surprisingly,the centereddifferenceyields the exact derivative at .v : 0.5. This is becausethe tbrmula basedon the Taylor seriesis equivalentto passinga fourth-orderpolynomialthroughthe datapoints.
I9.3
RICHARDSONEXTRAPOTATION To this point, we have seenthat there arc two ways to improve derivativeestimateswhen employing llnite diff'erences:(l) decreasethe step size or (2) use a higher-orderforrnula that eniploys more poiuts.A third approach,basedon Richardsonextrapolation,usestwo derivativeestimatesto computea third, more accurate.approximation. provideda meansto obtainan Recallfrom Sec. 18.2.i that Richardsonextrapolation ( improved integralestimateby the formula [Eq. I 8.4)] I : I ( r r : )* - - - - l
-
\.ttt/tt))- -
l
l r t t t l , t- r t h l \ l
(te.r8)
wlrere/(ft1) andI(h) areintegralestimatesusingtwo stepsizes:/r1 and /22.Becauseof its conveniencewhen expressedas a computeralgorithm, this formula is usually written for the casewherehz : ht 12, as in
4
456
DIFFERENTIATION NUMERICAL
FirslDerivoiive f ( x , r , )- f
-------;-;-
Error (.xi-)
o(h'1\
ln
-f (x,*2) + B/(iri+r) - Bf (.r,-,) + l (.t,-1)
o(h')
I ltr
Second Derivotive
f t.t,+t)- 2f $,) + f \x, )
0(h')
h2
f" (x):
-f (xi*) + l6/(.r,*,) * 3O/(ri) + 16/(:r,_,)- f (x, z)
o(hn\
l2tt2
Third Derivotive
-f (r,*r) * 2.f (t+) + 2f (xi r) -./(.r,,) f "'(x,)=
oft'1)
f"'(x) =
0(hn)
2hl - . f ( r i + r ) + B / ( x i + 2 )- 1 3 1 1 " r , * , + . ) l 3 / ( r r - r ) - B . f( x i _ z ) * f ( x ; _ )
8ft3
FourthDerivotive f (r,*) - 4f {x,n) + 6f (r,) - 4f (xi_)-l .f (xi z) f"" (x) =
oftI)
h4
f"" (x,1:
-_f (t,o.,)+ 12f (x.+2i\ + 39/(.rr+r)+ 56f (x,)- 39/(r,-r) + 12.f(x_r) 1-f (xi_z)
0(ho)
6h1
TIGUREI9.5
rvvgve s onso'e oresetred fo'rnulos: Cenie'ed['^rle-d'{Ie'ence for eochderivotive. Thelofier ^..1 ia a^nao^ 'anllrr mnro versionincorporotes moretermsof theToylorseries exponsion occurote. In a similarfashion.Eq. (19.19)can be written fbr derivativesas 4l D : : D ( h : t _ : D \ h1 t )
( I9.20)
-)J
For centereddifferenceapproximutions with O\h2), the applicationof this formulawill yield a new derivativeestimateof O(h+). E X A M P L E1 9 . 2
R i c h o r d s o nE x t r o p o l o t i o n thefirstdeProblem Stqtement. Using the samefunctionas in Example19.1.estimate r i v a t i v ea t . r : 0 . - 5e m p l o y i n gs t e ps i z e so f h 1 - 0 . 5 a n dh z : 0 . 2 5 . T h e nu s eE q ,( 1 9 . 2 0 ) to computean improvedestimatewith Richardsonextrapolation.Recallthatthetruevalue
is-0.9125. can be computed with centereddifferences as estimates Solution. The first-derivative 0 . 2- 1 . 2 D(0.5): -----:- : *1.0 tt : *9.67c I
and
D(0.25):
0 . 6 3 6 3 2 8- 1r . 1 0 3 5 t 6 -0.934375 : 0.5
€r : -2.47a
I9.4 DERIVATIVES OF UNEQUALLY SPACED DATA
457
The improvedestirnate can be determinedby applyingEq. ( 19.20)to give 4I D:4(-0.934315) -
-0.9125
J(-l): w h i c h l ' o rt h ep r e s e nct a s ei s e x a c t .
The previousexampleyielded an exactresultbecausethe function being analyzedwas a fourth-orderpolynomial.The exactoutcomewas due to the fact that Richardsonextrapolation is actuallyequivalentto litting a higher-orderpolynomialthroughthe dataand then evaluatingthe derivativesby centereddivided differences.Thus, the presentcasematched polynomialprecisely.For most otherfunctions,of course, the derivativeof the foLrrth-order this would not occur,and our derivativeestirnatewould be irnprovedbut not exact.Consequently,as was the casefor the applicationof Richardsonextrapolation.the approachcan be appliediteratively using a Rombergalgorithmuntil the resultfalls below an acceptable en'or criterion.
19.4
DERIVATIVES SPACEDDATA OF UNEGIUATTY The approachesdiscussedto this point areprimarily designedto determinethe derivativeof a given function. For the finite-differenceapproximationsof Sec. 19.2,the datahad to be l 9o.f3S, tehce. d a t a a l s o h a d t o evenlyspaced.FortheRichardsonextrapolationtechnique halvedintervals.Suchcontrolofdata spacbe evenly spacedand generatedfor successively ing is usuallyavailableonly in caseswherewe canusea functionto generatea tableof values. In contrast,ernpiricallyderived information-that is. data from experimentsor fic-ld studies-are often collected at unequal intervals.Such intormation cannot be analyzed with the techniques discussed to this point. One way to l-randlenonequispaced data is to fit a Lagrangeinterpolatingpolynomial [recallEq. (15.21)Jto a setofadjacentpointsthat bracketthe locationvalueat which you want to evaluatethe derivative.Remernberthat this polynomial does not require that the points be equispaced.The polynomial can thenbe diff'erentiated analyticallyto yield a formula that canbe usedto estimatethe derivative. For example.you can fit a second-orderLagrangepolynomialto threeadjacentpoints ( - t o , . r t i )(,. r i . I r ) . a n d ( . r u , . r . ' zD)i.f f e r e n t i a t i ntgh ep o l y n o m i ayl i e l d s : 2.r-.r1 -.t2
l'(x) : l'(-re)(xo -
xr )(xe - x;)
-t .f (rr), 2 x - . r g - . r 2 ( x r- r s ) ( x 1 -
2x-rg-x1
* lGz) ( , r :- , r o ) ( - l r-
rr)
12)
(le.21)
where-r is the value at which you want to estimatethe derivative.Although this equationis certainly more complicatedthan the first-derivativeapproximationfrorn Fig. 19.3through Fig. 19.5, it has some importantadvantages. First, it can provide estimatesanywhere within the rangeprescribedby the threepoints.Second,the points thernselvesdo not have to be equally spaced.Third. the derivativeestimateis of the sameaccuracyas the centered points,Eq. (19.21)evaluatedat t : xt redifferencet&q.9.25)1. In fact, for equispaced ducesto Eo.4.25\.
458
NUMERICAL DIFFERENTIATION
EXAMPLE I9.3 DifferentiotingUnequolly Spoced Doto Problem Stotement. As in Fig. 19.6,a temperaturegraclientcanbe measureddownintothe soil. The heatflux at the soil-airintert'acecan be computedwith Fourier'slaw (Tablel9.l):
q(::0t:-ft,
drl I a_
l.=u
whereq (r) : heatflux 1W/m2).k : coetllcientof thennal conductivityfbr soil [: 0.5W (m . K)1, I : temperature(K), &rd 1 : distancemeasureddown fron the surfaceintothe soil (m). Note that a positive valuefor f'lux rneansthat heatis transf-erred frorntheair tothe soil. Use numericalditTerentiationto evaluatethe gradientat the soil-air interfaceandenploy this estimateto determinethe heatflux into the ground. Solution. Equation(19.21)canbeusedtocalculatethederivativeattheair-soilinterfaceas
2(0)- 0.0r2s- 0.037-5
/ ' ( 0 ) : 1 3 . 5(0 -
+ t2 - 0.037-5) 0.0125.)(0
ffi
z(0)-0-0.0375
2(0)-0-0.0t25 - 0)(0.0375 - 0.012s) (0.037s : __1140 : - 133.333 K/m + 1440- 133.333 +10
whichcanbe usedto compute
w/ K\ w q ( z: 0 ) : - 0 . 5 , : 6 6 . 6 0-1 K ( - 1 3 3 . - l - 1_3J FIGUREI9.6 Tempercfure versusdepth info the soii
I9.5
DERIVATIVES AND INTEGRATS FORDATAWITH ERRORS Aside fronr unequalspacing,anotherproblernrelateclto diff'erentiatingernpiricaldatais that it usuallv includesmeasurementerror.A shortcomingof numericalcliff'erentiation
I 9.6 PARTIAL DERIVATIVES
4
dr
459
4I dl
F I G U R EI 9 . 7 (o)ootc with no lllustrolion of how smcrLl dotoerrorscre omplified by numericol difiereniioiion. numericol differenfiofion error,(b)theresuhing of curve{o),(cJdotc modifiedslightly, ond (d) the resrlting differenfiotion of curve(cJmcnifesting irrcrecsed voricbilityln controst, thereverse operoliono,fintegrotion Imovingfrorn(dJto (c)by tokingtheoreounder(d)]tendsto ottenuofe or srnooTn oolo errors.
Fig. 19.7ttshows smooth,error-fieedata that wlren numericallydilferentiatedyields a smoothresult(Fig. 19.7b).Incontrast,Fig. 19.7cusesthe samedata,but with alternating points raised and lowered slightly. This minor modification is barely apparentfiom Fig. 19.7c.However,the resultinget-fbctin Fig. 19.7tlis significant. The error arnplificationoccursbecausedifferentiationis sr-rbtractive. Hence,random positiveanclnegirtiveerrorstetrdto add.In contrast,the fact thatintegrationib a sumrring processmakesit very forgiving with regardto uncertaindata.In essence.as points lre summedto fbrm an integral,randompositive and negativeerrorscancelout. As might be expected.the primary approachfbr deterrniningderivativestbr imprecise regression datais to useleast-squares to fit a smooth.ditferentiable functionto the data.In the absenceof any other information,a lower-orderpolynomialregressionmight be a good first choice.Obviously,if the truefunctionalrelationship betweenthe dependent and independentvariableis known, this relationshipshouldlbrnr the basisfor the least-squares fit.
r9.6 PARTIATDERIVATIVES Partialderivativesalouga singledimensionarecomputedin the samefashionas ordinary derivatives.For example,supposethat we want to determineto partialderivativesfor a
I
460
N U M E R I C ADLI F F E R E N T I A T I O N
EX
approximatedwith centereddiff'erences: Af i.).r
-
Af
l(r-A"..u) 2A.r (.r, ,v* A.y) - ./(x,,t' - A,t') "f /(r*Ar,.r)-
2Ar
dl'
(19.22\ (19.23)
All the other fbrmulas and approachesdiscussedto this point can be appliedto evaluate partial derivativesin a similar fashion. t0 For higher-orclerderivatives.we mi-ehtwant to differentiarte a function with respect two or rnore ditl'erentvariables.The result is called a ntixedpartiaLderivcttive.Forexample, we rnight want to take the partialderivativeof/(-r,,r) with respectto both independent variables
#:*(#)
( l 9.21)
To developa finite-ditferenceapproximation,we can first form a diff'erencein .r of thepartial derivativesin r':
( t -. [ d-rrd I'
ja( tf ;) \'
I Ar..r') -
af
- A - l. r ' ) -1 (r i, r'
2Lx
(1 9 . 2 5 )
Then, we can usetlnite differcncesto evaluateeachof the partialsin r':
a2.f
- A . y ) l ( r - A x , . v * A y ) - / ( x - A - r . , r '- A y ) l(x * Ax, I * At') l(:r f Ax, I _ 2Ar lAr'
(i-r d 1'
2L.r
(19.26)
Collectingtermsyieldsthefinal result atJ 0x01
/(r * Ax,,r'* A1')- f(r + A,r,.y- Af') - "/("r- A.r,l' * A.r')* /(,t - A . r , r ' - A r ' ) 4A;rAy
(19.27)
WITH MATTAB 19.7 NUMERICAT DIFFERENTIATION MATLAB softwarehasthe ability to cleterminethe derivativesof databasedon twobuilti n f u n c t i o n sd: i f f a n dg r r a d i e n t .
19.7.1 MATTABFunction:ai.tr When it is passeda one-dimensionalvectorof lengthn, the di f f function retunlsa vector in the of length zr - I containingthe differencesbetweenadjacentelements.As described approxirnafbllowing example,thesecan then he employedto determinefinite-dift-erence t i o n so f f i r s td e r i v a t i v e s
.I9,2
E X A M P L1E9 . 4
461
WITH MATLAB DIFFERENTIATION NUMERICAL
Usingit i r i for Differentiotion ProblemStqtemenf. Explorehow the MAILAB dif f functioncan be employedto differentiate the function - 900x4 + 400r-5 l (x) : 0.2 -t 25x 20012l61513 from x : 0 to 0.8. Compareyour resultswith the exact solution: f
'(x) : 25 - 400x2+ 202512- 3600"t3* 2000.14
Solution. We can first express./('T) as an anonymousfunction: > > f = G ( x ) a . 2 + 2 5 * x - 2 0 0 * x . " 2 + 6 15 * x . ^ 3 9 0 0 * x . ^ 4 + 4 0 0 * x . ^ 5 ; We can then generatea seriesof equally spacedvaluesof the independentand dependent variables: >> x=0:0.1 :0.8; >> Y-f
(r) ;
The dif f function can be usedto determinethe differencesbetweenadjacentelementsof each vector.For examPle, >>
diff
(x)
l
Columns
0.1000
througth
5
0.1000
0.1000
Columns 6 through B 0.1000 0.1000
0.1000
0.1000
0.1000
As expected,the result representsthe differencesbetweeneachpair of elementsof x. To computedivided-differenceapproximationsof the derivative,we merely perform a vector division of the y differencesby the x differencesby entering >>
d=dif
f (y) . /drf
through -0.0100
5
6 through -11.010A 1.3900
B
Columns
1
f (x)
10.8900 Columns
3.1900
8.4900
8.6900
-21 .31"00
Note that becausewe are using equally spacedvalues,after generatingthe x values,we could have simply performedthe abovecomputationconciselyas >>
I
d=diff(f(r))i0.1;
The vector d now containsderivative estimatescorrespondingto the midpoint between adjacentelements.Therefbre,in order to developa plot of our results,we must first generatea vectorholding the x valuesfor the midpoint of eachinterval: >> n=Length >> xm=(x(1:n
(r) ; 1) +x(2:n)
J . /2;
462
N U M E R I C ADLI F F E R E N T I A T I O N
FIGURE I9.8 Comporisonof the exoct derivotive(line)wiih nunrericolesiimofes(circles) compufedwith MATIAB'sdif f function.
As a final step,we can computevaluesfor the analyticalderivativeat a finer levelof resolution to includeon the plot for comparison. r - ' >X a = O :. C l : . E ; > - i . ' a = 2 5 4 0 0 * x a + 3 * 5 7 5 * x . a . ^ 24 * 9 0 0 * r , r . ^ 3 + 5 * 4 O O * x a^. 4 ; A plot of the nurnericaland analyticalestimates canbe generated with >> plot (xm,d,'o',):a,ya) As displayedin Fig. 19.8,the resultscomparelavorablyfor this case. Note that asidefrom evaluatingderivatives,the dif f function comesin handyasa progranming tool fbr testingcertaincharacteristics of vectors.For example,thefollowing statementdisplaysan error nlessageand terminatesan M-file if it determinesthata vector :i has unequalspacing: if
any (drf
f (drf
f (x) )-=91 ,
error
( 'ur.iequal
spacing'
) ,
end
Another common useis to detectwhethera vectoris in ascendingor descending order. For example,the following code rejectsa vector that is not in ascendingorder(ntonotoni""'t,t:t:::::*.f,.,.r.:,:-f, erro]-(,n.r
E)
I 9 . Z N U M E R I C ADLI F F E R E N T I A TW I OI N T HM A T L A B
19.7.2
MATTAB Function: sradienr
The gradienr- functionalso returnsdifferences. However.it doesso in a mannerthat is more compatible with evalr.ratingderivativesat the valuesthemselvesrather tharr in the intervalsbetweenvalues.A simple representation of its syntaxis 'x
-
9.,'.1 .'l
(''
where f : a one-dimensional\/ectorof lengthn, and fx is a vectorof lengthn containing differencesbasedon f. Just as with the dirf function, the flrst value returnedis the differencebetweenthe first and secondvalue. However, for the intermediatevalues,a centered dif'fbrencebasedon the adiacentvaluesis returned .l,tr - .fi r (19.28) dtlIi : 2 The last value is then computedas the differencebetweenthe final two values.Hence,the resultsare akin to using centereddifferencesfor all the intermediatevalues,with forward and backwarddiflbrencesat the ends. Note that the spacingbetweenpoints is assumedto be one. If the vector represents equally spaceddata,the following versiondivides all the resultsby the interval and hence returnsthe actualvaluesof the derivatives, f;E = gradient
( f,
h)
where h : the spacingbetweenpoints. ,l9.5 EXAMPLE
Usingu, .riii+rL for Differentiotion Problem Stotement. Use the qiradient tunctionto diflbrentiatethe samef'unctionthat we analvzedin Example 19.4with the di tf function. Solution. In the samefashionas Exanrple19.4,we can generatea seriesof equally variables: and dependent spacedvaluesof the independent ^ 2 + 6 1 > > f - @( x ) A . 2 + 2 5 * x - 2 0 O * x . 5 * x . ^ 3 9 0 0 * x . ^ 4 + 40 t J * x .^ 5 ; > > x = 0 : 0 . 1 : O .B ; Y I rl; We can then use the graclient function to determinethe derivativesas >> dv_sradient1y,0.1) ^., Columns 1 through 5 5.44C\0 10.89t)0 Columns 6 through 9 -4.8100 5.0400
1.5900
5.8400
16.1500
-21.3100
8.5900
As in Example 19.4, we can generate values for the analytical derivative and display both tlre numerical and analytical estimates on a plot: >> xa=O:.01:.8; > > ! a = 2 5 , 10 0 * x a + 3 * 6 7 5 * x a . ^ 2 r.r,ya) >' plot(x,dy,'o',
^4; 4*900*xa. ''3+5*400*xa.
464
NUMERICAL DIFFERENTIATION
25 20
10
FIGUREI9.9 Compcrisonof the exccf derivotive(line)wiih numericoleslimotes(circles)computedwith MATLABs qradient function.
As displayedin Fig. 19.9,the rcsults arc not as accurateas thoseobtainedwith the dif f function in Example 19.4.This is due to the f'actthat gradient ernploysintervals that aretwo times(0.2)as wide as for thoseusedfor dl rf (0.1).
vectors,the gradient functionis particularlywell suited Beyond one-dinrensional mafor determiningthe partialderivativesof matrices.For example,for a two-dimensional trix, f, the function can be invoked as I fx, fyl
= gradient (f,
h)
where fx correspondsto the differencesin the r (column) direction,and 4.corresponds to the ditferencesin the y (row) direction, and .h : the spacingbetweenpoints.If .nis ornitted,the spacingbetweenpoints in both dimensionsis assumedto be one.In thenext section,we will illustratehow gradient can be usedto visualizevectorfields.
I9.8 CASESTUDY
465
VISUALIZINGFIELDS Beyond the determinationof derivativesin one dimension,the sradienr Bockground. function is also quite useful for determining partial derivatives in two or more dimensions. In particular, it can be usedin conjunction with other MATLAB functions to produce visualizationsof vector fields. To understandhow this is done, we can returnto our discussionof partial derivatives at the end of Section 19.1.1.Recall that we usedmountain elevationas an exampleof a two-dimensional function. We can representsuch a function mathematically as
z: fk,y) where z : elevation, x : distance measuredalong the east-westaxis, and "y : distance measuredalong the north-south axis. For this example, the partial derivatives provide the slopes in the directions of the axes.However, if you were mountainclimbing, you would probably be much more interested in determining the direction of the maximum slope.If we think of the two partial derivatives as component vectors, the answer is provided very neatly by
af
af
dx
dv
vf:+i++j
where V/ is referred to as the gradient of I This vector, which representsthe steepest slope, has a magnitude
(#)'. (#)' and a direction _r / Af/Ay\ 0^ : tan-' \ffi ) where I : the angle measuredcounterclockwisefrom the r axis. Now supposethat we generatea grid of points in the x-y plane and used the foregoing equations to draw the gradient vector at each point. The result would be a field of arrows indicating the steepestroute to the peak from any point. Conversely,if we plotted the negative of the gradient, it would indicate how a ball would travel as it rolled downhill from any point. Such graphical representationsare so useful that MATLAB has a special function, called quiver, to createsuchplots.A simplerepresentation of its syntaxis quiver(x,y,u,v) where x and y are matrjces containing the position coordinates and u and v are matrices containing the partial derivatives.The following example demonsffatesthe use of quiver to visualizea field. Employ the srradient function to determineto partial derivativesfor the following two-dimensionalfunction: /fr
rr\ :
1,-
y -7xZ
-2.xt,
-
tZ
466
NUMERICAT DIFFERENTIATION
continued from : -2 to 2 andy : l to 3. Thenusequiver to superimpose 1 a vectorfieldonacontour plot of thefunction. Solution.
Wecanfirst express/(.t, y) asananonymous function
> > f = G( x , y ) y - x - 2 * x . ^ 2 - 2 . * x . * y _ y . ^ 2 ; A seriesof equally spacedvalues of the independent and dependentvariablescanbe generated as >>
z_f (x,y-);
The gradient
function can be emproyedto determinethe paftiar
derivatives:
19.1 mati mati
We can then developa contourplot of the results: >>
cs-contour(x,y,z
);clabel
(cs);ho1d
on
As a final step,the resultant of the partial derivatives can be superimposed -' -"-- asvectorsonthe r contourplot: >> quirzer(x,y,_fx,,fy);ho1d
:
oft
FIGUREI9.IO
ffil$?":rui5r*n'".ruj:lof
tt",
o twodimensionol function with rheresulionr oftheporriol
J
2.6
2.6
t9.2 first E'np
19.3 finit, seco diffe In ei inter 2L.t equi first eacl cate 19.t riva and initi
l9.r l9.l of -r ro= the diff 19.' redr -1.6
-1.4
-1.2
-0.6
-0.4
-0.2
l9.i
PROBLEMS
467
Note that we have displayed the negative of the resultants, in order that they point "downhill." The resultis shownin Fig. 19.10.The function'speakoccursat x : *1 and y : 1.5 and then drops away in all directions.As indicatedby the lengtheningarrows,the gradient drops off more steeply to the northeastand the southwest.
PROBTEMS 19.1Compute forward and backward difference approxiof O(h) ard O(h]), andcentralclif'furencc approrirnations mationsof O(h)\ and O(h1l for the flrst derivativeof . l ' = c o s - ra t - r : r / 4 u s i n ga v a l u co f h : r 1 1 2 . E s t i m a l e thetruepercentrelative error rr fbr eachapproximation. approrinrrtionsto estimatethe 19.2Usecentereddifl'erence firstand secondderivativesof -r - e' ar x : 2 fot' ft : 0. I . Employboth O(h?) andO(ft4) formulasfbr your cstimates. 19.j Use a Taylor series expansion to derivc a ccntercd approximationto the third clerivalivethat is finite-difference second-order accurate.'Ibdo this,you will have(o use four expansionsfbr the pointsli :. ri I . r'r+t . ilnd,r;+2. diff'erent In eachcase,the cxpansionwill be aroundthe point -r;. The intervalA,r will be usedin eachcaseof i - i and i * l, and 2A.r will bc used in each casc of i - 2 antli * 2. The four' must then be combined in a way to climinate the equations first and secondderil'atives. Carry enou-{hterms along ilt eachexpansionto evaluatethe first term that will be truncatedto determinethe orilcr oi the approximation. 19.4Use Richardsonextrapolationto estimatethc first deof r' : cos.r at t : n llusing stepsizesof h 1: n f I rivative andh. : r/6. Ernploy centereddifl'erencesof O(/rr) for the initialestimates. 19.5RepcatProb. 19.4.but fbr the first derivativeof ln.r at . r = 5 u s i n gh t : 2 a r u lh z - - | . 19.6Employ Eq. (19.2[) to determinethe first dcrivative o f r : 2 - r a - 6 - t 3 - l 2 . r - 8 a t r : 0 b a s e do n v a l u e sa 1 x o= - 0 . 5 , J r : 1 , a t t dx 2 : 2 . C o m p a r ct h i s r e s u l tr v i t h thetruevalue and with an estimateobtainedusing a centered differenccapproxiniationbasedon /i : l. 19.7Provc that fbr cquispaceddata points, Eq. (19.21) reducesto Eq. 6.25) at .r : -r| . 19.8 Develop an M-fllc to apply a Romberg algorithm to -...:----!-
iL
-
-l-,-:,,^+:-,,.
.-f
^
ft;r,dh
{',,r,-ti,rn
19.9 Develop an M-filc to obtain flrst-tlerivativeestimates for unequallyspaceddata.Tcstit with thc fbllowin-edata:
15 0.6 t6 x f ( x ) 0 9 0 3 6 4 . 3 7 3 4 0 . 32 6l
25 35 o o B 4 2 2 0 0 15 9 6
rvhcre .l (r) : 5z-2'.r. Cornpareyour rcsulls with the true derivatives. 19.19 Devclop an M-file function that computcsfirst and secondclerivativc estirnates of oltler O1lir1brsedon the fbrmulas in Figs. 19.3 through 19.5.Thc function'sflrst line shouldbe sct up as f unctiotr
l r l l ' d x , O r O O * 2 I = f l i f f r z q( ; , - . ' 1
wherc x and y are input vectorsof length n containingthe valuesof the indepcndent and dependeutvariables,rcspectively, and dydx and dy2d.;2 are output vectorsof lcngth il estimatesat containing the first- and second-derivative cach value of thc indcpendcntvariable.The function shoulcl generatea plot of c1y'dxarrd dy2dx2 versusx. Havc your M-file returnan error messlgeif (a) the input vectorsare not varithe samelength,or (b) the valueslor the independent able are not equally spaced.Test your programwith the data f i o m P r o b .1 9 . Il . 19.11The fbllowing data was collectcdfor the distance traveledversustime fbr a rocket:
It
S
J, km
25 3'2
50 5B
75 7B
t00 92
125 r00
Usc numericaldiff'erentiationto estimatethe rocket's velocilr
468
N U M E R I C ADLI F F E R E N T I A T I O N
19.12 A jet fighter'spositionon an aircraftcarier's runway was timed during landin-u:
t,s x,m
0 153
o:.l85
lAA 2aB
t 75 2 37 '249 261
325 271
383 273
where x is the distancefrom the end of the carricr.Estimate (a) velocity @.rldt') and (b) acccleration(duldrl using nunrericaldifferentiation. 19.13 Use the fbllowing datato find the velocitl'and accelerationat / : l0 seconds:
lime,t,s O 2 Position,x,m O a7
4 IB
6 34
B 5l
l0 63
l2 l4 16 7:3 B0 B4
19.17 The fbllowing data was generatedftom the normal distribution:
-l -s -05 x 2 I 0 (1 { l 0 , s 3 9 9 O ) 2 9 5 2 a.2atq7 352070 3e894 /(x)
r.5 r 05 l 2 f ( x \ 0 . 3 5 2 A 74 2 4 1 9 7 4 . 1 2 9 5 20 0 5 3 9 9
I
2
J
4
5
6
7
B
9
10
Use sccond-order correct (a) centered flnite-diftcrence, y 1 4 2 ) 3 . 3 4 B 6 B 6 6 8 . 6 7 5 8 . 9 1 0 9 t 0 (b) fbrward finite-difl'erence,and (c) backward finitemethods. clifTercnce 19,19 The objcctive of this problem is to comparesecondlg.l1 A plane is being trackcdby radar,and data is taken orclel accurate forwiucl, backward, and centeredfiniteeverysccondin polarcoordinates I and r ditfbrenceapproxirrationsof thc first clerivative of a function to the actr.ralvalue of the derivativc.This will be donefor
t, s 2AA 0,(rodl A75 t', tn 5l 20
2A2 244 246 2 0 8 2 t O A72 o 7 a 0 6 8 4 57 4.66 5374 5560 5800 6 0 3 0 6 2 4 a
lG):e-2'-x
(a) Use calculusto dctclmine the corrcctvalueof thedenvativeatr:2. (b) Develop an M-f ile function to evaluatethe centered fi nite-difl'erenceapproximations, startingwith .r = 0.5. At 206 seconds,use the centercdfinite-diflerence(secondThus, for the first evaluation,the .r valuesfor thecenorder correct) to find the vector cxpressionslor vclocity il tered differencc approximationwill be .r : 2 t 0.5or and accelerationrl. The velocitv and acceleratiorrgiven rn of 0.1 polar coordinatesale r : 1.5 and 2.5. Then. decreascin increments down to a nrinimumvalueof A-r = 0.01. i : i a , + t 0 4 0 a n d a : G - r 0 : ) 2 , ,I \ r 0 + 2 i 0 ) A e (c) Repeatpart (b) lbr the second-orderfbrwardandback(Notethatthesecanbedoneatthesame ward difTerences. 19.15 Use regrcssionto estimatc the accelerationat each time thatthe centereddifl'erenceis computedin theloop.) time fbr the following data with second-,third-, and fburth(d) Plot the resultsof (b) and (c) versr.rs r. Includetheexact orderoolvnomials.Plot the results: rcsullon the plot lol eomparison. 19.20 You have to measurethe flo$,rate of waterthrougha 1 2 3.25 45 678 B 5 9 3 t 0 surallpipe. In order to do it, you placea bucketat thepipe's t u lO )2 ll 1 4 1 7 t 6 t 2 t 4 l 4 t 0 outlet and measurethe volumc in the bucketasa function of titnc as tabulatedbelow. Estimatethe flow rateat t = 7 s. 19.16 The nolrlal distributionis definedas - - r -I t - 1 z "f ( r ) J2tr Use MAILAB
. ,^
to determine the inflection points of this
Time, s Volume, cm3
0 0
T
::::: J,h U, ftl
Use MATLAB to estimatethe inflectionpointsof thisdata. 19.18 Usc the .tif f iy; commanclto developa MATLAB M-file tunction to computeflnite-difftrenceapproximations to the first and sccondderivativeat each-r valuein thetable below. Usc finitc-ditfbrenceapproximationsthataresecondordercorrcct.O (.r2):
,rO
Use i r (N/
B 164
19,21 The velocity rr (m/s) of air flowing pasta flat surface
t9.22 M
wher u n r t€ (cmz Ane centr derly
i:::
x, cn c, l(
Use t estirr conjr pollu (D= sedin the lr t9.24
'T:
t, mi v, lr
Calc order 19.2,
neer! temp
x
rrh T, "l
If thr
469
PROBLEMS
larl to determine the shear stress UseNewtott'sr:i.rczrsln' r (N/m:)at the surface(.v : 0),
19.25 Ths horizontalsurfaceareaA, (nr21of a lakeataparticulardepthcanbecomputedfionr volumeby difTerentiation:
in 4 . 1 ; 1:
L-lL-
aI
A s s u m e a v a l udeyonfa m i c v i s c o s i4t y: 1 . 8 ),h
0
u,m/s 0
0.002 0006 a2B7 0899
00r2 I915
x l0-5 N ' s/rn2.
0 0r B 3 048
A A?! 4 299
rvhere I./ : volur.ne(ml) and ; : tlepth (nr) as nreasured fiom the surfacedown to the bottom. The averageconcentlationof a substance thatvarieswith depth.- (g/m3),canbe computeclby integlatron:
.[,] r:Q)A,(z)ctz.
19'22pi711',e 71vstdiffir.sion/rnl stalesthat Massflux : -Dd' dr
(Pt9.22) whereZ:
where massflux : the quantity of massthat passesacrossa D : a dil-fusioncoefTicient unitareaper unit tirne (g/cmr/s.1. (g/cm3).and ,r : distance1cm). (cmr/s), c : concentration An environmentalengineer measuresthe fbllowing concentration of a pollutant in the pore watersof sedimentsunderlying a lake (.r : 0 at the sediment-waterinterfaceancl downward): increases
0 006
r, Cm c, l0-6 g/cm3
dV , (:) aa.
a32
1'de,s1az
the total depth (rr). Determinethe averageconcentralionbasedon the following data:
16 4 B 12 O z,n v, lo6m3 I8175 5 105I I 9635 Q3927 00000 lo.2 7.4 4.1 B5 5.2 c,9/m3 19.26 p;orr4^r's law characterizesthe r'oltagedrop across an inductor as
3 06
di Vt.: L; t7l
Usethebestnr,rmerical difl'erentiationtechniclueavailableto estimate the derivalive at "rr: 0. Ernploy this estimate in with Eq. (PI9.22) to colrlputethe massllux of corlunction pollutant out of the sedimentsand into the overlying walers Q - - 1 . 5 2 x 1 0 - 6 c m 2 / s )F. o r a l a k ew i t h 3 . 6 x 1 0 6r n 2o f sediments. how much pollutantu'ould be transportedinto thelakeover a year's time? 19.23fne fbllowing data was collectedwhen a large oil tankerwas loading: /, min I', 106 borrels
0 r 0 2 0 3 0 4 5 60 04 07 027 0BB r 05 \17
(:,
the flow rate Q (that is, tll/ ldt) fbr eachtirne to the Calculate orderof /rI . 19'24psudsp'slaw is usedroutinelyby architectural engineersto detennineheat flow through walls. The fbllowing are measuredfiorn the surface(.r : 0) into a temperatures
(i1 henrys: where V.: r,oltagedrop (V), L: indtrctance I H : I V. s/A), I : current(A), and I: time (s). Determine the voltagedrop as a functionof time from the fbllowing datafor an inductance ol4 H.
a 0
t i
0I 016
02 a32
03 056
05 084
0/ 2A
19.27 gxr.4 on Faraday'slaw 1Prob.19.26),usethe following voltagedatato estimatethe inductanceif a currentof 2 A is passerJ thloughthe inductorover'400milliseconrls. t, ms V, volts
0 t0 20 40 60 B0 12a lB0 280 400 35 26 15 7 a)829414946
l9.2tt 15srareof coolingof a body(Fig.P19.28) canbeexpressed as JT
klT
Tul
dT I,
lf,
T,'C
0 t9
008 l7
If theflux at x : 0 is 60 W/mr, cornputeft.
O.t U
whereI: temperatureof the body ("C), 4, : temperatureof conrnedium('C), andl:: a propoflionality the surrounding stant(per minute).Thus, this equation(calledNcruton's/aw of cooling) specifiesthat the rateof cooling is proportionalto
470
NUMERICAL DIFFERENTIATION madetbr air flowing over a flat plate where,r': distancenorrnal to thc surface:
0 900
Jr clll
T,K
I 480
35 270
dioxi deter lated and <
2t0 T'K
FIGURE PI9.28
the ditlcrencein thc tempcraturesol the body and of thc surrounding medium. If a metal ball heatedto 80 'C is dropped into water that is held constantat f,, : 20'C, the temperature of the ball changes,as in
Time, min
r, "c
0 80
5 r0 ,145 300
t5 24t
20 2t7
25 2A7
If thc plate's dimensionsare 200 cm long and 50 cm wide, and I : 0.028J/(s . Irr . K), (a) deterrninethe flux at thesurface.and (b) the heattransf'erin watts.Note that I J = 1W.s. I 9.3I The pressuregradientfor larninarflow througha constantradiustube is given by dn
8uQ
d.r
ft14
224 22 r35 tl 090 O68 o6t a54
250 47 2,\ I49 t2 099 475 4 . 6 75 06
282.5 523 27 I 5.5 I ).4
I03 a78 4.7 462
T, K C,, kll(kmol
300 4AO 500 600 82 BBB Il2 136 136933 157744
700 800 900 ' K) I Z5 036 ) 89 273 20A 9n
1000 210450
19.33 Tllg specificheat at constantpressureq, tJ/(kg. K)l of an ideal gas is relatedto enthalpyby dh
19.30 p61 fluid tlorv over a surface.the heat l]ux to the surface can be computedwith Fourier's law: ,y : distancenormal to the surf-acc(m). The tbllowing measurementsare
,,
reacti
-a
where p = pressureiN/rn:), -r : distancealongthetube\ centerline(m), fr : dynamicriscosity{N.s/nrr;,0 = flow (mr/s)anclr: radius(m). (a) Detcrminc the pressuredrop for a l0-cm lengthtuh = p= for a viscousliquid (p : 0.005N . s/mr,density I x I03 kg/mr) with a flow of l0 x 10-6 m3/sandthe fbllowing varying radii along its length:
T, K Cr,kl/lkmol.K)
19.3 react
d,
Utilize nunrericaldiff'erentiationto determinedTldr ateach value of tirnc. Plot dT f dt versw T - 7,, andemploy lir"rear regression to evaluatefr. r,cm O 2 4 5 710 6 19'29 Tl1senthalpyof a real gas is a function of pressureas l58 r,mm 2 135 1.34 l6 142 2 dcscribcclbelow. The data was takcn lbr a real fluid. Estimatethe enthalpyof the f'luid at zl00K and 50 atm (evaluate (b) Comparcyour result with the pressuredrop thatwould thc integral frorn 0. I atm to 50 atm). have occured if thc tubc had a constantradiuseoualto 7P .;lt/ ') the averagcradius. H : I (\ r ' r (\ 1 ' ),rp dT/pl (c) Dctelminc the averageReynoldsnuntberfor thetubeto Ju verit'y that flow is truly laminar (Re : pu D I 1t < 21N whercu: velocity). 19.32 the following data for the specificheatof benzene V,L was generatedwith an /rth-orderpolynomial.Usenumerical P, otm r=350K r=400K T=450K difl'erentiationto dcterminen.
0t 5 t0 20 25 30 4A 45 50
!
dT
whcre ft : enthalpy (kJ/kg), and I : absolutelemperature (K). The fbllowing enthalpiesare providedfor carbon
wher, reacti (minevalu logar lo
Thert log(slope meth, follor urea: t, mi c, ml
19.31
El(m:
PROBLEMS
471
dioxide(COr) at severaltemperatures.Use these valuesto determine the specificheat in J/(kg . K) fbr eachof the tabulatedtemperatures. Note that the atomic weights of carbon g/mol, r'espectively andoxygenare 12.01I and I -5.9994
Lid
I I
Water
T,K h,kt/kmol
754 29 629
800 32,179
H
1000 900 37,445 42,769 Sediments
19.34An nth-orderrate law is often usedto model chemical reactions that solely dependon the concentrationof a single reactant: dc
--
F I G U R EP I 9 . 3 5
: *kc"
dl
wherec : concentration (mole), I : time (rnin), n : reactionorder (dimensionless), and ft : reaction rate (min I moler "). The di.fferentialmethod can be used to evaluatethe parametersk and rt. This involves applying a logarithmictransformto the rate law to yield, . / - rlcr I o- g\ { , } = l o g k dt /
L,?log(
Therefore,if the nth-order rate law holds, a plot of the log(-dcldf) versuslog c should yield a straightline with a slopeof n and an intercept of logfr. Use the dilferential methodand linear regressionto deternrinet and n for the following data fbr the conversionof arnmoniumcyanateto urea: l, min c,mole
0 A750
5 o 594
l5 a 420
30 a 29t
dissolvedoxygencontentof a naturalwater.11is measured by placing a sediment core in a cylindrical conniner (Fig. P I 9.35).After carefullyintroducinga layer of distilled, oxygenatedwater abovethe sedinrents, the containeris coveredto preventgas transfer.A stirel is usedto rnix the water gently,and an oxygen probetracks how the water's oxygen concentrationdecreasesover time. The SOD can then be computedas SOD: -I1:
do at
where11: the depthof water (m). o : oxygenconcentration (g/mr),and t: time (d). Based on the following data and H : O.l m, use numerical differentiationto generateplots of (a) SOD versus lime and (b) SOD versusoxygenconcentralioll:
45 0.223 t , d 0 0125 425 0375 05 0625 475 o , m g / L 1 o 7 . l l 4.59 2 57 r r s 0 3 3 0 . 0 3
19.35The sedirnent oxygen demand [SOD in units of g/1mr. d.11is an important parametel in determining the
&
t f,
Ordinory Differentiol l-
r'
Equolrons
6.r ovERvrEw The fundamental laws of physics. mechanics,electricity, and thermodynamicsare usually' basedon empirical observationsthat explain variations in physical properties and statesof i' systems.Rather than describing the stateof physical systemsdirectly, the laws are usually_ .r" *F '* , couched in terms of spatial and temporal changes. These laws define mechallt** .oj ., momentum, differenliol change.When combined with continuity laws for enorgy,mass,or integrationof thesedifferentialequationsresultsin matheequationsresult.Subsequent matical functionsthat describethe spatialand temporalstateof a syslemin termsof energy, -, " '" mass,or velocity variations.As in Fig. FT6,1, the integrationcan be implementedanalyti* l,, cally with calculus or numericalli with the computer jumper introproblem The free-fallingbungee ducedin Chap. I is an exampleof the derivationof a ditferential equationfrom a fundamentallaw. Recall that Newton's second law was used to develop an ODE describingthe rate of changeof velocity of a'" falling bungeejumper: c,t t
du
i:r--:-u-
(PT6'l)
where g is the gravitationa.lconstant.rr is the mass, and c, is a drag coefficient.Suchequations,which are composedof an unknown functionand its derivatives. are called dffirential equntions,,:TheY ate sometimes referred to as rate equationsbecausethey expressthe rate of change of a variable as'a"'functionof variables and parameters. u In Eq. 1PT6.l), the quantitybeingdifferentiated is called the dependent variable. The quantity with respectto which u is differentiated t is calledthe independent variable. When the function involves one independentvariable, the equation is called an ordinary ' ,.::.:..1.r..
, r,i:,ir.,.,, t.,..
473
474
PART 6 ORDINARY DIFFERENTIAL EQUATIONS
Physicallaw
F=ma
II
I
I
f
du -^ -6
ODE
I
cJ..2
, aIm
n
Analytical (calculus)
I
/
V Solution
l o m
\ /
l o ^ ,
\
u - . / " " ' t a n h[ , / r y r ] \
t,j
Numerical (computer)
\V
/n
/
u r + r= u i + ( s - ' ! r ' )
n,
FIGUREPT6.I i I e s e o u e n r eo t e v e n t si n r h ed e u e l o p r r e no' ^ d s o l - l l o ^o l O D E sf o e n g - e e r i n go n d s c i e n c e Ihe exompleshown is for the velocityof the freefcliingbungee iumper.
dffirential equation (or ODE). This is in contrastto a partial dffirentiul equation (or PDE) that involves two or more independentvariables. Differentialequationsare also classifiedas to their orderFor example.Eq. (PT6.l) is called afirst-order equatio,?becausethe highestderivativeis a lirst derivative.Asecondorder equationwould includea secondderivative.For example,the equationdescribingthe positionx of an unforcedmass-springsystemwith damping is the second-orderequation:
,)
,
d'x
m -*t. dt'
dx
dt
*A-x:0
(PT6.2)
where ,fl is mass,c is a damping coefficient,and k is a spring constant.Similarly, an nthorder equationwould include an nth derivative. Higher-orderdifferentialequationscan be reducedto a systemof first-orderequations. This is accomplishedby defining the first derivativeof the dependentvariable as a new variable.For Eq. (PT6.2),this is doneby creatinga new variableu as the first derivativeof disolacement dx dt
(PT6.3)
where u is velocity.This equationcan itself be differentiatedto yield (l t'
o'-l
dt
dtz
(PT6.4)
Equations(PT6.3)and (PT6.4)can be substitutedinto Eq. (PT6.2)to convertit into a firstorder equation: du m7-tcu+kr:0
(PT6.s)
6.I OVERVIEW
475
As a final step,we can expressEqs.(PT6.3)and (PT6.5)as rateequations: dx
(Pr6.6)
l-
ut
tlu
'
dT
ck tn
nt
(PT6.t)
Thus, Eqs. (PT6.6) and (PT6.7) are a pair of first-order equationsthat arreequivalent to the original second-orderequation (Eq. PT6.2). Becauseother nth-order ciiff'erential equationscan be similarlyreduced,this part of our book fbcuseson the solutionof firstorderequations. A solutionof an ordinarydifl'erentialequationis a specificfunctionof the independent variable and parametersthat satisfiesthe original differential equation.To illustrate this concept,let us startwith a simple fourth-orderpolynomial, . t ,: - 0 . 5 1 4 + 4 . r 3 - l 0 x 2 + 8 . 5 . r* I
(Pr6.8)
Now. if we differentiateEq. (PT6.8).we obtainan ODE:
d.r - _2r3 * l2,rr - 20r * 8.5 dx
(Pr6.e)
This equationalso describesthe behaviorof the polynomial,but in a mannerdifferentfrom Eq. (PT6.8). Rather than explicitly representingthe values of ,r for each vahre of .r, Eq. (PT6.9)givesthe rateof changeof -ywith respectto.r (thatis, the slope)at everyvallle of .r. FigurePT6.2showsboth the functionand the deriv'atileplottedverslls.{.Noticehow the zero valuesof the derivativescorrespondto the point at which the original function is flat-that is, where it hiis a zero slope.Also, the maximum absolutevaluesof the derivativesale at the endsofthe intervalwherethe slopesofthe functionaregreatest. we can determinea differentialequationgiven the Altlrou-eh,as just denronstrated, originalfunction,the objecthereis tt.rdeterminethe originalfunctiongiventhedifferential eqr.ration. The original function then representsthe solution. Withoutcomputers,ODEs areusuallysolvedanalyticallywith calculus.For example, Eq. (PT6.9)could be multiplied by dx andintegratedto yield
t:
lr-ztt
1 l2.r'r- 2ox* 8.5)r/.r-
(PT6.I 0)
The righrhand sideof this equationis calledan indefiniteintegraLbecausethe limits of integrationare unspecified.This is in contrastto the definite integralsdiscussedpreviously in PartFive [compareEq. (PT6.l0) with Eq. (17.5)]. An analyticalsolutionfor Eq. (PT6.l0) is obtainedif the irrdefiniteintegralcan be evaluatedexactlyin equationfblnr. For this sinrplecase,it is possibleto do this witlr the result: r ' : - 0 . 5 1 4 * 4 . t 1 - 1 0 . 1+2 8 . 5 . r* C
(PT6. r r)
which is identicalto the original function with one notableexception.In the courseof differentiatingand then integrating,we lost the constantvalue of I in the original equation and gainedthe value C. This C is called a u)nstantof integrorion The f'actthat suchan arbitraryconstantappearsindicatesthat the solutionis not uniclue.In fact,it is but one ofan
476
PART 6 ORDINARYDIFFERENTIAL EAUATIONS
FIGUREPT6.2 P l o t so f ( o )\ ' v e r s u sr o n d l b )d - r . / r hv e r s u s r f o r t h e f u n c t i o n l , :- O 5 r a * 4 x 3- l O ; + B 5 j r + I
infinite numberof possiblefunctions(con'esponding to an infinitenumberof possiblevalues of C) that satisfythe differentialequation.For example,Fig. PT6.3 showssix possible functionsthat satisfyEq. (PT6.l I ). Therefore,to specifythe solutioncompletely,a ditTerentialequationis usuallyaccompanied by auxiliary conditions.For first-orderODEs, a type of auxiliary condition called ln initial value is requiredto determirrethe constantand obtain a uniquesolution.For example,the original diff'erentialequationcould be accompaniedby the initial condition thatat r: 0 . . 1 ' : l . T h e s ev a l u e sc o u l d b e s u b s t i t u t eidn t o E q . ( P T 6 l. l ) t o d e t e r m i n e C : l. Therefore,the unique solution that satisfiesboth the differentialequationand the s p e c i l ' i eidn i t i a lc o n d i t i o ni s -r': -0.5x1 f 4rr - l0rr -r-8.5r * I Thus,we have"pinneddown" Eq. (PT6.l1) by forcingit to passthroughthe initial condition, and in so doing, we havedevelopeda uniquesolutionto the ODE and havecornefull circle to the originalfunction[Eq. (PT6.8)]. Initial conditionsusually have very tangibleinterpretationsfor diff'erentialequations derived from physical problem settings.For example.in the bungeejumper problem, the
ORGANIZATION 6.2 PART
477
C=3 ( =Z
C=1 C=0
FIGUREPT6.3 of -2-r'rI l2x2- 2Or* 8.5.Eochconforms to o differenl Sixpossibe solutions fortheiniegrol vo ue of lhe constontof inteqrolionC.
initial conditionwas reflectiveof the physicallact that at time zero the verticalvelocity was zero.If the bungeejumperhad alreadybeenin verticalnlotiorlat time zero,the solution would havebeenmodified to accoLllltfor this initial velocity. When dealingwith an nth-orderdifferentialequation,n conditionsare requiredto obtain a unique solution.If all conditionsare specifiedat the samevalue of the independent variable (fbr example,at,v or t : 0), then the problem is called an initial-valueproblem. This is in contlastto boundary-valueproblemswhere specificationof conditionsoccursat difTerentvaluesof the independentvariable.Chapters20 and 2l will fbcus on initial-value problems.Boundary-value problemsarecoveledtn Chap.22.
6.2
PARTORGANIZATION Chapter 20 is devotedto one-stepmethodsfor solving initial-value ODEs. As the name suggests,one-stepmethodscomputea futurepredictiony;a1, basedonly on information at a single point r; and no other previous information.This is in contrastto multistepapproachesthat useinlbrmation from severalpreviouspointsas the basisfor extrapolatingto a newvalue. With all but a rninorexception,the one-stepmethodspresented in Chap.20 belongto wh:rt are called,Runge-Kuttatechni4lle.s. Although the chapterrnight havebeenorganized aroundthis theoreticalnotion, we haveoptedfor a more graphical.intuitive approachto introduce the rnethods.Thus, we begin the chapterwrth Euler's ntethod,which has a very straightforwardgraphicalinterpretation.In addition,becausewe have alreadyintroduced Euler's methodin Chap. l, our emphasishereis on quantifyingits truncationeror and describineits stabilitv.
478
PART 6 ORDINARY DIFFERENTIAL EQUATIONS
Next, we usevisually orientedargumentsto developtwo improvedversionsof Euler's nretlrod-tlre Heun and the midpoint techniques.After this introduction,we formally develop the conceptof Runge-Kutta(or RK) approachesand demonstratehow the foregoing techniquesare actuallyfirst- and second-orderRK methods.This is followed by a discussion of the higher-orderRK fbrmulations that are fi'equentlyused fbr engineeringand scientific problem solving. In addition, we cover the applicationof one-stepmethodsto .t-y.r/erns oJ ODEs.Note that all the applicationsin Chap.20 are limited to caseswith a fixed stepsize. ln Chop. 21, we cover more advancedapproachesfor solving initial-valueproblems. First. we describeadaptiveRK ntethodsthat automaticallyadjustthe stepsize in response to the truncationerror of the computation.Thesemethodsare especiallypertinentas they are employedby MATLAB to solve ODEs. Next, we discussnrulti,stepmethods.As mentionedabove,thesealgorithmsretain infbrn'rationof previousstepsto more ell'ectivelycapturethe trajectoryof the solution.They alsoyield the truncationerror estimatesthat canbe usedto implementstep-sizecontrol.We describea sinrplemethod-the non-self-startingHeun method-to introducethe essential f-eatures of the multistepapproaches. Finally, the chapterends with a descriptionof ,stiffODEs. Theseare both individual and systemsof ODEs thathaveboth fastand slow components to their solution.As a consequence,they require specialsolution approaches.We introducethe idea of an implicit solution techniqueas one commonly usedremedy.We also describeMMLAB's built-in functionsfor solving stiff ODEs. ln Cha1t.22, we focus on two approachesfor obtaining solutionsto boundary'-value problems:theshootingandJinite-dffirencemethods.Aside from demonstratinghow these techniquesare implemented,we illustratehow they handleclerivatiyeboundorvconditions and nonlinear ODEs.
ti'.iili, 'l '';
'j:''
':
'ii.
l:io ":i'
*l-\,
Initiol-Volue Problems
CHAPTEROBJECTIVES The primary objectiveof this chapteris to introduceyou to solving initial-value problemsfbr ODEs (ordinarydifferentialequations).Specificobjectivesand topics coveredare t .
. .
Understandingthe meaningof local and global truncationerrorsand their relationshipto stepsize for one-stepmethodsfor solving ODEs. Knowing how to implementthe following Runge-Kutta(RK) methodsfor a singleODE: Euler Heun Midpoint Fourth-olderRK Knowing how to iteratethe coffectorof Heun'smethod. Knowing how to irnplementthe followingRunge-Kuttamethodsfbr systems of ODEs: Euler Fourth-orderRK
YOU'VE GOT A PROBLEM e startedthis book with the problemof simulatingthe velocity of a free-falling bungeejumper.This problemamountedto formulatingand solving an ordinary diffbrentialequation,the topic of this chapter.Now let's return to this problem and make it more interestingby computingwhat happenswhen thejumper reachesthe end of the bunseecord.
480
INITIAL-VALU E PROBLEMS To do this, we should recognizethat the jumper will experiencedifferentforcesdependingon whetherthe cord is slack or stretched.If it is slack,the situationis thatoffree fall where the only forces are gravity and drag. However, becausethe jumpel cannow move up as well as down, the sign of the dragforce mustbe nrodifiedso that it alwaystends to retardvelocity,
du g - srgn(r,)?u2 dt
(20.1a)
where u is velocity (m/s), t is time (s), g is the accelerationdue to gravity (9.81m/s'),c7is the drag coefficient(kg/m), and m is mass(kg). The .signum Jilndion.t sign,returnsa - I or a I dependingon whetherits argumentis negativeor positive,respectively. Thus,whenthe jumper is falling downward (positive velocity, sign - l), the drag fbrce will be negative and hence will act to reduce velocity. In contrast.when the jumper is moving upward (negativevelocity.sign: -l), the dragfbrce will be positiveso that it againreduces the velocity. Once the cord beginsto stretch,it obviouslyexefis an upward force on thejumper.As donepreviouslyin Chap.8. Hooke's law can be usedas a first approximationof thisforce. In addition,a dampeningforce shouldalso be includedto accountfor frictional effectsas the cord stretchesand contracts.Thesef'actorscan be incorporatedalong with gravityand drag into a secondfbrce balancethat applieswhen the cord is stretched.The resultis the f o ll o w i n ed i f f e r e n t i ael o u a t i o n : du al
:,9
-
c,t
) Slgn(u)-U- tn
k
-(r ilt
-
I.l -
y
-U m
(20,\b)
wherek is the cord's springconstant(N/m), x is verticaldistancemeasureddownwardfrom the bungeejump platform (m). L is the lengthof the unstretchedcord (m), and 7 is a dampeningcoefticient(N . s/m). BecauseEq. 120.1b)only holds when the cord is stretched(x > L), the springforce will always be negative.That is, it will always act to pull the jumper back up. The dampening force increase.s in magnitudeas the jumper's velocity increasesand alwaysactst0 slow the jumper down. If we want to simulatethejumper's velocity,we would initially solveEq. (20.1a)until the cord was fully extended.Then, we could switch to Eq. (20.1b) for periodsthatthecord is stretched.Although this is fairly straightforward,it rneans that knowledgeof the jumper's position is required.This can be done by formulartinganotherdifferentialequation lbr distance: dx
(20.2)
dt
Thus, solving tbr the bungeejurrrper'svelocity amountsto soiving two ordinarydifferentialequationswhereone of the equationstakesdifferentforms dependingon thevalue
I Some computerlanguagesrepresentthe signum functionas sgr (x). As represented here,MATLAB usesthe n o m e n c l a t u r sei o n 1 x ) .
20.2 EULER'S METHOD
48r
of one of the dependentvariables.Chapters20 and21 exploremethodsfor solving this and similar problemsinvolvingODEs.
20.t ovERvrEw This chapteris devotedto solvine ordinarv diflerentialequationsof the form d.y
dr
: I (l' -Y)
(20.3)
In Chap. l, we developeda nnmericalmethodto solve suchan equationfor the velocity of the free-falling bungeejumper. Recall that the methodwas of the generalform New value : old value* slope x stepsize or, in mathematicalterms, .Vi+r: )'i*Qh
(20.4)
where the slope@is calledan incrcnzent Junction.Accordingto this equation,the slopeestimate of @is usedto extrapolateliom an old value -yrto a new value y;11 over a distance /2.This fornrula can be appliedstep by stepto traceout the trajectoryof the solution into the future. Suchapproaches are calledute-ste1tntethodsbecausethe value of the increment function is basedon information at a single point i. They are also refelred to as Rwrgewho first discussedthem in the early Kutta ntethodsafter the two appliedmathematicians 1900s.Another class of rnethodscalled multistepmethodsuse information from several previous points as the basisfor extrapolatingto a new value. We will describemultistep rnethodsbriefly in Chap.21. All one-stepmethodscan be expressedin the generalform of Eq. (20.4),with the only differencebeing the mannerin which the slopeis estimated.The sirnplestapproachis to use the differentialequationto estimatethe slopein the fbrm of the first derivativeat /;. ln other words, the slopeat the beginningof the interval is taken as an approximationof the averageslope over the whole interval.This approach,called Euler's method,is discussed next. This is followed by other one-stepmethodsthat employ alternativeslope estimates that result in more accuratepredictions.
2O.2 EULER's METHOD The first derivativeprovidesa direct estimateof the slopeat /i (Fig. 20.I ): q : f ftr,),i) where /(t;,1;) is the differentialequationevaluatedat /; and _r'i.This estimatecan be substitutedinto Eq. (20.1t: )'i+r: t"i* fGi,v"t)h
(20.s)
This formula is ref-enedto as Euler's rnethod(or the Euler-Cauchyor point-slopemethod). A new value of -"-is predictedusing the slope (equalto the first derivativeat the original valtte of r) to extrapolatelinearly over the stepsize h (Fig.20.1).
482
INITIAL-VALUE PROBTEMS
FIGURE 20.I Euler s meihod.
E X A M P L E2 0 . . l
E u l e r ' sM e t h o d ProblemStotement. Use Euler'smethodto integrate!' - 4ro s' - 0.5.r'from r : 0 to 4 with a stepsizeof 1. The initial conditionat / : 0 is y : 2. Note that the exactsolutioncan be determinedanalvticallvas ^' * e o5r')12n-o5r ) : r r-lrtrut' Solution. Equation(20.5)can be usedto implementEuler'smethod: , ] , ' ( l ): ) , ( 0 )+ J ( 0 , 2 ) ( l ) where 1,(0) :2
andthe slopeestimateat / : 0 rs
. f ( 0 , 2 ) : 4 e o- 0 ' 5 ( 2 ) : 3 Therefbre, ](l):2+3(l):5 The true solutionat r : r :
4
,-,
1 is
r o * r tt tr ( e u s t - e 0 5 ( l ) )+ 2 e - 0 5 n \ : 6 . 1 9 4 6 3
Thus.the percentrelativeerroris t6.19463-51 F, : l-l x 1007o: l9.28Vo 6.19463 l For the secondstep: )(2) : y(l) * /(l. s)(l) : - 5+ f 4 e 0 8 ( r- t 0 . 5 r 5 t l 0 ) :
11.40216
483
METHOD 20.2 EULER'S TABLE20.l
voluesof the integrol of trueond numericol Comporison q1 o f . r ": 4 e ( t 8- t 0 . 5 , vw, i t ht h ei n i t i ocl o n d i t i o tnh o t y : 2 Euler's using computed volueswere r : O.Thenumericol methodwith o stePsizeof I ' Jt"u"
0 I 2 3 4
2 00000 6 19463 I4.84392 3 36 7 7 l 7 75 . 3 3 8 9 6
JEur".
lerl (%)
2 00000 5 00000 t 1.442t6 25 51321 568493l
1 92 8 2319 24 24 24.54
T r u es o l u t i o n
20.2 FIGURE o! me,tf usingEulers solution wiiho numericol solurion of fhe.rrue Comporison -* !,]^"I59|.1 ?t 2 T h J l n i t i oclo n d i l i oont / : O i s i",\;,-- 6 i" f,.r ;:O;4
w 1 ho s i e ps i z eo f I o
1':
relative error is The true solution at t : 2.0 is 14.84392and, therefore,the true percent 1 and Fig' 20'2' 23.l97o.Thecomputationis repeated,and the resultscompiledin Table20' the error Note that althoughthe computationcapturesthe generaltrend of the true solution, using a by is considerable.As discussedin the next section,this error can be reduced smaller step size.
20.2.1 Error Anolysis for Euler'sMethod 4): The numericalsolutionof ODEs involves two typesof error (recallChap' employed l. Truncation,or discretization,errorscausedby the natureof the techniques to approximatevaluesof Y. canbe retained 2. Roundnfi'enofscausedby the limited numbersof significantdigits that by a comPuter. effor The truncationerrors are composedof two parts.The first is a local truncation is The second step' a single over question that resultsfrom an applicationof the methodin the during produced a propagatetl truncation error that results from the approximations
484
INITIAL-VALUE PROBLEMS previous steps.The sum of the two is the total enor. It is referredto as the gbbal truncation error. Insight into the magnitudeand propertiesof the truncationerror can be gainedby deriving Euler's methoddirectly from the Taylor seriesexpansion.To do this, realizethatthe differential equation being integratedwill be of the general fbrm of Eq. (20.3), where dyldt : -y', and t and y are the independentand the dependentvariables,respectively. If the solution-that is, the function describingthe behaviorof y-has continuousderivatives, it can be representedby a Taylor seriesexpansionabout a starting value (/;,.yr). as in [recallEq. (4.13)]: t
]i+r: ti *yitt +*n'+ :!
ln|
" + ' r ' i ' 7 , "* 4 n
(20.6)
nl
where/u : ti+t - /; and Ra : the remainderterm, definedas R,,' -
,,0t*l)191
j------::-/r"+l ( t z* 1 ) !
(20.7)
where f lies somewherein the interval from /; to /; 11. An alternativeform can be developedby substitutingEq. (20.3)into Eqs.(20.6)and (20.7)to yield f'(tj'-Yi) ,,2 * .yr+r: )'i t f (ti,l')h + where O(h'+l) specifiesthat the local truncationetror is proportional to the stepsize raisedto the (n * l)th power. By cornparingEqs. (20.5) and (20.8),it can be seenthat Euler's rnethodconesponds to the Taylor seriesup to and including the term /(r;,,yi)/r. Additionally, the comparison indicatesthat a truncationenor occursbecausewe approxirnatethe true solutionusinga finite numberof termsfrom the Taylor series.We thustruncate,or leaveout, a part of thetrue solution.For example,the tluncationerror in Euler's methodis attributableto the remaining terms in the Taylor seriesexpansionthat were not included in Eq. (20.5). Subtracting Eq. (20.5)from Eq. (20.8)yields
E ,- [ 1 + P h 2+ . . .+ o ( h , + Ii
t20.ql
where E, : the true local truncationerror.For sufficientlysmall lz,the higher-orderterms in Eq. (20.9) are usually negligible,and the result is often representedas
^
F
-
,\r,, .f ) , ), , r
- h '
(20.l0)
or Eo : O(h2)
(.20.1)
where E, : the approximatelocal truncationeffor. According to Eq. (20.ll), we seethat the local error is proportionalto the squareof the step size and the first derivative of the differential equation.It can also be demonstratedthat the global truncationerror is O(/r)-that is, it is proportionalto the stepsize
20.2 EULER'S METHOD
485
(Carnahanet al., 1969).Theseobservations lead to someusefulconclusions: 1. 2.
The global error can be reducedby decreasingthe stepsize. The method wiil provide error-freepredictionsif the underlying function (i.e., the solution of the differential equation)is linear, becausefor a straightline the second derivativewould be zero.
This latter conclusionmakesintuitive sensebecauseEuler's methodusesstraight-linesegments to approximatethe solution. Hence, Euler's method is referredto as a first-order method. It should also be noted that this generalpatternholds for the higher-olderone-step methodsdescribedin the following pages.That is, an nth-ordermethodwill yield perfect resultsif the underlying solution is an rrth-orderpolynomial. Further,the local truncation errorwill be O (h"+1) and the globalerror O (h').
20.2.2 Stobility of Euler'sMethod ln the precedingsection.we learnedthat the truncationerror of Euler's methoddependson the step size in a predictableway basedon the Taylor series.This is an accuracyissue. The stability of a solutionrrrethodis anotherimportantconsiderationthat must be consider-edwhen solving ODEs. A nurnericalsolution is said to be unstableif enors grow exponentiallyfor a problem for which there is a boundedsolution.The stability of a particular applicationcan dependon three f-actors:the differential equation,the numerical method,and the stepsize. Insight into the step size required for stability can be exanined by studying a very simpleODE: dt' -ay ; at If r(0) - )0, calculuscan be usedto determinethe solutionas V :
ltr€
(20.12)
-''
ffrrr, tf," *fution startsat 1,eand asymptoticallyapproacheszero. Now supposethat we useEuler's methodto solve the sameproblernnumerically: - y i + l: l i + ,
d)', ,
S u h s t i t u t i nEgq .t l 0 . I 2 t g i v e s -Yr+l:
ji -
\'ih
or - ) ' i + r : 1 , i ( l- a h )
(20.13)
The parentheticalquantity 1 - ah is called an cunplifictttion foctor. If its absolutevalue is greaterthan unity, the solutionwill grow in an unboundedfashion.So clearly,the stability dependson the step size /r. That is, if /z > 2la,lyil --+ oo as i -+ oo. Basedon this analysis, Euler's methodis said to be conditionall,vstable. Note that thereare certainODEs where errorsalways grow regardlessof the method. SrrchODEs are called ill-conditioned.
486
INITIAL.VALUE PROBLEMS Inaccuracyand instability are often confused.This is probablybecause(a) bothrepresentsituationswhere the numericalsolutionbreaksdown and (b) both are affectedby step size.However,they are distinct problems.For example.an inaccuratemethodcanbe very stable.We will retum to the topic when we discussstilTsystemsin Chap.21.
20.2.3
MATLAB M-file Function: eulode
We have alreadydevelopeda simple M-file to implementEuler's method for the falling bungeejumper problem in Chap.3. Recallfrom Section3.6, that this function usedEuler's nethod to compute the velocity after a given time of tiee fall. Now. let's developa rnore general,all-purposealgorithm. Figure 20.3 showsan M-file that usesEuler's methodto computevaluesof thedependentvariabley over a rangeof valuesof theindependentvariablet . The nameof thefunction holding the right-hand side of the differential equationis passedinto the function as the
FIGURE 20.3 AnMJileto implemeni Euler's method = eulode(dydt, tspan,y0,h,,rarargin) f u n c r ,i o n It,y] B eulode: EUIET O-DE SOIVCT z [ r , y ] . e u .l o d e r d y d t , r s p o n , y 0 , h , p l , p l , . . . , : method to integrate z uses Euler's an ODE % rnput: 96 dydt = name of the M-file Chat evaluates the ODE z tspan = lti, tfl where ti and tf = initial and z final values of independent variable value of dependent variable z y0 = initial 9a h = step size z pI,p2, . . . = additional parameters used by dydt % output : z t - vector of independent variable z y = vector of solution ior dependent variable if
nargin.4,elror = tspan(1);tf ' lLf>ri),error
ri if
(ti:h:tf)';
t
=
z z
if
necessary,
q^
rh)l
1a"-.4
{'ar
least 4 inpuu argumen-s reqL rq.6'r,end tspan(2); {'Jpper lrmit musl be grearer Lhar' .ower'l,end n = length(t-); add an additional value of t -
^^ac
rIOm
L
-
ri
tO
Ct
r (n)
end
);
20.3 IMPROVEMENTS METHOD OF EULER'S
487
variabledydt. The initial and llnal valuesof the desiredrangeof the independentvariable is passedas a vector tspan. The initial value and the desiredstepsizearepassedasy0 and h, respectively. The lunction first generatesa vector t over the desiredrangeofthe dependentvariable using an incrementof t'r.In the eventthat the stepsizeis not evenly divisible into the range, the last value will fall short of the final value of the range.If this occurs,the final value is addedto r so that the seliesspansthe completerange.The length of the r vector is determined as n. In addition, a vector of the dependentvariabley is preallocatedwith n values of the initial condition to improve efficiency. At this point, Euler'smethod(Eq. 20.5)is implementedby a simpleloop: y(i+1)
= y(i)
+ dydt(t(i),v(i))*(t(i+l)-t(i),var);
end
Notice how a function is usedto generatea value for the derivativeat the appropriatevalues of the independentand dependentvariables.Also noticehow the time stepis automatically calculatedbasedon the differencebetweenadjacentvaluesin the vectort. The ODE being solvedcan be setup in severalways.First, the differentialequationcan be definedasan anonymousfuncfionobject.For example,fbr the ODE from Example20.I : > > t i 1 ' c i t = G( t , V ' )
4*exp (0. 3*t)
o.5"yt
The solutioncan then be generatedas >> disP(tt,Yl
)
with the result(comparewith Table20.1): 0 1.0000 2.4400 3.000a 4.0000
2.0000 5.0000 1r.4422 25.4132 55.8493
Although using an anonymousfunction is feasiblefor the presentcase.there will be more complex problemswherethe definition of the ODE requiresseverallines of code.In suchinstances,creatinga separateM-file is the only option.
20.3
TI,IETHOD IMPROVEMENTS OF EULER'S A fundamentalsourceof error in Euler's methodis that the derivativeat tlre beginningof the interval is assumedto apply acrossthe entire interval. Two simple modificationsare availableto help circumventthis shorlcoming.As will be dernonstratedin Section20.4, both modifications(as well as Euler's methoditself) actuallybelongto a largerclassof solution techniquescalledRunge-Kuttamethods.However,becausethey have very straightforward graphicalinterpretations,we will presentthem prior to their forrnal derivationas Runse-Kuttamethods.
488
INITIAL-VALUE PROBLEMS
2O.3.1 Heun's Method One methodto improve the estimateof the slopeinvolves the determinationof two derivativesfor the interval-one at the beginningand anotherat the end.The two derivatives are then averagedto obtain an improved estimateof the slopefor the entire interval.Thisapproach,calledHeun'smethod,is depictedgraphicallyin Fig. 20.4. Recall that in Euler's method.the slopeat the besinnins of an interryal )"i : .f (ti,-vi)
(20.r4)
is usedto extrapolatelinearly to y111:
(20.r5)
))+r:litJ'Gi,Yi)h
For the standardEuler methodwe would stopat this point. However,in Heun'srnethod the r ' , 1 *c,a l c u l a t e c l i n E q . ( 2 0 . 1 5 ) i s n o t t h eat 'ni nsawl e r . b u t a n i n t e r n r e d i a t e p r e c l i c t i o n . T h i s i s why we havedistinguishedit with a superscript0. Equation(20.15)is calledaprcdictorequatiott. It providesan estimatethat allowstlrecalculationof a slopeat the end of the interval:
)'1+r: ,f (rt*t,l'j!,)
(20.l6)
Thus, the two slopes[Eqs.(20.14)and (20.16)] can be combinedto obtainan averageslope for the interval; - , _ / ( r , , - r ' r )* , f ( r r + r , r ' , 9 + r )
2 This averageslope l s then used to extrapolatelinearly frorn -1';to li+r using Euler's method: , f ( r , . y i l ' . l Q i - t . . , ' 'l )' , )rrl:)'ir-------
,)
/l
which is called a corrector eguation.
FIGURE20.4 Grophicol depiciiono[ Heun'smelhod (o) Predictorond (b) correctof
= .f(ri* r. YP*r)
: f(ti,y i)
(20.r7)
489
OFEULER'S METHOD 20.3 IMPROVEMENTS
FIGURE 20.5 of iterotingfhe correctorof Heun'smethodto obloin cn improved Grophicol representotion estimcte.
The Heun method is a pretlic'tor,(orrectorapproach.As just derived, it can be expressedconciselyas Preclictor(Fig.20.4a):
y,o*r: y j " *
Corector (Fig. 20.4b):
y/+r : l'i' I
(20.r8)
f (ti,yi)h
l'(rt,yi)+ / ( t , * , , 1 ' , 1 * ' ' )
h
(20.r9)
(for;:1.2""'m) Note that becauseEq. (20.19)haslial on both sidesof the eclualsign,it can be appliedin an iterativefashionas indicated.That is. an old estimatecan be usedlepeatedlyto provide The processis depictedin Fig. 20.5. an improvedestimateof -v;11. As with similar iterativemethodsdiscussedin previoussectionsof the book, a termination criterion for conversenceof the correctoris frrovidedby
tc
t-
t#l"ouo"
where1,f,r and.v/*1 are the resulttiom the prior and the presentiterationof the corrector, respectively.lt should be understoodthat the iterative processdoes not necessarilyconverge on the true answerbut will convergeon an estimatewith a finite truncationerror,as demonstratedin the following example.
20.2 Heun's EXAMPLE Method 8' Problem Stotement. Use Heun's methoclwith iterationto integrate.]'/ 4e0 - 0.5y f r o m t : 0 t o 4 w i t h a s t e ps i z eo f L T h e i n i t i a lc o n d i t i o n aIt : 0 i s . r : 2 . E m p l o ya s t o p ping criterion of 0.00001c/o to terminirtetlre correctoriterations. Solution. First,the slopeat (16,,rg)is calculatedas ! ' o : 4 e " - 0 . 5 ( 2 ): 3 Then,the predictoris usedto computea valueat 1.0: v9:2*3(t):5
490
INITIAL-VALUE PROBLEMS
TABTE20.2
Comporison of trueond numericol voluesof the integrolol tr - 4e0E 0 . 5 1 'w , i t ht h e i n i t i ocl o n d i t i o tnh o t - y : 2 o t t : O .T h en u m e r i c vo ol l u e s were computedusingthe Eulerond Heunmethods with o stepsizeof 1. TheHeunmethodwos imolemented bothwithoutqnd with iierqtion o[ thecorrector. With lterotion
Without lterqlion Jt.uo
0 I '2 3 4
Jtiur".
200000 6. I 9463 l4 84392 3 3. 6 7 7l 7 75 33896
200000 5 00000 t 140216 2 5. 5 I3 2 1 56 84931
le,l("/"1 l9 28 2319 24 24 2454
l€,1("/"1 2 00000 6 7010B I6 31978 37 19925 83 33777
B.tB 994 la 46 t4.62
.VH"un
le,l ('hl
2 00000 636487 268 I534224 309 3 47 4 3 2 8 3 l l 777351A 3 tB
Note that this is the result that would be obtainedby the standardEuler method.The true velluein Table 20.2 showsthat it correspondsto a percentrelativeerror of 19.28a/o. Now, to improve the estimatefor .v;..1,we usethe value.r'f'to predict the slopeat the end of the interval t . i : . / ( x r , r , l ' ): 4 e 0 8 ( r ) - 0 . 5 ( 5 ): 6 . 4 0 2 1 6 1 which can be combined with the initial slope to yield an averageslope over the interval f r o n rf : 0 t o l : r':
3 + 6.102164 :4.101082 ^ L
This result can then be substitutedinto the corrector[Eq. (20.19)]to give the predictionat t -
l,
t , i : 2 + 4 . 7 0 1 0 8 2 ( :l )6 . 7 0 1 0 8 2 which representsa true percentrelative error of -8.18c/c.Thus, the Heun methodwithout iterationof the correctorreducesthe absolutevalue of the error by a f'actorof about2.4as comparedwith Euler's method.At this point, we can alsocomputean approximateerroras l670roR)-sl l' r" .' r :- l l
6.70r082 l^'""
Now the estimateof r'1 can be refined by substitutingthe new result back into the right-handsideof Eq. (20.19)to give 3 -2 -- ' '> -t.t
.'l
14"o
t r t D- 0 . 5 ( 6 . 7 0 1 0 8 2 ) I : 6 . 2 7 5 8I1
whichrepresents a truepercentrelativeerrorof 1.3I percentandan approximate enor0f il - 6.70r0821 16.2758 l c .l ,: l - l x 6 . 1 7 5 811 | I
1 0 0 1: 6 . 1 1 6 ( / t
20.3 IMPROVEMENTS OF EULER'S METHOD
491
The next iterationgives )l
) ^-
L-r
- 0.5(6.275g1 3 a 4rrt'xrtt 1) I :6.382129
which representsa true error of 3.03voandan approximateerror of 1.666vc. The approxirnateerror will keep droppingas the iterativeprocessconvelgeson a stable final result. In this example,after l2 iterationsthe approximateerror falls below the stoppingcriterion.At this point, the result at r : I is 6.36087,which representsa rrue relative error of 2.687c.Table 20.2 showsresultsfor the remainderof the computafionalong with resultsfor Euler's methodand for the Heun methodwithout iterationof the corrector.
Insight into the local error of the Heun methodcan be gainedby recognizingthat it is relatedto the trapezoidalrule. In the previousexample,the derivativeis a function of both the dependentvariabley and the independent variable/. For casessuchas polynomials. where the ODE is solely a function of the inclependentvariable, the predictor step [Eq. (20.l8)] is not requiredand the correctoris appliedonly oncefbr eachireration.For suchcases,the techniqueis expressedconc.iselyas .l',
-l :
. .ltri)-f [(ri.r).
l', +
/,
2
(20.20)
Notice the sirnilaritybetweenthe secondterrnon the right-handsideof Eq. (20.20)anc1the trapezoidalrule [Eq. (l7.ll)]. The connectionbetweenthe two methodscan be formally demonstratedby startingwith the ordinary differentialequation tl 1'
-: dt
fut
(20.2t)
This equationcan be solvedlbr.r'by integration: f
,,',
dt:l
I J
f t,-t J r,
.t',
frndt
(20.22)
[(t\dt
(20.23)
Ittlrlt
(20.24\
which yields
. r ' i , r- . t ' :,
. \ . i + :l l . i +
ft'tl
J,, ft'+t
/
JI,
Now, recall that the trapezoidalrule [Eq. ( l 7 . l l ) l i s d e f i n e da s
'' ["
f ( t i ) + J Q i + r ), ' f ( r ) d t : ---2
(20.25)
492
INITIAL VATUE PROBLEMS where/r : !i+1 - 4. Subsriruring Eq. (20.25)into Eq. (20.24)yields . l(titl J/rt-.tr---2
l ( t ; ,-1 1 .
(20.26)
tr
which is equivalenttoEq. (20.20).For this reason,Heun's methodis sometimesrefemed t0 as the trapezoidalrule. BecauseEq. (.20.26)is a direct expressionof the trapezoidalrule, the local truncation error is given by [r'ecallEq. (17.14)l J
r
'
f " ( \t \r
'
'
I
(20.27 )
1a
wheref is betweenti afidt;y1. Thus, tl.remethodis secondorder becausethe secondderivative of the ODE is zero when the true solution is a quaclratic.In acldition.the local and global errors are o(h3) and o(lf), respectively.Therefbre, decreasingthe step size decreasesthe error at a lasterrate than fbr Euler's rnethod.
20.3.2 The Midpoint Merhod Figure 20.6 illustratesanothersimple modificationof Euler's method.Called the nitlpoittt ntethod,this techniqueusesEuler's method to predict a value of .y af the miclpointof the interval(Fig.20.6a): , r ' i + 1 ,:/ r ) ' i * J ' ( t , . y | L
(20.28)
Then, this predictedvalue is usedto calculatea slopeat the midpoint: )"t+r1:. : .f (.ti + r lz,t-i +t p)
(20.29)
FIGURE 20.6 Grophico depictionof Heun'smethod.(o) Predictorond (b) corrector v
=
J ( . 1 i 1 11 2),' i 1 1 1 2 )
493
20.4 RUNGE.KUTTA METHODS
which is assumedto representa valid approximationof the averageslope for the entire interval.This slope is then usedto extrapolatelinearly from /1 to /111(Fig. 20.6b): r'r+t :
)'i I
(:0.-r0)
f Q i* t , ' , , ) ' i+ t n ) l t
Observethat because-r';aI is not on both sides,the corrector[Eq. (20.30)]cannotbe applied iterativelyto improvethe solutionas was donewith Heun'smethod. As in our discussionof Heun's method,the rnidpoint rnethodcan also be linked to Newton-Cotesintegrationlbrmulas.Recallfron Table 11.4 that the simplestNewton-Cotes open integrationtbrmula, which is calied the nridpointmethod,can be rcpresentedas rh
t..
I
J'trl dx =- (b - a)J'(r)
(20.31)
Jt
fbr the presenlcase, where,r1is the midpointof the interval(n, D).Usingthe nomenclature h e e x p r e s s e a d s it can
|r l t t t . . f]tdrZh.t'ftia1P) I Jti
Q0.32)
Substitutionof this formula into Eq. 120.24)yields Eq. (20.30). Thus, just as the Heun method can be called the trapezoidalrule, the midpoint method gets its name from the underlyingintegrationformula on which it is based. The midpoint methodis superiorto Euler's methodbecauseit utilizes a slopeestimate at tlre miclpointof the predictioninterval.Recallfrom our discussionof numericaldifTerentiation in Section4.3.4 that centeredfinite differencesarebetterapproximationsof derivatives thaneitherlbrward or backwardversions.In the samesense.a cenleredapproximation suchas F.q.(20.29)has a local truncationeror of O1h:1in comparisonwith the forward approximationof Euler's method,which has an en'orof O (h). Consequently,the local and global erors of the midpoint method areO(h3) andO(h2), respectively.
METHODS 2O.4 RUNGE-KUTTA Runge-Kutta (RK) methods achieve the accuracyof a Taylor series approachwithout requiring the calculationof higher derivatives.Many variationsexist but all can be cast in the generirlizedfbrm of Eq. (20.4): (20.33)
)r+r : )'i l- Qh
slope where@is called an incrcmentfunction, which can be interpretedas a repl'esentative over the interval. The incremerrtfunction can be written in generalform as Q : a r k t I a z k z+ ' ' ' ' - l a , , k , ,
(20.i4)
where the a's are constantsand the k's are (20.34a)
k 1 : J ' Q ; ,t ' i ) l ; . : f ( t i * p 1 h .) ' i * q r r k r l t ) k . , : j ' ( t i - t p z h , - , *Ii q t t k t l t i q n k z h ) : : k , , f ' ( t ; * p , t h , t - i* q , r . r k t hI q u , t . z k z hI
(20.34b) (20.3:lc)
* q,-t.u-:k,,rh)
(20.34d)
494
PROBTEMS INITIAL-VALUE Thatis, wherethep's and 4's areconstants.Notice that the ft's arerecurrencerelationships. ft1 appearsin the equationfor k2, which appearsin the equationfor ftr, and so forth.Becauseeach ft is a functional evaluation,this recurrencemakes RK methodsefficientfor computercalculations. Various types of Runge-Kuttamethodscan be devisedby employing differentnumbers of terms in the increment function as specifiedby n. Note that the first-orderRK methodwith n : I is, in fact, Euler's method.Oncen is chosen,valuesfor the a's,p's,and Thus, q's areevaluatedby settingEq. (20.33)equal to terms in a Taylor seriesexpansion. theorderof at leastfor the lower-orderversions,the numberof termsn usuallyrepresents the approach.For example,in Section20.4.1,second-orderRK methodsusean increment function with two terms (n : 2). Thesesecond-ordermethodswill be exactif thesolution to the differentialequationis quadratic.ln addition,becauseterms with /rr andhigherare droppedduring the derivation,the local truncationenor is O1ft31and the globalenoris O (h2). ln Section 20.4.2,the fourth-orderRK method (n : 4) is presentedfor whichthe g l o b a lt r u n c a t i o e n r r o ri s O t f t l t .
20.4.1 Second-Order Runge-Kuttq Methods versionof Eq. (20.33)is The second-order .li+l :li
l(a.k1 +a2k)h
(20.35)
where ky : k2:
J'Qi' 1t;) -t pth, li i qlhh) f Qi
(20.35c) (20.35b\
The valuestbr ar, a2, pt, and qy1 are evaluatedby settingEq. (20'35)equalto a the second-orderTaylor series.By doing this, three equationscan be derivedto evaluate are (see The three equations 2006, fbr details). Chapraand Canale, four unknown constants a 1I c t 2 : l
(20.36)
a z P r:
l12
(20.37\
a z Q t r:
I12
(20.38)
Becausewe have three equationswith four unknowns,theseequationsaresaidto be We, therefore,must assumea valueof one of the unknownsto determine underdetermined. the other three.Supposethat we specify a value for a2. Then Eqs. (20.36)through(20.38) can be solvedsimultaneouslyfbr (20.39)
ut:l-o:
I
(20.40)
/42
Becausewe can choosean infinite numberof valuesfor a2, thereare an infinitenumber of second-orderRK methods.Every versionwould yield exactlythe sameresultsif the resolutionto the ODE were quadratic,linear,or a constant.However,they yield different most Three of the sults when (as is typically the case)the solution is more complicated. commonlv usedand preferredversionsare presentednext.
20.4 RUNGE.KUTTA METHODS H e u n M e t h o d w i t h o u f l t e r o t i o n( o , : 1 / 2 ) If a2 is assumedtobe ll2, Eqs.(20.39) and (20.40) can be solvedfor a1 : 1/2 ancJ pt : Qu : L Theseparameters,when substitutedinto Eq. (20.35),yield
/t .\'ijr:.t'i+[;ftr \z
+ ljt , -\l1 , -
/
(20.4r )
where f t r: . / ( r , , , r ' i ) -t kth) k2: f ft;* /1,,r,;
(20.41a) (20.4tb)
N o t e t h a t f r il s t h e s l o p e a t t h e b e g i n n i n g o f t h e i n t e r v a l a n d k 2 i s t h e s l o p e a t t h e e n d o f t h e interval. Conseqr-rently, this second-orderRunge-Kutta nethod is actually Heun's technique without iferationof the corrector. T h e M i d p o i n t M e t h o d( o z: and Eq. (20.35)becomes
1 1 . l t a 2 i s a s s u m e d t o bl ,et h e n a 1: 0 , p r : 4 t r : l 1 2 ,
.l'i+r: )'i'l k2h
(20-42)
where / < 1:
/(t;, r,;)
k2: f fti + h/2, J'i* krhl2)
(20.42a) (20.42b)
Thisis themidpointmethod. Rqlston'sMethod lo, : 2/3). Ralston(1962)and Ralstonand Rabinowitz(1978)determined that cltoosingo? :213 provides a minimum bound on the truncationeror for and RK algorithrus.For this version,(11: lf3 and 7r1:7tt:314, the second-order Eq. (20.35)becomes
: r',+ (]t, +f,rr) n .r',+r
(20.43)
where f t 1: / ( , t ; , r ' ; )
(20.13a)
k t : l ( t , n ' o r , r , ,l+r ' l )
(20.43b)
20.4.2 ClqssicolFourth-OrderRunge'KuttoMethod there The most popularRK methodsarefourth order.As with the second-orderapproaches, are an inlinite numberof versions.The following is the most commonly usedtbrm, and we thereforecall it the classicalfourth-orderRK ntetlrcd: . v , + :i ) ' i *
I
*2k:*2h*k+)h U(kr
(20.44)
496
PROBLEMS INITIAL-VALUE
i
*'/-*
a: a: a.
u: a'.
/'
"l
,!-)' ,tI
'.k'
*,
-t-l+
\Alo
\
FIGURE20.7 RKmelhod thefourth-order comprising of theslopeestimotes C.pt-,i.otdepiction
where k1: f fti,!i)
(20.44a)
kz: f (',*)r,t, +!;',n)
(20.44b)
t.: f (t,*)r,r,+1,,n)
(20.44c)
ka:f(tith,y,+hh)
(20.44d)
Notice that for ODEs that are a function of / alone, the classical fourth-order RK method is similar to Simpson's1/3 rule. In addition,the fourth-orderRK methodis similar to the Heun approachin that multiple estimatesof the slope are developedto comeup with an improved averageslopefor the interval.As depictedin Fig. 20.7, eachof theft's representsa slope.Equation(20.44) then representsa weightedaverageof theseto arrive at the imProvedsloPe. E X A M P L E2 0 . 3
r KM e t h o d C l o s s i c q lF o u r t h - O r d eR Problem Stqtement. Employ the classicalfourth-order RK method to integratey'= 4 r 0 8 t- 0 . 5 y f r o m / : 0 t o l u s i n g a s t e p s i z e oI f w i t h y ( 0 ) : 2 . Solution. For this case,the slopeat the beginningof the interval is computedas :
(0.2\ -
-
: 3
20.4 RUNGE-KUTTA METHODS
497
This valueis usedto computea valueof .r and a slopeat the midpoint: l'(0.5):2+3(0.5):3.5 k r : l ' ( 0 . 5 .3 . - 5 :) 4 c 08 ( 05 )- 0 . - 5 ( 3 . 5: ) 4 . 2 1 1 2 9 9 This slope in turn is usedto computeanothervalue of ,vand anotherslopeat the midpoint: r(0..5): 2 + 1.211299(0.5): 4. 108649 5) - 0.-5(4.108649): 3.912974 k7 : .f (0.5,4.108649)- 1e08(0 Next. this siopeis usedto computea valueof .r,and a slopeat the end of the interval; , y ( 1 . 0 ): 2 + 3 . 9 1 2 9 1 4 ( 1 . 0: ) 5 . 9 1 2 9 7 4 k a : f ( 1 . 0 , 5 . 9 | 2 9 7 4- ) 4 e 0 8 ( t 0-)0 . 5 ( 5 . 9 1 2 9 7 4 ) : 5 . 9 4 5 6 7 7 Finally,the four slopeestimates arecombinedto yield an averageslope.This averageslope is thenusedto rnakethe final predictionat the end of the interval. Q:
u
[3 + 2(1.2112e9)+ 2(3.912911) + 5.945611] : 4.201031
.\'(1.0) : 2 + 4.201031(1.0) : 6.201031 which cornpares favorablywith the true solutionof 6.19463I (e, :0. 103Vo).
It is certainly possibleto developfifth- and higher-orderRK methods.For example, Butcher's (196,+)fiflh-order RK methodis written as )'i+r : )'i *
I
,(7kr
* - l l A r + 1 2 k 1 * - l 2 A s+ 7 k ) h
(20.45)
where k 1: f G i , ) , i ) k::
.t(,, * ),,-r, - ]l a/
(20.45a) l)
(20.4sb)
t, : /(,' * Io,r',+ jr'r + jr,r,)
(20.45c)
k+: .r(,, * + )0,t, \r,'n hh)
(20.4sd)
rr5:y(ti*ton.,., + ftr,n+ft*.n) r o : f ( , , * h , t --i) r , , + 2 r r ' n + |r.n - lr^o +]r,rn)
(20.45e) (20.4sf)
Note the similaritybetweenButcher'smethodandBoole'srule in Table17.2.As expected, this rnethoclhas a global truncationerror of O (h5). Although the fifth-order versionprovidesmore accuracy,noticethat six tunction evaluationsarerequired.Recall thatup throughthefourth-orderversions,n flnction evaluations
PROBLEMS INITIAL.VALUE
498
are requiredlbr an nth-orderRK method.Interestingly,for ordershigher than four,oneor Becausethefunctionevaluationsacc0unt two additionalfunction evaluationsarenecessary. relfive and higherare usuallyconsidered methods of order time, for the most computation for the main reasons This is one of the versions. fourth-order less than the efficient atively popularityof the fourth-orderRK method.
20,5 SYSTEMS OF EQUATIONS Many practicalproblemsin engineeringand sciencerequirethe solutionof a systemof simay multaneousordinary differentialequationsratherthan a singleequation.Suchsystems be representedgenerallyas d)'t i
: / 1 ( t . . r ' 1 . . r ' . . . \. t. i.)
4z : J : ( r .l t . ) . : . . . . .) t , )
(20.46)
" : ,t,, :!:! dt
:.f,,(1.
. r . r .) . : . . . . . ) ) i )
The solution of such a systemrequiresthat n initial conditionsbe known at the stafting value of /. An exampleis the calculationof the bungeejumper's velocity and positionthatwe set at the beginning of this chapter.For the free-fall portion of the jump, this problem up amountsto solving the following systemof ODEs:
dx ' dt du dt
(20.47) wat -6
)
(20.48)
m
If the stationaryplatform from which the jumper launchesis defined as r : 0, the initial conditionswould be x(0) : u ( 0 ) : 0 .
20.5.1 Euler'sMerhod All the methodsdiscussedin this chapterfor single equationscan be extendedto systems In of ODEs. Engineeringapplicationscan involve thousandsof simultaneousequations. the involves applying simply of equations each case,the procedurefor solving a system one-steptechniquefor every equationat eachstepbeforeproceedingto the next step.This is best illustratedby the following examplefor Euler's method. E X A M P L E2 0 . 4
S o l v i n gS y s t e m so f O D E s w i t h E u l e r ' sM e t h o d Problem Stotement. Solvefor the velocity andpositionof the free-fallingbungeejumper usingEuler'smethod.Assumingthat at / : 0, r : u : 0, and integrateto / : l0switha stepsizeof 2 s.As was donepreviouslyin Examples1.1and 1.2,the gravitationalacceleration is 9.8I m/sr, and thejumper hasa massof 68.I kg with a dragcoefficientof 0.25kg/m.
20.5 SYSTEMS OFEAUATIONS Recallthat the analyricalsolurionfor velocityis tEq. (1.9)l: i::^
t
i-l-
'
r,(/) : l!!! 61nn(lE!!,\ V c.i
\V
,z )
This result can be substitutedinto Eq. (20.41) which can be integratedto dererminean a n a l y t i c asl o l u t i o nf o r d i s t a n c a es Gr, ll
m f t . r ( I ) : - I n l c o s h { . / a rl rl cl L \V,n'J) Use theseanalyticalsolutionsto conpute the true relativeerrorsof the results. Soluiion. The ODEs can be usedto computethe slopesat / :0
as
dr -:U dt
du :::9.81 dt
0.25 -::::(142:9.81 68.1
Euler's methodis then usedto computethe valuesat t : 2 s, .r:0*0(2):0 u : 0 * 9 . 8 1 ( 2:)1 9 . 6 2 The analyticalsolutionscanbe computedas.t(2) : 19.16629and r'(2) : 18.72919. Thus, the percentrelativeeffors are 1007oand4.1567a,respectively. The processcan be repeatedto computethe resultsat t : 4 as ,r:0*19.62(2):39.24 u:
/ 0.25 .\ 19.62+ f 9.8t - lf' r 19.62)2 )z : 30.+t.lo8 68.t \ l
Proceedingin a like mannergives the resultsdisplayedin Table 20.3.
TABTE2O.3 Distonce ond velocityof o freeJolling bungeeiumperos computed numericolly with Euler'smethod. rt.u"
0 2 4 6 B t0
0 1 91 6 6 3 7t 9304 147.9462 2 3 75 ) 0 4 3 3 4) 7 8 2
ur"u*
00 t 8.7292 33 il tB 42 4762 469575 49.4214
rEul".
0 39.240A I t2 A674 2A4 664A 3 0 54 2 4 4
uEul".
o r9 6200 36.4137 46 2983 50 I 802 5r 3t23
e, (x)
r 0 00 0 % 45 45% ?425% ]3 83% B72%
e, (a)
476% 9.97% 1003% 6.86% 3 B3:d
soo
INITIAL.VALUEPROBLEMS for Although the fbregoingexampleillustrateshow Euler'smethodcanbe implemented In addition, large step size. systemsof ODEs, the resultsarenot very accuratebecauseof the the resultsfor distanceare a bit unsatisfyingbecause;r does not changeuntil the second iteration.Using a much snrallerstepgreatlymitigatesthesedeficiencies.As describednext, using a higher-ordersolverprovidesdecentresultsevenwith a relativelyiargestepsize.
20.5.2 Runge'Kutfo Methods of Note rhat any of the higher-orderRK methodsin this chaptercan be appliedto systems helpful in is equations.However.caremust be takenin determiningthe slopes.Figure 20.7 develop is, we first visualizing the proper way to do this for the fourth-ordermethod.That 's) Theseslopes(a setof ft1 arethenusedto make slopesfbr all variablesat the initial vah-re. predictionsof the dependentvariableat the midpoint of the interval.Thesemidpointvalues are i1 turn usedto cornputea setof slopesat the midpoint (the ft2's).Thesenew slopes are then taken back to the startingpoint to rnakeanotherset of midpoint predictionsthat leaclto 1ew slopepredictionsat the midpoint (the l-1's).Theseare then employedto make predictionsat the end of the interval that are usedto developslopesat the end of the interval (the fta's). Finally, the fr's are combined into a set of increment functions [as in Eq. (20.41))rhat are brought back to the beginningto make the final predictions.The fbllowing exarnpleillustratesthe approach. EXAMPLE20.5
Solving Systemsof ODEs with the Fourth-OrderRK Method Problem Stotement. Use the fburth-orderRK methodto solve fbr the sameproblemwe addressedin Example 20.4. Solution. First, it is convenientto expressthe ODEs in the functional lbrmat of Eq. (20.46)as dx
E 0Il
,tr
: f i ( t , , r ,u ) : 1 , Ct
: .fzj, x, u) : I - :u' tn
t
The first stepin obtainingthe solutionis to solve for all the slopesat the beginningofthe interval: kr.r :
0. 0) : 0 J'r(O,
k 1 . :2 1 2 ( 0 . 0 ,0 ) : 9 . 8 1- 9 J 1 t': {'t,l 2 : 9 . 8 1 68.1 where ki..; is the lth value of ft for theTth dependentvariable.Next, we must calculatethe first valuesof .r and u at the rnidpointof tlre first step:
r ' ( l ): r ( 0 )r f , . , ! : O + O i : o 22
u ( r ): u ( o + ) r r . r t:]o + 9 . 8 1 ]: n . t t
20.5 SYSTEMS OF EQUATIONS
501
which can be usedto computethe first set of midpoint slopes: k z : : . f t( 1 ,0 , 9 . 8 1 ): 9 . 8 1 0 0 k 2 . 2: f 2 ( 1 . 0 ,9 . 8 1) : 9 . 4 5 6 1 Theseare usedto deternine the secondsetof midpoint predictions:
: 19 . 8 l o o . r ( l ): . r ( 0 )* t . . , ! : o + 9 . 8 1 0 0 ;; u ( l ) : u ( 0 )+ t z : i
0+9.4567;:9.4561
whichcanbeused,o.oilpur.thesecond'set of midpointslopes: : 9.4561 kt.t : .fr(1.9.8100, 9.456't) :9.4817 k3..: fi(1, 9.8100, 9.4561) Theseareusedto determine thepredictions at theendof theinterval: x(2) : r(0) + k3.1h 0 + 9.4567 (2) : 18.9134 u ( 2 ) : u ( 0 )+ k 3 . 2 l t : 0 + 9 . 4 8 1 1 ( 2 )1: 8 . 9 6 3 4 wltichcanbe usedto computetheendpointslopes: : )18.9634 k q . t: J t ( 2 ,1 8 . 9 1 3 1. 18,. 9 6 3 4 : : ka.2 t'.(2,18.9134, 18.9634) 8.,1898 Tlrevaluesof t canthenbe usedto compute[Eq.(20.4a)l: x ( 2 ): 0 +
I t 0 + 2 ( 9 . 8 1 0+09 . 4 5 6 1+) 1 8 . 9 6 3 4 1 2 : 1 9 . 1 6 5 6 6
u \ 2 ): 0 + ; [ 9 . 8 1 0 0 + 2 ( 9 . 4 5+6 9 7 . 4 8 1 7+) 8 . 4 8 9 8 1 2 :1 8 . ] 2 5 6 6Proceedingin a like manner for the remaining stepsyields the values displayedin Table 20.4. In contrastto the resultsobtainedwith Euler's method, the fourth-orderRK predictionsare much closerto the true values.Further,a highly accurate,nonzerovalue is computedfor distanceon the first step.
TABLE2O.4
Distonce ond velocityof o freeJolling bungeeiumperos computed numericolly with thefourth-order RKmethod.
rt.u"
0 2 4 6 8 t0
0 1 91 6 6 3 71 9304 147.9462 2 3 75 1 0 4 3 3 41 7 8 2
ut.u"
0 )B7292 331|8 42.0762 46.9575 4 44 2 \ 4
JnKr
0 19.r656 7 ) . 9 3 )I t47 9s21 2 3 75 t O 4 3 3 41 6 2 6
unx{
0 I8.7256 330995 42.4547 46.9345 49.4427
e,(x)
e, (u)
0.004% 0 00r% 0 004% 0 000?{ 0 005%
0.0r 9% oo37% 0 051% a a49iL 0 038%
502
INITIAL-VALUE PROBLEMS
20.5.3
MATLAB M-file Funclion: rk4sys
Figure 20.8 showsan M-file called rk4 sys that usesthe fourth-orderRunge-Kuttamethod to solve a systemof ODEs. This code is similar in many ways to the functiondeveloped earlier (Fig. 20.3) to solve a singleODE with Euler's method.For example,it is passed the function name defining the ODEs throughits argument.
FIGURE2O.8 AnMJilelo implemenl iheRK4meihod foro system oltCDEs. € , 1 h - F ; ^ h
--
f f ^ . ' ^ l t L I l ' y . | , l
I ^ ^ - h L - I J a r t '
- l t a ' t - , J . ' i ' ! N l b y 5 \ u l q L ,
. , 4 y v '
L t t t
' . - ' r - ^ i - \ r r r r v o r a r v
g rk4sys: fou:-th-order Runge-Kurta for d system ol ODts = rk4sys(dydt,tspan,y0,h,p1,p2,...) % : int.eqrates lt,yl Z a system of ODEs wit.h fourth-order RK meEhod ? inpuL: ?' dydt = name of the M-file that evaluaLes the ODES tspan = tti, ? t.fl; initial times wiLh output. and final generaLed ac interval Z of h, or = ltO t1 ... tfl; % times where solution output specific y0 = initial ? values of dependent variables % h = step size p1 ,p2, . .. = additi"onal parameters used by dydt % % output: % tp = vector yp = vector %
of of
it
nargin.4,error('ar
r!
arrJ -
h
landfh/f
\ F + 1, -^-LrlJarrr
F
-
_ _
redsL 4 - _O
I
a:
/^r
/
input 'rcn:r LJPcrr
= tspan(n) ;
f
L = (ti:h:tf)'; n = length(t); if t (n)
argumenrs requ ired' t, end not ascendi ng order') , end
c^rr\
= tspan(1,);tf
ti I
I );
independent variable solution for dependent variables
;);
.
while(1) Eend = t(np+1); hh = g(nP+1) - t(nP);
(Continued)
503
OF EAUATIONS 20.5 SYSTEMS
i f hh>h, h]-i = h; end while(1) i f ct+hh>tend, hh = tend-tt,'end : })' ; :),vararqin{ kl = dydt (tt.y1i, ymid = y(1,:) + kL.*hrh,./2; : ] )' ; k2 = dydt |.tt+hY\/2,ymid,varargini ymid = y(i,:) + k2*hrh/2; k3 - dydt (tt+hh,/2,ymid,vararqrin{ : } )' ; +k3*hh; Vei-rd=y(i,:) ; } )' ; k4 = dydL(tL+hh,yend,varargin{ phi = (k1+2*(k2+k3) +k4) /6; = y(i,:) y(i+l,:) + phi*hh; r ts
ts |
rlih
.
if tt>=Lend, brea]:, end end np = np+1; tp(nP) = tt; YP(nP,:) i f tt>=t f, break, end end
= Y{1,:)
FfGURE20.8 lContinued)
However,it has an additionalfeaturethat allows you to generateoutput ln two ways, dependingon how the input variabletspan is specified.As was the casefbr Fig. 20.3,you can set rspan = tri rf l, whereri arld tf are the initial and final times,respectively. lf done in this way, the routine automaticallygeneratesoutput valuesbetweentheselimits at equalspacedintervalsh. Alternatively,if you want to obtainresultsat specifictimes.you can define r span = i t 0 , t 1 , . . . , t f I . Note that in both cases,the L span valuesmust be in ascendingolder. We can employ rk4sys to solvethe sameproblemas in Example20j. First,we can developan M-file to hold the ODEs: f unction cty = dydtsys (t, y) ^21; d y = l V ( 2 , \; 9 . B \ - 0 . 2 ' : / 6 8 . 1 * Y ( 2 ) wherey ( 1) : distance(:r)and y 12) : velocity (u). The solutioncan thenbe generatedas
>>
disp(
[1-' ]'(:,1)
0 2.0000 4.0000 6.0000 8.0000 10.0000
Y( t,2))\
0 19. r656 1L.93II !47.952r a??
(141
334.1 526
0 tB.12a6 33.0995 42 .0541 45.9344 49 .4427
504
INITIAL.VALUE PROBLEMS
We can also use t span to generate results at specific values of the able. For example, >> tsPan=t0
5 101;
>> disp(lt'
y(:,1)
y(:,2)))
000 6.0000 141.9521 10.0000 334.1626
42.A5t11 49.4021
PREDATOR.PREY MODELSAND CHAOS Engineers and scientistsdeal with a variety of problems involving sysBockgrcund. tems of nonlinear ordinary differential equations.This case study focuseson two of these applications. The first relates to predator-preymodels that are used to study speciesinteractions.The secondare equationsderived from tluid dynamics that are usedto simulatethe atmosphere. Predator-prey models were developedindependentlyin the early part of the twentieth century by the Italian mathematician Vito Voltena and the American biologist Alfred Lotka. These equations are commonly called Lotka-Volterra equations.The simplestversion is the following pairs of ODEs: dx
(20.49)
E:ax-oxY dv
(20.50)
E--'Y+axY
where x and y - the number of prey and predators,respectively,a = the prey growthrate, c : the predator deathrate, and b andd : the ratescharacterizingthe effect of the predatorprey interactions on the prey death and the predator growth, respectively.The multiplicative terms(that is, thoseinvolving xy) are what make suchequationsnonlinear. An example of a simple nonlinear model basedon atmosphericfluid dynamicsis the Lorenz equations createdby the American meteorologist Edward Lorenz: dx - - 6 x - o y
E
u
--rx-)-Jcl.
dt
dz :-bz*xy ,tt Lorenz developed theseequations to relate the intensity of atmospheric fluid motionr to tAmhArotrrra
rrariotinnc
1r enrl
7 in
thp
qnA
20.6 CASESTUDY
505
coniinued with the predator-preymodel, the nonlinearitiesstem from the simple multiplicative terms: xz and xy. Use numerical methods to obtain solutions for these equations. Plot the results to visualize how the dependentvariableschangetemporally. In addition, graph the dependent variables versus each other to seewhether any interestingpatternsemerge. The following parametervalues can be used for the predator-prey simulaSolution. tion: a : 1.2,b :0.6, c: 0.8, andd: 0.3. Employ initial conditionsof _r:2 andy : I and integratefrom t: 0 to 30, using a step size of h:0.0625. First, we can develop a lunction to hold the differential equations: function yp = predprey {t,y, a,b. c,d) y p = [ a * y ( 1 ) - b * y ( 1 ) * y ( 2 ) ; - c " y ( 2 ) + d * y ( L J* y ( 2 ) ] ; The following script employs this function to generate solutions with both the Euler and the fourth-order RK methods.Note that the function eulersys was basedon modifying the rk4sys function (Fig. 20.8).We will leavethe developmentof suchan M-file as a homework problem. In addition to displaying the solution as a time-series plot (.r and y versus /), the script also generatesa plot ofy versus x. Such phase-plane plots are often useful in elucidating featuresof the model's underlying structure that may not be evident from the time series. h = 0 . 0 6 2 5 ; t s p a n = t 0 4 0 1; y 0 = [ 2 i l i a = L . 2 ; b = 0 . 6 ; c = 0 . B; d = 0 . 3 ; I t y l = e u l e r s y s ( @ p r e d p r e y ,L s p a n , y 0 , h , a , b , c , d ) ; ) subplot (2,2,L) ;plot(t,y(:,L),t,Y {:,2),'--' (a) Euler tine plot' ) legend('prey','predaror' ) ;title(' subplot \2,2,2) ;plot (y( :, 1),v ( :,21 | title (' (b) Euler phase plane plot') tt V) = rk4sys(GpredPreY,tspan,y0,h,a,b,c,d) ; subplot (2,2,3 ) ;plot(t,y ( , ,1) , t ,Y (: ,2 ) , '--') title (' (c) RK4 rime plot') subplot \2,2,4 ) ;plot (y ( : ,1) ,V (: ,21 j title ( ' (d) RK4 phase plane plot') The solution obtained with Euler's method is shown at the top of Fig. 20.9. The time series (Fig. 20.9a) indicates that the amplitudes of the oscillations are expanding. This is reinforced by the phase-planeplot (Fig. 20.9b). Hence, theseresults indicate that the crude Euler method would require a much smaller time step to obtain accurateresults. In contrast, because of its much smaller truncation error, the RK4 method yields good results with the same time step. As in Fig. 20.9c, a cyclical pattern emergesin time. Because the predator population is initially small, the prey grows exponentia\. At a certain point, the prey become so numerous that the predator population begins to grow' leads Eventually, the increasedpredatorscausethe prey to decline. This decrease,in turn, the expected' as that, to a decreaseofthe predators.Eventually, the processrepeats.Notice
PROBTEMS INITIAL.VALUE
s06 ti, :i,
.
i
-+"t '.ii*
continued
10 20 30 ( a )E u l e tri m ep l o t
, II
I I I
I ll tl
tl tl ll t t
I I I I I
I I I
t\
I
r I
,
lt tl
,] I I
I I I t
10 20 30 (c)RK4time plot
5 10 ( b ) E u l e rp h a s ep l a n ep l o t
I I I I
t I I I
40
24 (d) RK4phaseplaneplot
FIGURE2O.9 Solution fortheLoiko-Volterro model. Euler's method ond{b)phose-plone plots, ond {o)fime-series (cJtime-series plots. RK4meihod ondld)phcse-plone
predator peak lags the prey. Also, observe that the processhas a fixed period-that is, it repeatsin a set time. The phase-planerepresentationfor the accurateRK4 solution (Fi9.20.9Q indicatesthat the interaction between the predator and the prey amounts to a closed counterclockwise orbit. Interestingly, there is a resting or critical point at the center of the orbit. The exactlocation of this point can be determined by setting Eqs. (20.49) and (20.50) to steadystate (dy ldt : dx /dt : 0) and solving for (x, y) = (0, 0) and (cld, alb). The formeris the trivial result that if we start with neither predatorsnor prey, nothing will happen.Thelatter is the more interesting outcome that if the initial conditions are set at x:cld and : populations the derivatives will zero, will remain constant. a be and the ! /b, Now, let's use the sameapproachto investigatethe trajectoriesof the Lorcnz equations with the following parametervalues:a = 10,b = 8I 3,and r : 28. Employ initial conditions
20.6 CASESTUDY
507
confinued Lorenz model "{ versus r
20 E
15
5 . 0 0 1r,' - : : 5
10
-5 -10 -
tc
-20 FIGURE20.IO Time-domoin representotion of x versus t for fheLorenz equotions. Thesolidtimeseriesis for the initiolconditions for.r is perturbed {5, 5, 5).Thedoshedlineis wheretheinitiolcondttion slighily i s 0 0 r, 5 , 5 )
of"r:.)':z:5andintegratefrom/:0to20.Forthiscase,wewillusethefourth-order RK methodto obtain solutionswith a consranttime stepof h :0.03125. The results are quite different from the behavior of the Lotka-Volterra equations.As in Fig. 20.10,the variable.r seemsto be undergoingan almostrandompatternof oscillations, bouncing around from negative values to positive values. The other variablesexhibit similar behavior. However, even though the pattems seem random, the frequency of the oscillation and the amplitudesseemfairly consistent. An interestingfeatureof suchsolutionscan be illustratedby changingthe initial condition for "r slightly (from 5 to 5.001).The resultsare superimposedas the dashedline in Fig. 20.10.Although the solutionstrack on eachother for a time, after about / : l5 they diverge significantly.Thus, we can see that the Lorenz equationsare quite sensitiveto their initial conditions.The term chaotic is used to describesuch solutions.In his original study, this led Lorenz to the conclusionthat long-rangeweatherforecastsmight be impossible!
508
INITIAL.VALUE PROBLEMS
continued (a) y versusr
10 n
(b) z versus-r
45 40 35 30
JU
a25
25 20
-10 1q
-20
10 5 -20
(c) . versus!
45 40
l5
-10
0 "r
10
10 5 -40
-20
0 )
20
FIGURE20.I I represeniot;on Phose-plone for fhe LorenzeqL.rotions (oJrr, (bl .r:, ond (cJyl proiections
The sensitivityof a dynamicalsystemto small perturbationsof its initial conditions is sometimescalled the butte(Iy ffict.The idea is that rhe flapping of a butterfly'swings might induce tiny changesin the atmospherethat ultimately leads to a large-scaleweather phenomenonlike a tornado. Although the time-series plots are chaotic, phase-planeplots reveal an underlying structure. Because we are dealing with three independent variables, we can generate projections.Figure 20.11 showsprojectionsin the x.y,xz, and the yz planes.Noticehowa structure is manif-estwhen perceivedfrom the phase-planeperspective.The solutionforms orbits around what appearto be critical points. Thesepoints are called strangeattractorsin the jargon of mathematicianswho study suchnonlinearsystems. Beyond the two-variableprojections,MATLAB's p1or3 function providesa vehicle to directly generatea three-dimensional phase-pianeplot: >> plot3 (y(: ,L) ,y(:,2) ,y(:,2) ) >> xlabel('x' ) ;ylabel('y' ) ;z1abe} ('2, ) ;grid
As was the casefor Fig. 20.1i, the three-dimensional plot (Fig20.12) depictstrajecrories cycling in a definite pattem arounda pair ofcritical points. As a final note, the sensitivity of chaotic systemsto initial conditions hasimplications for numericalcomputations.Beyond the initial conditionsthemselves,differentstepsizes or different algorithms (and in some cases,even different computers)can introducesmall differencesin the solutions.In a similar fashion to Fig. 20.10, thesediscrepancies will eventuallylead to large deviations.Someof the problemsin this chapterand in Chap.21 are designedto demonstratethis issue.
20. intt res
(a) (b) (c) (d)
PROBLEMS
conrinued
,l*lin
K,
509
t'\
30
--l
20
\l
10
--l
I I
ltI l-I I
-\|
v0
tI
_10
rl
-t_ t-
-20 -2n
40
- 4 0 -20
FIGURE2O.12 Three-dimensionol phose-plone representoiion forlheLorenz equolions generoted withMATLABs p}ot 3 iunction.
PROBLEMS 20.1 Solve the following initial value problem over the interval frorn / : 0 to 2 where r'(0) : l Display all your resultson the samegraph. dr'
-:\,/--l
20.2 Solve the following problem over the interval from -r : 0 to I usinga stepsizeof 0.25 where-l (0) - l. Display all your resultson the sameglaph. -:-
l\.
-- " r I I r ' r) '. v\ /
/i;
,1r
dt
(a) (b) (c) (d)
Analytically. Using Euler'smethodwith /i : 0.5 and 0.25. Using the midpointmethodu'ith ft : 0.5. Usingthe fburth-orderRK methodwith i : 0.5.
(a) (b) (c) (d) (e)
Analytically. Using Euler'smethod. Using Heun's methodwithout iteration. UsineRalston'smethod. Using the fburth-orderRK method.
5ro
INITIAL.VALU E PROBLEMS
2{).3 Solve the iollowing problem over the interval fi'om I :0 to 3 using a stepsizeof 0.-5wherey(0) : l. Display all your resultson the samegraph.
This is referredto as the logistic model.The analyticalsolution to this modelis y-P\l
d:"
Pmax
Po * (2nr""
Po)e
kc^t
dt
Simulate the world's population from l950 to 2050 using (a) the analytical solution, rnd (b) the fourth-orderRK Obtain your solutions with (a) Heun's method without method with a step size of 5 years. Employ the tbllowing iteratingthe conector, (b) Heun's method with iteratingthe initial conditionsand parametervaluesfor your simulation: correctoruntil s, < 0.1%. (c) the midpoint method, and px (in 1950):2,55-5 million people,kgtn:0.0261yr,and (d) Ralston'smethc)d. pmrr : 12,000million people.Display your resultsasa plot has rrany en20.J The growth of populationsof or,ganisms along with the datafiorn Prob. 20.4. gineeringand scicntiflc applications.One of the simplest 20.6 Supposethat a projectileis launchedupwardflom the rnodelsassumesthat the rate ofchange ofthe population7' is earth'ssurface.Assumethat the only force actingon theobproportionalto the existing populationat any time /: ject is the downward force of gravity. Under theseconditions. a fbrce balancecan be usedto derive dLt (P20.4.1) - : kpp ul dt, K-dlf)\o"'(R+.r12 tlt where ftq : the growth rate. The world population in millions from 1950throush2000was
where u : upward velocity (m/s), t : time (s),x: altitude (m) measuredupward from the earth's surface,g(0) : the r 1 9 5 0 r 955 I 960 r 965 lsTa 1 9 7 5 gravitationalaccelerationat the earth'ssurface(? 9.8rnAr). p 2 5 5 5 2 7 8 0 3 0 4 0 3 3 4 6 3743 4AB7 e n d R : t h e e l r t h ' sr a d i u s{ } 6 . 3 7 z 1 0 6m ) . R e c o g n i t i n g that dxldt: u. use Euler's method to deterrninethe I t 9 B 0 t 9 B 5 I 990 I 995 2000 maximum height that would be obtained if u(t = 0; = p 4 4 5 4 4 8 5 0 5276 5686 6079 1400m/s. 20,7 Solve the lbllowing pair of ODEs over the interval (a) Assumingthat Eq. (P20.4.1)holds, use the data from frorr r : 0 to 0.4 using a step size of 0. L The initial conditions are .y(0) : 2 and ;(0) : 4. Obtain your solutionwith I 950 throughI97Oto estimateks. (b) Use the fburth-orderRK method along with the results (a) Euler's methodand (b) the fourth-orderRK method.Disof (a) to stimulate the world population from 1950 to play your resultsas a plot. 2050 with a ste-psizeof 5 yeals.Display your simulation (/l' :-21.t4et resultsalongrvith the dataon a plot. dl 20.5 Although the model in Prob. 20.4 works adequately -yawhen population growth is unlinrited,it breaksdown when ,h 3 factors such as fbod shortages,pollution, and lack of space inhibit growth. ln such cases,the growth rate is not a con20.tt The van der Pol equation is a model of an electronic stant.but can be formulatedas circuit that aroseback in the days of vacuumtubes: k, : kr,r(l - p/p,,,,,,) where t,rr, : the maximum growth rate under unlimited conditions, p - population, and /max: the rnaximurn population.Note that 2n,", is sontetimescalled the curn-ing t'upucin. Thus, at low popr"rlationdensity p {{ p^u,, k, -+ kr,,. As p approaches7r,,,"*.the growth rate approacheszero. Using this growth rate fbrmulation, the rate of changeof populationcan be modeledas dp dt
: * c , , ( l - p f 1 t , , , , 1, *t )
.t2,,
---(l at'
.t,,
-.r'') . f t.:0 dt
Given the initial conditions,1(0) : )'(0) : 1, solvethis equationfrom 1 : 0 to 10 using Euler's methodwith a step sizeof (a) 0.2 ancl(b) 0.i. Plot both solutionson thesame grrph. =6, 20.9 Given the initial conditions,)(0): I and 1,'10; solve the fbllowing initial-valueproblem from t : 0 to 4: r/l l -lQ\.-O dtt
ot
for Pl so 20 m( pl( po 5' 2tJ mi of uIr 5. 2A fb al Pr 2( Er
rh
2( pe is b1
ta Si h:
5lt
PROBLEMS Obtainyour solutionswith (a) Euler's rr.rethodand (b) the fourth-orderRK method.In both cases,usea stepsizeoi 0. I . Plot both solutionson the same graph along with the exact solution.t' : cos34. 20.10 Developan M-file 1tlsolvea singleODE with Heun's methodwith iteration.Design the M-file so that it createsa plot of the results.Test your programby using it to solvefor populationas describedin Prob. 20.5. Employ a step size of 5 1'eiirsand iteratethe correctol'until c., < 0' l7o. 20.11 Develop an M-tile lo solve a single ODE with the miclpointmethod.Designthe M-file so that it createsl plot of theresults.Testyour prograrnby using it to solvefbr population as describedin Prob. 20.5. Employ a step size of
FIGURE P2O.I5
(a) tntegratethe Lotka-Volterraequations (Section 20.6) liom 1960through2020 using the tbllowing coeflicient 5 years. v a l u e sa: : 0 . 2 3 , 0 : 0 . 0 1 3 3t,: : 0 . ' 1 a, n d d : 0 . 0 0 0 4 . 20.12 Develop an M-file to solve a single ODE with the your simulation with the data using a tinreCompare tburth-orderRK method.Design the M-file so that it creates plot series and determinethe sum of the squaresof the it solve a plot of the results.Tesl your program by using to your model and the data for both the residuals between l. size 0. a step of Prob.20.2.Employ the wolves. nrclose and 20.13 Develop an M-file to solve a systemof ODEs with (b) Developa phase-plane plot olyour solution. Euler'smethod.Designthe M-tile so that it createsa plot ol a damped spring-nrass system of motion The 20,15 20.7 theresults.Test your programby using it 1osolveProb. (Fig. P20.l5) is describedby the fbllowing oldiniuy difl-er$ith a stepsizeof 0.25. archi- entialequation: 20.14 Isle RoyaleNationalPark is a 2[0-square-mile pelagocomposedof a sinele large islandand many snrall dx d2x lr-,;+t - +k.i:0 islandsin Lake Superior'.Moose arrived around 1900, and LltdI vegeravaging b1 l9-10.their populationrtpprouched.i000. (m)' t : tation.In 19.19,wolves crossedan ice bridge from Ontario. wherex : displacementfrom equiliblium position : coefficient :20-kg damping (s), c the and mass, rn Sincethe late 1950s,the numbersof the mooseand wolves time (N . s/m). The damping coefficientc takes on three values havebeen tracked.
Yeor
I 959 I 960 196l 1962 I963 1964 I 965 1966 1967 I968 1969 t97a ) 9 7l 1973 1974
Moose
563 610 628 639 663 747 733 765 9)2 la42 I 268 )2c)5 I 439 1,193 143-5 1467
Wolves
2A 22 22 23 20 26 2B 26 22 22 l7 IB 20 23 24 3l
Yeor 5 197 1976 1977 t97B 1979 I980 r 9 8I t9B2 I9 8 3 lsB4 I985 I986 l9B7 ]9BB I989 r 990
Moose
I001 I028 9r0 863 872 932 I038 ll t5 ) t92 t268 r3 3 5 I 397 1216
Wolves
4) 44 34 40 43 50 30 l4 23 24 22 2A t6 t2 t2 l5
Yeor l 991 1992 1993 1994 I 995 1996 1997 I 998 I 999 2000 200l 2AO2 2003 lvJ4
2005 2046
Moose
t3r3 r590 lB79 l77A 2422 I 163 500 699 750 B5O 900 I 100 400 7 5Ct 544 450
Wolves
12 l2 t3 l7 t6 22 24 l4 25 2a t9 t7 l9 29 30 30
INITIAL-VALUE PROBLEMS
512
Supposethat the temperatureof the corpsewhen it wasdiscoveredwas 29.5 "C, and that two hours later,it is 23.5"C. The ambient temperatureis 20 'C. (a) DetermineK and the time of death. (b) Solve the ODE nunerically and plot the results. 20.18 The reaction A --+ B takes place in two reactorsin series.The reactorsare well mixed but are not at steady massbalancefor eachstirredtank state.The unsteady-state reactoris shownbelow: ICA' I :-(CAu-CA:)-kCAr ; atr varl(
dCB, I : -CBtIkCAt ;
FIGUREP2O.I6 A sphericol tonk.
aIt
J
dCA, I --j:-(CAt-CA)-kCAt of 5 (underdamped),40 (critically damped),and 200 (overdamped).The spring constantk : 20 N/m. The initial velocity is zero, and tlre initial displacement,r : I m. Solve this equationusing a numericalmethodover the time period 0 < / < 15 s. Plot the displacementversustirne for eachof the three valuesof the dampingcoefficient on the sameplot. 20.16 A sphericaltank has a circular orifice in its bottom through which the liquid flows out (Fig. P20.16).The flow rate through the hole can be estimatcdas
dt
T
dCBt ----:-- :
I -tCBt - CB:-l - kCB:
dl
T
where CA6 : concentration of A at the inlet of the first reactor.CA 1 - s6nqsrtration ofA at the outlet of the first reactor (and inlet of the second),CA2 - ssnssrtrationofA at the outlet of the secondreactor. CB r : concentrationof B at the outlet of the first reactor(andinlet of the second), CB2= concentrationof B in the secondreactor,r : residence time : the rate constantfor reactionofA to each fbr reactor. and k AnEii C Q.,r: produceB.lf C:Aois equalto 20, find the concentrations of,4 where Qou,: outflow (mr/st. C : an empirically derived and B in both reactorsduring their first 10 minutesof operacocfficient.A : the areaof the oril'ice(mr).3 : the gnrvita- tion. Use t : 0. l2lmin and r : 5 nrin and assumethatthe lional constant(:9.81 m/srt.und h: the depthof liquid in initial conditions of all the dependentvariablesarezero. the tank. Use one of the numericalmethodsdescribedin this 20.19 A, nonisothermal batch reactor can be describedby chapterto determinehorvlong it will takefor the waterto flow the following equations: out of a 3-m diametertank with an initial height of 2.15 m. Note that the orifice hasa diameterof 3 cm and C: 0.55. dC _ --_r\-t)/(T+273)) g 20.17 ln the investigationof a honricideor accidentaldeath, dl it is often important to estimatethe time of death.From the dT -;experimeutalobservations,it is known that the surfacetem: 1 0 0 0 e ( - r 0 / ( r + 2 7 3 ) ) C- l 0 ( T - 2 0 ) dI peratureof an object changesat a rateproportionalto the differencebetweenthe temperatureof the object and that of the where C is the concentrationof the reactantand I is thetemsurrounding environment or ambient temperature.This is peratureof the reactor.lnitially, the reactoris at I 5 andhas "C known as Newton's law of cooling. Thus, if f(/) is the tem- a concentrationof reactantC of 1.0 gmoyl-. Find theconcenperatureof the object at time /, and i".,is the constantambi- tration and temperatureol' the reactor as a function of time. ent temperature: 2(1.20The following equationcan be usedto modelthededT . :-K(T
flection of a sailboatmast subiectto a wind force:
-7.)
dt
t )
. .
(1-)'
where K > 0 is a constantof proportionality.Supposethat at time I : 0 a corpse is discoveredand its temperatureis nreasuredto be I,. We assumethat at the time of death,the oa-
-
-
tl:: where /(l)
./ (a ) -
-1,
-
)
11-
?E I : wind force, E : modulus of elasticity, I = -^A
I
-
ff^'.^n+
^f
:-^-1:-
\T^r^
]L^*
+L^.^-^^
Calc Use I: 20.2 Und diff(
! (
whe A(h gra\ outl area min h(0" h,n A(h
20.2 gair enc sho this stru dev
5r3
PROBLEMS
-_-_t_ e
d
FIGUREP2O.2I varieswith height accordingto 200;
"
n: = 8,000 kg
) ,1,,
L--]"il1\
ftr
1800kN/m
k.
2400kN/m
k1
3000kN/m
-/ T.,
C a l c u l a tteh e d e f l e c t i o ni f r ' : 0 a n dd v l d z : 0 at z = 0. Ure paranreterl'alues of L :30, E : 1.2-5x 108, arrd / : 0.05 for your computation. 20.21A ponddrainsthrougha pipeas shownin Fig. P20.21. Undera numberof simplifying assumptions. the fbllowing differentialequationdescribeshow depthchangeswith time: .tt. utt -:
--\
-,t) .tu
h, ffr
6
A(h), m2
I l7
543210 a97
A67
445
A32 01B 0
20.22 Engineersand scientistsuse mass-springmodelsto gain insight into the dynamics of structuresunder the influence of disturbancessuch as earthquakes.Figure P20.22 shorvssuch a representationfbr a three-storybuilding. For this case,the analysisis limited to horizontal motion of the canbe structure. UsingNewton'ssecondlaw,forcebalances developedfor this systemas d--tr
ltr
-rrr
,+:
dl'
Ilt|
( r{ 2 l- 1.
t ^) ---:(.\r
-
=
dt: t ) t
A'x3 /1t-
tn: K1 . m1
qT0-\-/ 7 r r= 1 2 1 0 0 0k g
qTn'-r
f--., /-q\tl Le)
' 1A(h)' u,her-e /i : depth (m), r: time (s),al: pipediarneter(nr), A\h): pond surf'aceareaas a functionoldepth (m2t.g : grar.'itational constant(: 9.81 m/s2),and c : depthof pipe outletbelow the pond bottom (m). Basedon the following area-depthtable, solve this diff'erentialequation to determine how long it takes tbr the pond to empty, given that l , ( 0 ) : 6 r n , d : 0 . 2 5m , e : I m . dr
In2= 10,000kg
(l * j('r: -.rr)
]tl|
- -\\)-
t. ^l -{.tt
m2
-
)l)
FIGUREP2O.22 Simulatethe dynamicsof this structurefrom , : 0 to 20 s, siven the initial condition that the velocity of the ground floor is d.r1/th : I m/s, and all other initial values of displacementsand velocitiesare zero. Presentyour results as two time-seriesplots of (a) displacements and (b) velocities. phase-planeplot of In addition,developa three-dimensional the displacements. 20.23 Repeattlre the sarnesirnulationsas in Section 20.6 for the Lorenz equationsbut generatethe solutionswith the rnidpointmethod. 2(1.24Perfornl the samesimulationsas in Section 20.6 for the Lorenz equationsbut use a value of r :99 .96. Compare vour resultswith thoseobtainedin Section20.6.
AdoptiveMethods ond StiffSystems
CHAPTEROBJECTIVES The primary objectiveof this chapteris to introduceyou to more advancedmethods for solving initial-valueproblemsfor ordinary differentialequations.Specific objectivesand topics coveredare r r o o . o
2I.I
Understandinghow the Runge-KuttaFehlbergmethodsuse RK methodsof difl'erentordersto provide error estimatesthat are useclto adjustthe stepsize. Familiarizing yourselfwith the bLrilt-inMATLAB lunctions for solvingODEs. Learninghow to adjustthe optionsfor MATLAB's ODE solvers. Learninghow to passparametersto MATLAB's ODE solvers. Understandingthe differencebetweenone-stepand multistepmethodsfor solving ODEs. Understandingwhat is meantby stiffnessand its implicationsfor solvingODEs.
METHODS ADAPTIVERUNGE.KUTTA To this point, we have presentedmethodstor solving ODEs that employ a constantstep size. For a signiticant nuinber of problems,this can representa selious limitation.For in example,supposethat we are integratingan ODE with a solution of the type depicted gradually. Fig. 21.1.For most ofthe range,the solutionchanges Suchbehaviol'suggests that a fairly large step size could be employedto obtain adequateresults.However,for a localizedregion from t : | .l 5 to 2.25,the solutionundergoesan abruptchange.Thepractical consequenceof dealing with such functions is that a very small stepsizewouldbe required to accuratelycapturethe impulsive behavior.If a constantstep-sizealgorithm were employed,the srnallerstep size requiredfor the region of abruptchangewouldhave a much smallerstepsizethan to be applied to the entire computation.As a consequence. of necessary-and, therefore,many more calculations-would be wastedon the regions gradualchange.
I
21.1 ADAPTIVE RUNGE-KUTTA METHODS
5r5
F I G U R E2 I . I A ^ e x o m oe o i o s o- r ' o r o f o n C D I t h a le x h i b i t so n o b r u p tc h o n g e .A u t o m o i i cs t e p s i z e nli,r
hn< nrnnt
^l,r^^,^^o.
{^,
suchcoses.
Algorithms that automaticallyadjustthe stepsizecan avoid suchoverkill and hencebe of great advantage.Becausethey "adapt" to the solution'strajectory,they are said to have adaptivestep-sii.econtrol. Implementationof suchapproachesrequiresthat an estimateof the local truncationerror be obtainedat each step.This error estimatecan then serveas a basisfor either shorteningor lengtheningthe stepsize. Betbre proceeding,we should mention that aside from solving ODEs, the methods describedin this chaptercan also be usedto evaluatedefinite integrals.The evaluationof the definite integral rb
_t
t:
I
J$tdx
Ja
is equivalentto solving the differentialequation dy -/rr, , ,o, ,lr'Urgiventhe initial conditiony(a) : 0. Thus, the fbllowing techniques can be employed to efficiently evaluate definite integrals involving functions that are generally smoothbut exhibit regionsof abruptchange. There are two primary approachesto incorporateadaptivestep-sizecontrol into onestepmethods.Stephalving involves taking eachsteptwice, once as a full stepand then as
516
ADAPTIVE METHODSAND STIFF SYSTEMS two half steps.The differencein the two resultsrepresentsan estimateof the localtruncation eror. The step size can then be adjustedbasedon this error estimate. In the secondapproach.calledetnbeddedRK methods,the local truncationenor is estimatedasthe differencebetweentwo predictionsusingdiff'erent-order RK methods.These are cun'entlythe methodsof choicebectrusethey ale more efficient than stephalving. The embeddedmethodswere first developedby Fehlberg.Hence.they aresometimes refen'edto as RK-Fehlbergmethods.At facevalue,the ideaof usingtwo predictionsof different order might seemtoo computationallyexpensive.For example,a fburth- andfifthorder predictionamountsto a total of I 0 function evaluationsper step[recallEqs.(20.44) and 120.45)1.Fehibergcleveriy circur.nvented this problernby deriving a fifth-orderRK methodthat employsmost of the samefunctionevaluationsrequiredfor un accompanying fourth-orderRK rnethod.Thus, the approachyieldedthe error estimateon the basisof only six functionevaluationsl
2l.l.l
MATTABFunctionsfor Nonstiff Systems
Since Fehlbergoriginally developedhis approach.other even betterapproaches havebeen developed.Severalof theseare availableas built-in functionsin MATLAB. ode23. The ode23 function usesthe BS23 algorithm (Bogacki and Shampine,1989; Shampine,1994),which simultaneouslyusessecond-and third-orderRK fbrmulasto solve the ODE and make error estimatesfor step-sizeadjustment.The formulas to advancethe solution are J'i+r : li -t
I n(.2h
-t 3kz 1- kt)lt
( 2r . 1 )
where
t1 : ./(r;,,r;)
(2Lla)
r,:f(,,*)r,r,
lr,r)
(2t.tb)
t r : f ( r ,* ' r r , r ,
ir")
( 21 .c1)
The error is estimatedas Er+t :
I * (*5kr -t 6kz + 8k1- gk+)h
(71.2)
/l
where k q : . f ( t i + t ,) ' i + t )
(2t.2a)
Note that althoughthereappearto be fbur function evaluations,thereare really only three becauseafter the first step,the k1 for the presentstepwill be the ftafrom the previousstep. Thus, the approachyields a predictionand error estimatebasedon threeevaluations rather
2I .I ADAPTIVE RUNGE-KUTTA METHODS
5r7
than the five that would ordinarily result tiom using second-(two evaluations)and thirdorder (tlrreeevaluations)RK formulasjn tandem. After each step,the error is checkedto determinewhetherit is within a desiredtolerance.If it is, the value of y;a1 is accepted,and k+ becomesk1 lbr the next step.If the error is too large. the stepis repeatedwith reducedstepsizesuntil the estimatederror satisfies E < n t a x ( R e l T oxl l y l . A b s T oIl
(21.3)
j) where RelTol is the relativetolerance(default: l0 andAbsTol is the absolutetolerance (default : l0-6). Observethat the criteriafor the relativeeffor usesa fractionratherthan a percentrelativeerror as we havedone on many occasionsprior to this point. ode45. The ode45 functionusesan algorithmdevelopedby Dormandand Prince(1990), which simultaneouslyusesfburth- andfifth-order RK formulasto solvethe ODE and make error estimatesfbr step-sizeadjustment.MATLAB recommendsthat ode45 is the best function to apply as a "first try" for most problems. odeL13. The ode113 functionusesa variable-order Adams-Bashforth-Moulton solver.It is useful for stringentemor tolerancesor computationallyintensiveODE functions.Note in Section21.2. that this is a multistepn'rethod as we will describesubsequently Thesefunctionscan be calledin a numberof differentways.The simplestapproachis Ir,
- t . ' 1= o c l e 4 5 ( o d e f u n , t - s p a n , - r . ' d )
wherey is the solutionarraywhereeachcolumn is one of the dependentvariablesand each row corresponds to a time in the column vector t, odefun is the name of the function returninga column vectorof the right-hand-sides of the difTerentialequations,t span specifies the integrationinterval,and y0 : a vectorcontainingthe initial values. Note that . s-pancan be lbrmulatedin two ways.First. if it is enteredas a vectorof two numbers. f-spar -
Ltt
tf);
the integrationis perfornredfrom ti to tf. Second.to obtain solutionsat specific times t0, tl tn (all increasingor all decreasing), use t-qpal
=
ltA
t1
tn);
Herc is an example of how ode45 can be used to solve a single ODE. .r": 4e08t- 0.-5.r, fiom /:0 to 4 with an initial conditionof y(0):2. Recallfrom Example 20.I that the analyticalsolutionat t:4 is 75.33896.Representing the ODE as an anonymousfunction, ode4 5 can be usedto generatethe sameresultnumericallyas >> dydt=G(t,y) 4*exp(0.8*L) -0.5*y; >:, It,y1=66"4!.(dydt,lA 4),21 ; >> y(Length(tt
)
75.3390 As described in the following example, the ODE is typically stored in its own M-file when
5r8 EXAMPLE2l .l
ADAPTIVE METHODSAND STIFF SYSTEMS
Using MATLABto Solve o Systemof oors ProblemStotement. Employ ode45 to solvethe following setof nonlinearODEsfrom t:0to20: dt''
dv,
- 0 . 8 1 ' *, 0 . 3 y 1 y 2 ;:l.2ytE: where .r'1: 2 and.1'2: 1 at 1 : 0. Suchequationsarerefered to aspredator-pre)'cquations. 0.6)rlz
Solution. Before obtaininga solutionwith MATLAB, you must createa functionto compute the right-handside of the ODEs. One way to do this is to createan M-file as in r uflCLi on yp = pr eopi e/ ( L,y) *y\2\ = l'1.2*y lI) 0.6*y(1)
yp
;-A.B"y(2)+0.3*y(1)
*y(2)
I ;
We storedthis M-file underthe name:predprey. m. Next, enter the following commandsto specify the integrationrange and the initial conditions: '. spdn fn 20 ; >> ya = 12, 7l; The solvercan then be invoked by
This command will then solve the differential equationsin predprey.m over the range definedby tspan usingthe initial conditionsfound in yo. The resultscanbe displayed by simply typing >1
plot(t,y)
whiclr yieldsFig. 21.2.
F I G U R 2EI . 2 Solutionof predotor-prey moo'elwithMATLAB
5r9
METHODS RUNGE-KUTTA 2I .I ADAPTIVE
In addition to a time seriesplot, it is also instructiveto generatea phase-planeplotthat is, a plot of the dependentvariablesversuseachother by >> plot (y( :, 1),v (:,2) ) w h i c h v i e l d sF i s . 2 l . 3 .
F I G U R2 EI . 3 modelwiihMATLAB Sloiespoceplotof predofor-prey
?.
!.2
1
oot
?
y1
to control varAs in the previousexample,the MAILAB solverusesdefaultparameters ious aspectsofthe integration.In addition,thereis alsono controloverthe differentialequaTo havecontroloverthesefeatures,additionalargumentsareincludedasin tions' parameters. = ode45(odefun, tspan, yA, options'
lt,yl
p1' p2,"')
where options is a datastructurethat is createdwith the odeset function to control feathat you want to passinto odefun. turesof the solution,andpJ , p2, .. . areparameters The odeset function has the generalsyntax options
= odeset
('parr'
,vaf,,'par
'
,vaf
' - -.)
where the parameterpar, has the value vaf .. A completelisting of all the possibleparameterscan be obtainedby merely enteringodeset at the commandprompt. Some commonly usedparametersare 'RelTol
'
'AbsTol '
,rnirialsrep' 'MaxSf
'
Allows you to adjustthe relativetolerance. Allows vou to adiustthe absolutetolerance the initial step.This optionaldetermines The solverautomatically your own. you to set lows The maximum stepdefaultsto one-tenthof the t span interval.This
520
EXAMPLE 2I .2
ADAPTIVE METHODSAND STIFF SYSTEMS
U s i n g o d e s e t t o C o n i r o l I n t e g r o t i o nO p t i o n s ProblemStotement. Use ode23 to solvethe followine ODE,from I : 0 to 4: /t al
2 dt
:
10e
{r l)
[ 2 r 0 . 0 7 5| r-
0.6t
where1(0) :0.5. Obtainsolutionsfor the default(10-3)and for a more stringent(l0r) relative error tolerance. Solution. First,we will createan M-file to computethe right-handsideof theODE: function yp = dydt (t, y) y P = 1 O * e x p( - ( L - 2 I * ( t - 2 ) / ( 2 * . A 7 5 ^ 2 ) ) 0 . 6 * y t Then, we can implementthe solverwithout settingthe options.Hencethe defaultvaluefor the relativeenor ( l0 3) is automatically used: > > o d e 2 3 ( @ c 1 y d r ,t A 4 l , 0 . 5 ) ; Note that we have not setthe function equalto output variables tr , y I . When we implement one of the ODE solversin this way, MATLAB automaticallycreatesa plot of the resultsdisplayingcirclesat the valuesit hascomputed.As in Fig. 2 l .4a, noticehow ode23 takes relatively large stepsin the smoothregionsof the solufion whereasit takessmaller stepsin the region ol'rapid changearoundI : 2. We can obtailt a more accuratesolutionby using the ocleser function to settherelat i v e e r r o rt o l e r a n c teo l 0 a : >>
optlons=odeset
>>
ode23(@dydt,
( 'ReLTol t0,
4),
'
, Ie-4) ; 0.5, options);
As in Fig. 2l.4b.the solvertakesmore snall stepsto attainthe increasederccuritcy.
F I G U R 2EI . 4 Solution of CDEwithMATLAB For(b),o smoiler relotive errortoleronce isuseoononence monv no'esteo:oerol,er'.
( b ) R e l T o l= 1 0 - a
2I .2 MULTISIEP METHODS
21.2
52r
MUITISTEPMETHODS The one-stepmethodsdescribedin the previous sectionsutilize information at a single point tr to predict a value of the dependentvariable.y,+rrt a future point 1;11(Fig. 21.5a). Alternative approaches, calledmultistepmethods(Fig. 21.5b), arebasedon the insightthat. once the computation has begun, valuable infbrmation from previous points is at our command.The curvatureof the lines connectingthesepreviousvaluesprovidesinformation regardingthe trajectoryof the solution.Multistep methodsexploit this informationto solve ODEs. In this section,we will presenta simple second-ordermethod that servesto demonstratethe generalcharacteristics of multistepapproaches.
21.2.1 The Non-Self-SforfingHeun Method Recallthat the Heun approachusesEuler'srnethodas apredictor[Eq. (20.15)]: . ) ) + r: ) ' i * f Q i , v ) h
(21.4)
rule as a corector [Eq. (20.17)]: and the trapezoidal - 1 ' ; 1 g:
J'; *
f (t,.y,)+ f (t,*,,y!*,) n
( 2 1. s )
Thus, the predictorand the correctorhavelocal truncationerors of O(/rr) and O(h3), respectively.This suggeststhat the predictoris the weak link in the methodbecauseit has the greatesterror.This weaknessis signilicant becausethe efficiency of the iterativecorrector stepdependson the accuracyof the initial prediction.Consequently,one way to improve Heun'smethodis to developa predictorthathasa local errorof Othi). This can be
FIGUR2 EI . 5 ond{b)multlstep beiween drfference of thefundomentol Grophicol depiction {o)one-step merhodsfor solvingODLs
522
ADAPTIVE METHODSAND STIFFSYSTEMS accomplished by usingEuler's methodand the slopeat,'!;, and extrainformationfroma p r e v i o u sp o i n t l ' r , r , a s i n lj+r :
(2r.6)
-l'r-r * .f (ti,t'i)2h
This fornrulaattainsO(h3) at the expeuseof employinga largerstepsize2h. In addition, note that the equationis not self-startingbecauseit involvesa previousvalue'of thedependent variable)r-r. Sucha valuewould not be availablein a typicalinitial-valueproblem. Becauseof this fact, Eqs. (21.5) and (21.6) are called the nort-selfsterting Heunmethod. As depictedin Fig. 21.6,thederivativeestimatein Eq. (21.6) is now locatedat themidpoint ratherthan at the beginningof the interval over which the predictionis rnade.This centering inrprovesthe local errorof the predictorto O1h31.
F I G U R2 EI . 6 A grophicol depiction of lhenonself-sfirrling Heunmethod(o)Themidpolnt method thotisused rrlethotisemployed oso predictor osc corrector {b)Thetropezoidol S l o p e- / ( r , * , . r , f * , )
S4-ri
EXA
2I .2 MULTISTEP METHODS
523
Thenon-self-starting Heunmethodcanbe summarized as (Fig.21.6a): f'j'*r: .)';'lr+ t' (t,, yi)Zn Predictor '
Corrector (Ftg.2l .6b):
1 l/+r : _1',ltr
(2t.1)
J ( r ' ' ' 1 " -) ' t (' t " ' ' t / ' " )'
/r
( 2 1. 8 )
(for.l:1,2,""nr) where the superscriptsdenotethat the correctoris applied iteratively fiom 7 : I to la to obtain refined solutions. Note that -r'j" and y'l_., are the final results of the corrector iterationsat the previoustirne steps.The iterationsare terminatedbasedon an estimateof the approximateerror, I t i-l -'i+l lo t_1.'/+l r.trr-li .!i+ r I
( 2r . 9 )
x l00lo
When lenI is lessthana prespecifiederror tolerancee., the iterationsareterminated.At this p o i n t ,T : m . T h e u s eo f E q s .( 2 1 . 7 )t h r o u g h( 2 1 . 9 ) t os o l v ea n O D E i s d e r r r o n s t r a tiendt h e followine examnle. E X A M P L E2 l . 3
N o n - S e l f - S t o r t i nHge u n ' sM e t h o d Problem Stotement. Use the non-self-starting Heun methodto performthe samecomputationsas were peltbrmedpreviouslyin Example20.2 using Heun's method.That i.s, - + n " * ' - 0 . - 5 t f r o m / : 0 t o 4 w i t h a s t e ps i z eo f l . A s w i t h E x a m p l e2 0 . 2 , integrate l" the initial conditionat / : 0 is -t': 2. However,becausewe arenow dealingwith a multistepmethod,we requirethe additionalinfbrmationthat1 is equalto -0.3929953 at / : -1. Solution. The predictortEq. (2 1.7)lis usedto extrapolate linearlyfrorn r : -l to 1: .r| : -0.3929953+ [4"'r'iu' - 0.5(2)] 2 : 5.607005 The conectorIEq. 121.8)]is then usedto conpute the varlue:
vi:2+
- 1 " 0j l i t t -r 0 . 5 ( 2 )J - l e r ) x ' I ' - t - t . s t s . o o z o o - s t
I : 6.-s493-31
2
which representsa true percentrelativeerror of -5.139o(true value :6.194631). This error is somewhatsmallerthanthe valueof -8.1Solcincun'edin the self-startinsHeun. Now. Eq. (21.8)canbe appliediterativelyto irnprovethe solution:
3 1 4.,osrtr- 0.5(6.54933 l)
l^ .tl
-
z-T
1
I : 6.313149
which representsan error of -1.927o.An approximateestimateof the error can be determined usingEq. (21.9): l" Lr((ir-
- 6.54933 6.313749 r 6.313149 | lx |
r 0 0 ":/3" . J 7 c
524
ADAPTIVE METHODSAND STIFFSYSTEMS Equation(21.8)canbe appliediterativelyuntil e,,falls belowa prespecified valueofe..As was the casewith the Heun method (recall Example 20.2), the iterationsconvelgeon a value of 6.36087(et : -2.68%). However,becausethe initial predictorvalueis more accurate.the rnultistepmethodconvergesat a somewhatfasterrate. For the secondstep,the predictoris sitt - 0.5(6.36087)2: .]'l': 2 * l4ett
13.44346
€t :9.43a/c
wlrich is superior to the prediction of 12.0826(et : l87o) rhar was computedwith the original Heun method.The first correctoryields 15.76693(e, - 6.8vc),and subsequent iterationsconvergeon the sameresultas was obtainedwith the self-startingHeunmethod; 15.30224(t, : -3.097o).As with tlre previousstep.the rate of convergenceof theconector is somewhatirrprovedbecauseof the betterinitial prediction.
21.2.2 Error Estimqtes Aside from providing increasedefficiency,the non-sell-startingHeurrcan alsobe usedto estimatethe local truncationernrr.As with the adaptiveRK methodsin Section21.1,the error estimatethen providesa criterionfbr changingthe sfepsize. The error estintatecan be derivedby recognizingthat the predictoris equivalent to the midpointrule. Hence,its localtruncationerroris (Table17.4) t,, I 6 'r. :1-- /1r ' . r ' ' " ' { € , , t : - l t ' f " t € , , )
(21.10)
where the subscript7r designatesthat this is the error of the predictor.This errorestinrate can be cornbinedwith the estimateof )'i+ r from the predictorstepto yield
Truevalue:t.,1r+
fir,.u,,,r{,,r
(21.il)
By recognizingthat the correctoris equivalentto the trapezoidalrule, a similarestimate of the local truncationerror lbr the correctoris (Table 17.2) ll E, : -ih
) ' t ' ( { ,) :
--ht
l"l€, 1
(2t.l2l
Thiserrorestimate canbecombinedwith theconectorresult.1';11 to give Truevalue: )',1r- ift3.r.(3){6,)
(21.13)
(2I.1I) .un b. ,ub,ructed Equation fromEq.(21.13) ro yield
- r,9+r 0: .y,1ir
*r,tr,t',U,
(21.14)
where{ is now between r; 1 andf . Now,dividingEq.(21.14)by 5 andreanangingthe resultgives ) , 9 +-r. r i i ' : - 1 7 r - r u r r r 1 6 ) <
It
(2r.15)
525
2I .3 STIFFNESS
Notice that the right-handsidesof Eqs. (21.12) and (21.15) are identical,with the exception of the argumentof the third derivative.If the third derivativedoesnot vary appreciably over the interval in question,we can assumethat the right-handsidesare equal, and therefore,the left-handsidesshouldalso be equivalent,as in E ' 5-
.,0
- ) i-t
_
1'ri
ri-t
(l l.16)
Thus, we have arrived at a relationshipthat can be usedto estimatethe per-steptruncation error on the basis of two quantitiesthat are routine by-productsof the computation:the predictor(1,9*r) anAthe corrector("vlir) .
EXAMPLE 21.4
Error Estimqte Truncotion of Per-Steo Problem Stotemenf. Use Eq. (21.16) to estimatethe per-steptruncationerror of Example 21.3. Note that the true valuesat / : I and 2 are6.194631and 14.84392,respectively. Solution. At /;a1 : l, the predictorgives5.607005and the comectoryields 6.360865. Thesevaluescan be substitutedinto Eq. (21.l6) to give
- 5.607005 6.360865 : -0.150722
F_
which compareswell with the exacteffor, Et : 6.194631- 6.360865:
-0.1662341
At /;11 :2, the predictorgives 13.44346and the correctoryields 15.30224,which can be usedto comDute F
-_
- 13.44346 15.30224 : -0.31176
which also compares favorably with the exact error, Et : 14.84392- 15.30224: -0.45831.
The foregoing has been a brief introduction to multistep methods. Additional information can be found elsewhere(e.g., Chapra and Canale, 2006). Although they still have their place for solving certain types of problems,multistep methods are usually not the method of choice for most problemsroutinely confrontedin engineeringand science.That said, they are still used.For example,the MATLAB function odel13 is a multistep method.We have thereforeincluded this sectionto introduceyou to their basic principles.
2r.3
STTFFNESS Stiffnessis a specialproblem that can arisein the solution of ordinary differentialequations. A stiff systemis one involving rapidly changingcomponentstogetherwith slowly changing ones. In some cases,the rapidly varying componentsare ephemeraltransients that die away quickly, after which the solutionbecomesdominatedby the slowly varying
526
ADAPTIVE METHODSAND STIFF SYSTEMS
F I G U R 2EI . 7
D l o 'o r o , l r [ s o i t ' o n o r o r i n g l eC D L Althoughlhe solutionoppeorsto slorlot l, thereis octuo ly o fost ircnsienifrom y : g 1e I thct occursin lessthnn the 0 OO5lime unil.Thistronsienl . p e ' c e pi b l e e n l , w h e n h e r e : p o n s eis viewed on the finertimescolein the insel.
components.Although the transientphenomenaexistfor only a shortpart of theintegration interval,they can dictatethe time steplbr the entire solution. Both individual and systemsof ODEs can be stilT.An exampleof a singlestitTODEis ,/ r' ;
:
- 1 0 0 0 r '* . 1 0 0 0 - 1 0 0 0 e '
If I (0) : 0. the analyticalsolutioncan be developedas . l : 3 0 . 9 9 8 e - l { ) { x- t 2 . o o 2 e '
(2r.17)
(21.18)
As in Fig. 21.1,the solutionis initially dominatedby the fast exponential term(e-rmr). After a shortperiod (t < 0.005), this transientdies out and the solutionbesomesgoverned b y t h e s l o we x p o n e n t i a( le / ) . Insight into the stepsize requiredfor stability of such a solutioncan be gainedby expart of Eq. (21.17): aminirrgtlrehor.nogeneous dt'
(2t.t9)
dt
If f'(0) - )t), calculuscan be usedto determinethe solutionas -t' : -1'ue
ttl
Thus, the solution startsat ).0and asymptoticallyapproacheszero. Euler'smethodcan be usedto solvethe sameproblemnumerically: -]i+t :)'it
dt,, , n dt
S u h s t i t u t i nEgq .t 2 | . l 9 t g i v e s .l't+l :)'i
-alih
EX
2I .3 STIFFNESS
527
or . \ ' i + r:
t'i(l - ah\
e| .20)
The stabilitl'of this formulaclearlydependson the step sizeh.That is. ll - a/zl mLrstbe l e s st h a n l . T h u s ,i f h : , 2 / a , l . y r l - - o o a si - + o o . For the fast transientpart of Eq. (2 I . I 8 ), this criterioncan be usedto show that the step size to maintainstabilitymust be <2/1000:0.002. In addition.we sl.rouldnote that, whereasthiscriterionmaintainsstability(i.e.,a boundedsolution).an evensmallerstepsize would be requiredto obtainan accuratesolution.Thus,althoughthe transientoccurs1oronly a sntallfi'actionof theintegrationinterval,it controlsthemaximumallowablestepsize. Ratherthan usingexplicit approaches, implicit methoclsof'teran alternativeremecly. Such representations are called intplicit becausethe unknown appearson both sidesofthe equation.An implicit forrn of Euler's method can be developedby evaluatingthe derivative at the future tirne: -\','l:\';-|-
. d.r.;rr, 'dt
ll
This is calledthe baclovutd,or inrltlicir,Euler'smethotl.Substituting Eq. (2 I . l9) yielcls ,),+t :
)'i
-ali+th
which can be solvedfbr \ij
.\l-l:-
I t(tn
1ll.2ll
-+ 0 as r ,> oo. Hence, the approach is For this case, regardless of the size of the step, l_r,'i | called unc on di t i o n al I v stubl e.
EXAMPLE 2I 5
E x p l i c i ot n d l m p l i c i E t uter ProblemStotemeni. UseboththeexplicitandrmplicitEulermethods ro solveEq.(21.11), wherer'(0) : 0. (a) Use theexplicitEulerwith stepsizesof 0.000-5 and0.0015to solvefor ,t'between/ : 0 and 0.006.(b) Use the implicit Euler with a stepsizeof 0.05 to solve fbr I' between0 and0.4. Solution. (a) For this problem,the explicitEuler'smethoclis J,r+r: .r,rf (_1000],i * 3000 _2000e ')h The result for h:0.000-5 is displayedin Fig. 21.8a along with the analyticalsolution. Although it exhibitssometruncationelror, the resultcapturesthe generalshapeof the analytical scllution. In contrast,when the stepsizeis increased to a valuejustbelow the stability limit (ft : 0.0015).the solutionmanifestsoscillations. Usingft > 0.002would resulrin a totally unstablesolution-that is, it would go infinite as the solutionprogressed. (b) The implicit Euler'smerhodis .J,r+r: .yi* (_1000],i+rf 3000 _ )eeeg-ti+t1p
s28
ADAPTIVE METHODSAND STIFFSYSTEMS
.v
1.5
I
0.5
0
0.002
) z
0
0.1
0.2 (bl
0.3
0.4
FTGUR.E 2r.8 S o i u t i oonf o s t i fC f D Ew i t h( o )t h ee x p l i c o i t n d ( b )i m p l i c E i t u l em r ethods
Now becausethe ODE is linear.we can rearrangethis equationso that y;11is isolated on the left-handside: Li't ]i -F 3000/r- )QQQlls
''t'r-
l+roooft
The resultfor h :0.05 is displayedin Fig. 2l.8b alongwith the analyticalsolution.Notice that even though we have useda much bigger stepsize than the one that inducedinstability for the explicit Euler, the numericalresult tracksnicely on the analyticalsolution. Systemsof ODEs can also be stifT.An exampleis dyt __:_-_5r,.r1.,^
dr dv -; : dr
- -l0ly2 1001'1
(21.22a) (2t.22h)
For the initial conditions.rr(0) : 52.29and l,:(0) : 83.82,theexactsolutionis - g.67r-302.{\10tt lt : 52.96e-3.e8eet + 65.gge-302.01t)1t lz : lJ .83e--r.e8eel
(21.23a) (2t.23b)
2I .3 STIFFNESS
529
Note that the exponentsare negativeand dilTerby abouttwo ordersof magnitude.As with the singleequation,it is the largeexponentsthatrespondrapicllyandareat the heartofthe system'sstiffness. An implicit Euler's methodfbr systemscan be formulatedfor the presentexarnpleas , \ ' r . i + l:
l r i * ( - - 5 , t ' ri + r f 3 . v z . i + r ) / l
. \ ' 2 . i +:tr ' 2 1 f ( 1 0 0 r ' 1 ; 1 1 - 3 0 1 . y 2 . ; " . 1 ) / z
(21.24a) (21.24b)
Collecting terms gives ( l + - 5 / z ; . 1 ' 1 . -' * r3 ) ' : . i + t :
)t i
- 1 0 0 , r ' 1; 1 r * ( l t 3 0 1 / z ) 1 ' 1 . i :1 1 1'.,;
(21.25a) (2L25b)
Thus. we can seethat the problem consistsof solving a set of simultaneousequationsfbr eachtime step. For nonlinearODEs,the solutionbeconresevenrnoredilficult sinceit involvessolvinga systemof nonlinearsimultaneous equations(recallSection12.2).Thus,althoughstabilityis gainedthroughimplicit irpproaches. a priceis paid in tlreform of addedsolutionconrplexity.
21.3.1 MATTABFunclionsfor Stiff Systems MMLAB
hasa numberof built-infunctionstbr solvingstiff systemsof ODEs.Theseare
odel-5s. This function is a variable-ordersolverbasedon numericaldifferentiation formulas.lt is a multistepsolverthat optionally usesthe Gear backwarddifferentiation forrnulas.Tltis is usedfor stiff problemsof low to medium accuracy. ode2 3 s. This function is basedon a modified Rosenbrockformula of order 2. Becauseit is a one-stepsolver,it miry be more efficientthanode15s at crudetolerances. It can solvesomekindsof stifTproblents betterthanodel5s. ode23t. This function is an implementationof the trapezoidalrule with a "free" interpolant.This is usedfor moderatelystiff problemswith low accuracywhereyou needa solutionwithoutnumericaldanrping. ode23tb. This is an implementation of an implicit Runge-Kuttaformulawith a first stagethat is a trirpezoidalrule and a secondstagethat is a backwarddifferentiationformula of order2. This solvermay alsobe more efficientthanode15s at crudetolerances.
EXAMPLE 21 6
MATTAB for StiffODEs Problem Stotement. The van der Pol equationis a model of an electroniccircuit that aroseback in the daysofvacuum tubes,
#
-p(r- li)! *,-,:o
(E21.6.1)
The solutionto this equationbecomesprogressively stifferas 1r getslarge.Given the init i a l c o n d i t i o n s.,r ' r ( 0 ): h ) l l d t : 1 , u s e M A T L A B t o s o l v et h e l b l l o w i n g t w o c a s e s : ( a ) f b r I L : l . u s eo d e 4 5t o s o l v e f r o m / : 0 t o 2 0 ; a n d ( b )f o r 1 t : 1 0 0 0 , u s eo d e 2 3 st o solvefrom/:0to6000.
s30
ADAPTIVE METHODSAND STIFFSYSTEMS
Solution. (a) The first stepis to convertthe second-orderODE into a pair of first-order ODEs by defining 1/l'r -
,
dT
-f I
Using this equation.Eq. (E2l .6.1)can ['rewrittenirs d)':
: s ( l - r ' i ) . u- .. \ , r: 0
dt
An M-file can now be createdto hold tlris pair of diff'erentialequations: un=
yp
, v.nd-rpo r ,\',1u, ion )p ^2,)*y (2) --{ (r) ly (2) ;mu* (1-y(1) I ;
N o t i c e h o w t h e v a r l u e1o.firs p a s s e d a s a p a r a m e t e r ' . A s i n E x a m p l e 2 l . l ,5ocdaen,b1e i n voked and the resultsplotted: '
>> piof, (t,y(:,1), >>
'
,i-,y (:,2)
,'
')
lesend('yl','y2');
Observethat becausewe are not specityingany options,we must useopenbracketsll as a place holder.Tlre srnoothnatureof the plot (Fig. 21.9a)suggeststhat the van derPol equationwith p : I is not il stiff system. (b) If a standardsolverlike oc1e45is usedfor the stiff case(p : 1000), it will fail miserably (try it, if you like). However,ocle,tI s doesan efficientjob: >> plot(t,y(:,1))
F I G U R 2EI . 9 forvonderPol's formsolved withode45ond(b)stiffformsolved Soutions equolion. io)Nonstiff wilh otl.:21s.
?
-l
-z
it
' lt
I I
z 1
0 *1
{
tl I
_3 1 0
--. y2
fl
1 0
/l i it
2
{
*3 1
0
2l .4 MATLABAPPLICATION: BUNGEE IUMPERWITHCORD
53t
We have only displayedthe r1 componentbecausethe resultfor .yzhasa much larger scale. Notice how this solution(Fig.2l .9b)hasmuch sharperedgesthanis the caseinFig.21 .9a. This is a visual manifestationof the "stiffness"of the solution.
21.4
MATTABAPPLICATION: BUNGEEJUMPERWITH CORD In this section.we will use MATLAB to solve for the vertical dynamicsof a jumper connected to a stationaryplatform with a bungee cord. As developedat the beginning of Chap. 20, the problem consistedof solving two coupled ODEs fbr vertical position and velocity.The differentialequationfor positionis dr
---u
(2t.26)
dt
The diff'erentialequationfor velocityis differentdependingon whetherthejumper hasfallen to a distancewhere the cord is fully extendedand begins to stretch.Thus, if the distance fallen is lessthanthe cord length,thejumper is only subjectto gravitationaland dragfbrces, dtt
' : g - signlx)31rr
(2t.27a)
nt
dT
Once the cord beginsto stretch,the spring and dampeningfbrcesof the cord must also be included: dr t'.t - 1 : 8 - s i g n ( uj)u . atntmm
t
k
v
- -(,r - L) -:-u
(21.2'7b)
The following exampleshowshow MATLAB can be usedto solvethis problem. E X A M P L E2 l . 7
B u n g e eJ u m p e rw i t h C o r d Problem Stotement. Determine the position and v^elocityof a br.rngeejumper with t h e f o l l o w i n g p a r a m e t e r sI: : 3 0 m , g : 9 . 8 1 r t / s t , n r : 6 8 . 1 k g . . , r : 0 . 2 5 k g / m . k :40 N/m, and / : 8 N.s/rn. Perfom the computationfrom t :0 to 50 s and assume that the initial couditionsarer(0) : u(0) : 0. Solution. The following M-file canbe setup to computethe right-handsidesof the ODEs: f unct ion n
Q
dyclt - bunqee (t , y, L, cd, m,k, garnma )
Ql.
cord = 0; if y(1) > L %determine lf the cord exerts a force c o r d = k / m * ( y ( 1 ) - L ) + q a m m a , / m * y( 2 ) ; end - cordl; g - sign(y(2))*cdr'm*y(2)^2 dydt - ly(2li Notice that the derivatives are returned as a column vector because this is the format .^n,'irert hrr the MA'l'T,AR solvefs.
s32
METHODSAND STIFFSYSTEMS ADAPTIVE Becausetheseequationsarenot stiff, we can useode4 5 to obtainthe solutionsanddisplay them on a plot: >> plot(t,-y( >> leqend('x
:' )
:,1),'-',t,y(:,2),' (m) ','v (m/s)')
As in Fig. 2l.10, we havereversedthe sign of distancefor the plot so that negativedistance is in the downward direction.Notice how the simulationcapturesthe jumper's bouncing motion.
F I G U R E2 I . I O Plotof distonce ond ve ocilyof o bungeejumper 40 20 0 -20
l\,",''\' l"ta-rr
-ar
-i.
-r-
-40 -60
*too'
it
10
20
30
FOUNTAIN PLINY'SINTERMITTENT Bockground. The Roman natural philosopher,Pliny the Elder, pulportedly hadanintermittent fountain in his garden.As in Fig. 21. 11, water entersa cylindrical tank at a constant flow rate Qrnandfills until the water reaches)r,igr,.At this point, water siphonsoutof the tank through a circular dischargepipe, producing a fountain at the pipe's exit. Thefountain runs until the water level decreasesto )1oy,whereuponthe siphon fills with air andthe andthe fountain stops.The cycle then repeatsas the tank fills until the water reaches)rrigh, fountain flows again. When the siphon is running, the outflow Qo,, can be computed with the following formula basedon Torricelli's law: -eou, c J2gyn r2
(21.28)
2I.5 CASE STUDY
s33
continued
----------)=)high
F I G U R2 EI . I T An intermittent founioin Neglecting the volume of water in the pipe, compute and plot the level of the water in the tank as a function of time over 100 seconds.Assume an initial condition of an emptv tank )(0) - 0, and employ the following parametersfor your computation:
Rr: 0'05m
r:
0.007m
C:0.6
)r,ign:0'l m
.Iro*: 0.025m
g = 9.81m/sz
0,' : 50 x 10-6m3/s Solution. When the fountainis running,the rate of changein the tank's volume V(m3) is determinedby a simple balanceof inflow minus the outflow:
dv 4,
:
Qi"-
(2r.29)
Qou,
where V : volume (m3).Becausethe tank is cylindrical, y : r Rll.substituting this relationshipalong with Eq. (21.28)into Eq. (21.29)gives
d1, dt
Q,,- CJTfrrrz
(21.30)
1TR;
When the fountain is not running, the secondterm in the numerator goes to zero. We can incorporate this mechanismin the model by introducing a new dimensionlessvariable siphon that equals zero when the fountain is off and equals one when it is flowing; d1, dt
Qin-siphonxCJTfinrz TR?
(21.3 r)
In the presentcontext,siphon can be thoughtof as a switch that turns the fountainoff and on. Such two-state variables are called Boolean or loqical variables, where zero is equiva- r -_-_ :._ ^^.,:,,^t^-+
+^ +rra
534
ADAPTIVE METHODSAND STIFFSYSTEMS
rl
Next we must relate siphon to the dependent variable y. First, siphon is set to zero whenever the level falls below.)!,,.. Conversely, siphon is set to one whenever the level rises above yn,*n.The following M-file function follows this logic in computing the derivative: dy = Plinyode(L,y) function g i .o b a I s i p h o n Rt - 0.05; r = 0.007; yhr - 0.1; ylo = 0.025; g = 0.6; I = 9.81; Qin = 0.00005; if y(1) <= y1o srphon = O; y (f; >= yhi elseif sipiron = 1; end * pi * r * g * y(1)) Qout - siphon * C * sqrt(2 ^ 2); dy = (Qin - Qout) ,/ (pi * Rt
"'2;
Notice that because its value must be maintained between function calls, siphon is declared as a global variable. Although the use of global variables is not encouraged (particularly in larger programs), it is useful in the present context. The following script employs the built-in ode45 function to integrate Plinyode and generate a plot of the solution; globai siphor-i siphon = 0; y0 - 0; tspan = l0 f00l; ( G P l i l t 1 z e 6 s ,t s p a n , l / 0 ) ' = o d e 4 5 Itp,l'p] plot (tp,yp) (s)') xlabel('l.iine, (m) ') ylabel('water Le.rel in tank, As shown in Fig. 2I.12, the result is clearly incorrect. Except for the original filling period, the level seems to start emptying prior to reaching -vn,rn.Similarly, when it is draining, the siphon shuts off well before the level drops to,v1qp,. At this point, suspecting that the problem demands more firepower than the trusty ode4 5 routine, you might be tempted to use one of the other MATLAB ODE solvers such as ode2 3 s or ode2 3 tb . But if you did, you would discover that although these routines yield somewhat dift'erent results, they would still generate inconect solutions. The difficulty arises because the ODE is discontinuous at the point that the siphon switches on or off. For example, as the tank is filling, the derivative is dependent only on the constant inflow and for the present parameters has a constant value of 6.366 x l0-l m/s. However, as soon as the level reaches yn,*n,the outflow kicks in and the derivative abruptly drops to -1.013 x l0-2 m/s. Although ihe adaptive step-size routines used by MATLAB work marvelously for many problems, they often get heartbum when dealing with such discontinuities. Because they infer the behavior of the solution by comparing the results0f different steps, a discontinuity represents something akin to stepping into a deep potholeon q drrk
qtreet
s35
2I .5 CASESTUDY
continued u.tz
0.1
E 0.08 J
o
c) c)
0.06
q) G
0.04
0.02
U
80
010203040506070
90
100
T i m e( s )
21.12 FIGURE The level in Pliny'sfounloinversustime os simuloiedwith ode45.
At this point, your first inclinationmight be to just give up. After all, if it's too hard for MAILAB, no reasonablepersoncould expect you to come up with a solution. Because professionalengineersand scientistsrarely get away with suchexcuses,your only recourse is to develop a remedy basedon your knowledge of numerical methods. Because the problem results from adaptively stepping across a discontinuity, you might revert to a simpler approachand use a constant,small step size. If you think about it, that's precisely the approachyou would take if you were traversing a dark, pothole-filled street. We can implement this solution strategy by merely replacing ode45 with the constant-steprk4sys function from Chap.20 (Fig. 20.8).For the scriptoutlinedabove,the fourth line would be formulated as Itp,yp]
= r k 4 s y s ( @ P l i n y o d e ,t s p a n , y 0 , 0 ' 0 6 2 5) ;
As in Fig. 21.13, the solutionnow evolvesas expected.The tank fills to yn'rnand then empties until it reaches)1o*,when the cycle repeats. There are a two take-home messagesthat can be gleaned from this case study. First, although it's human nature to think the opposite, simpler is sometimesbetter.After all, to paraphraseEinstein,"Everything shouldbe as simpleas possible,but no simpler."Second, you shouldnever blindly believeevery result generatedby the computer.You've probably "sarbase in. out" in referenceto the impact of data quality
536
ADAPTIVE METHODSAND STIFFSYSTEMS
continued
2l.l Plin
0.1 0.09 0.08
thre 21.2 ane
?c 0.07
:
I
J € 0.06
: I
a o.os
:
0)
5 o.oc
whe the r rate. (a)
> 0.03
0.02 0.01 -0
10
20
30
40
50 60 T i m e( s )
70
80
90
100
FTGUR.E 2r.r3
(b)
Thelevelin Pliny's {ounloin timeos simuloted fie versus witho smoll,consiont stepsizeusing . 0.8) r k 4 s y s f u n c l i o(nF i g 2
on the validity of computer output. Unfortunately, some individuals think that regardlessof what went in (the data) and what's going on inside (the algorithm), it's always"gospeloul" Situationslike the one depictedin Fig. 21.12 areparticularlydangerous-thatis, althou$ the output is incorrect,it's not obviously wrong. That is, the simulationdoesnot go unst& ble or yield negative levels. In fact, the solution moves up and down in the mannerof an intermittent fountain, albeit incorrectly. Hopefully, this case study illustrates that even a great piece of software suchas MATLAB is not foolproof. Hence, sophisticatedengineersand scientistsalwaysexamine numerical output with a healthy skepticism based on their considerableexperiencead knowledgeof the problemsthey are solving.
; 21.3
inter
Use ofC v(2. Corr the e and. 2r.4 inter
! a
Use
PROBLEMS
537
PROBLEMS 2l,l Repeatthe the samesinrr.rlations as in Section21.-5for Pliny's fountain. but generatethe solutionswith c.i1e2j, ode23s, and ode113. Use :subplot to developa vertical plot of the time series. three-pane 21,2 The following ODEs havebeenproposedas a model of a ne p i d e m i c : dS - : -/r.sI at dt
, :aSI at
-rl
21.5 Given tly
;
: -1oo'ooo)' r 99.999e-'
(a) Estimate the step size requiled to maintain stability r"rsing the explicitEulermethod. (b) lf .v(0) : 0, use the implicit Euler to obtain a solution from / : 0 to 2 using a step size of 0. l. 21.6 Given z1t' , : 3 f l r s i n1 . r ) + 3 c 0 s 1 tlI
dR --:rl at whereS : the susceptibleindividuals,/ = the infected,R : the recovered,a : the int'ectionrate. and r: the recovery rate.A city has 10,000people,all of whonr are susceptible. (a) If a single infectious individual entersthe city at . : 0, computethe progressionof the epidenricuntil the number of infectedindividualsfalls below | 0. LIsethe following . week) and r : 0.15/d. parameters:a : 0.0021@erson Develop time-seriesplots of all the statevariables.Also generatea phase-planeplot of S versus1 versusR. (b) Supposethat after recovery,there is ir loss of immunity thatcausesrecoveredindividualsto hecomesusceptible. This reinfection mechanism can be computed as pR, where p : the reinf'ection rate. Modify the model to include this mechanismand repeatthe computationsin (a)usingP:0.03/d. 21.3 Solve the following initial-valueproblem over the intervalfiomr:2to3:
If f (0) : 0, use the implicit Euler to obtain a solutionfrom t : 0 lo 4 usinga stepsizeof 0.4. 21.7 Given cl.tt dt
:999xt -f 1999x2
dr: : -1000rr - 2000rr dr to 0.2 I f ; r 1 ( 0 ) : x u ( 0 ) : l . o b t a i na s o l u t i o nf r o m t : 0 using a stepsizeof 0.05 with the (a) explicit and (b) implicit Euler rnethods. 21.8 The fbllowing nonlinear.parasiticODE was suggested by Hornbeck(1975): dt' - :5(l'-
r'r
AT
If the initial condition is -r.'(0): 0.08. obtain a solutionfrom /:0to5: dv (a) Analytically. - - -0.5r' l-e-' (b) Using the fourth-orderRK methodwith a constantstep dt sizeof 0.03l2-5. Use the non-self-startingHeun method with a step size ( c ) U s i n gt h e M A T L A B f u n c t i o no c t e 4 . of 0.5 and initial conditions of -r,-(1.5):5.222138 and ( d ) U s i n gt h e M A T L A B l ' u n c l i o n ooe) : . .r(2.0) : 4.143883. Iterate the corrector to s. :0.17o. (e) Usingthe MATLAB functionode23tb. your resultsbasedon Computethe percentrelativeenors for Presentyour resultsin graphicalform. theexactsolutionsobtainedanalytically:v(2.5) : 1.27-1888 21.9 Recallfrom Example I7.5 that the following humps and -r'(3.0): 2.577988. function exhibitsboth flat and steepregionsover a relatively 21.4 Solve the following initial-value problem over the short.r range, interval from I : 0 to 0.5:
dy dt
-
-r'tZ
-
+
Use the fourth-orderRK methodto predict the first value at Heun method to t:0.25. Then use the non-self:starting *-r.^
rlr^ ^--Ji^r;,-n
-o
('I-0.3)?+0.01 (x-0.9)2f0.04
)
.t
| -
n 5
NTntp. rrtO)
-
I
Determinethe value of the deflnite integral of this function between.r:0 and I using(a) the quad and (b) the ode45
s38
ADAPTIVE METHODSAND STIFFSYSTEMS
21.10 The oscillationsof a swingingpendulumcan be sitnulated with the following nonlinearmodel: ,t)a
(a) 1
l (b) (
o
:: + I sind:0 tlt) I
1
2l.l' (Fie cant
wherc 0 : the angle of displacernent,.q : the gravitational constant,and / : the pendulum lcngth. For small angular displacements.the sin d is approxirnatelyequal to d and the model can be linearizedas L1 r, uo d12-
s, _,,
F I G U RP E2 I . I 3
t
radls.Solveusinganymethod Let fr(0):0 and A1O): CI.ZS studicd in this chapter.Plot the angle versustime andthe thesecondangularvelocity versustime. (Hint: Decompose order ODE.)
Use ode45 to solve fbr d as a function of time for both the linear and nonlinear models wherc I :0.6 m and g:9.81nr./s2. First, solve for thc case where the initial condition is fbr a small displaccment (d : :r/8 and d0 I dt :0). Then repcatthe calculationfbr a largedisplace- 21.14 Given the first-ordcrODE: ment (A : r 12 andd0 I dt : 0). For eachcase,plot the linear and nonlinearsimulationson the sameplot. dt 21.11 The following system is a classic cxample of stiff -r(t:0):4 ODEs that can occur in the solution of chemical reaction kinetics: Solve this stiff diff'erential equation using a numerical method over the time period 0 < r < 5. Also solveanalyti9:-o.ut,cr -looor:rc3 cally and plot the analytic and numericalsolutionfor both dt the tast transientand slow transitionphaseof thetimescale. dc.- -2-500crcr 21.L5 Solvc the following differential equationfrom dr t :0Io 2 -'* : dt
- 0 . 0 1 3 t . -r 1 0 0 0 r ' r c -r 2 5 0 0 c 2- rt
Solve theseequationsfrom l : 0 to 50 with initial conditions c1(0) : cr(0) : I and c.(0) : 0. If you have accessto MAILAB software,use both standard(e.g.,oce45) and stiff (e.g.,ode23s) functionsto obtainyour solutions. 21.12 The following second-orderODE is consideredto be stiff: 11
r
-a-. r_- t o o r ; - t o oor ox Solve this differential equation (a) analytically and (b) numerically fbr .r : 0 to 5. For (b) use an implicit apploach with /z : 0.5. Note that thc initial conditionsare .v(0) : 1 and 1'(0) : 0. Display both results graphically' 21.13 Consider the thin rod of length 1 moving in the x-,r'planeasshownin Fig. P2l .13.Thc rod is fixed with a pin on one end and a massat the other.Note that I : 9.81 m/sl and 1 : 0.5 m. This systcmcan be solvedusing ..e e-L0
-n
dt -l0r - : ot with the initial condition 1'(0): l. Use the followingtech(b) theexniquesto obtain your solutions:(a) analytically, For plicit Euler rnethod,and (c) the implicit Eulermethod. (b) and (c) use/r : 0. I and 0.2. Plot your results. 21.16 The Lotka-Volterra equations describedin Secthat tion 20.6 havebeenrefined to includeadditionalfactors impact predator-preydynamics. For example,over and above predation,prey population can be limitedby othet factors such as space.Spacelimitation can be incorporated into the model as a carrying capacity (recallthe logi$ic model describcdin Prob. 20.5) as in dx
-
r
,{r
:rr(l-V)x-bxt
dt
d), dr
: -61' f r/_ry
where K : the carrying capacity. Use the sameparameter valuesand initial conditionsas in Section20.6to integrate : 0 to 100usinsode45.
4( ! d
wher the ut pute r the fc
s39
PROBLEMS (a) Employ a very large value of ;g: 10" to validatethat you obtainthe sameresultsas in Section20.6. (b) Conrpare(a) with the more realisticcarryingcapacityof K : 2QO.Discussyour results. 21.17Two massesare attachedto a wall by linearsprinus tFig.P2l. I 7). Forcebalancesbasedon Newton'ssecondlaw canbe written as d--{I
: -If
,l,t d2.rt
dt'
mt -
k.
1 , r -, t . t ) , 1 ! ( r ,
----:(-t2 m2
n 1|
Ll
w1
L2
w2
- . \ 1- w ' - L 2 ) F I G U R EP 2 I . I 7
-.t1 -trrl -12)
t t ) , : y l . : 5 , a n dL t : L t : 2 . S e tt h ei n i t i a cl o n d i t i o nass : n-rass, wherek: the springconstants.,n L: the lengthof "\t: Lt tutdx, : L, I u, + L2+ 6. Perfbrrnthe simulation theunstretchedspring.and w : the width of the mass.Com- from / : 0 to 20. Constructtime-seriesplots of both the disputethe positionsof the massesas a functionof time usin-t placementsand the velocities.In adclition,producea phasethefbllowing parametervalues:t, : ft: : 5, mt : m):2, planeplot of .r, versusr,.
i"
:i; :li' i, ,'
Boundory-Voue Problems
CHAPTER OBJECTIVES The primary objectiveof this chapteris to introduceyou to solving boundary-value problemsfor ODEs. Specific objectivesand topicscoveredare . o . . . . . .
Understandingthe differencebetweeninitial-valueand boundary-valueproblems Knowin-qhow to expressan nth-orderODE as a systernof n first-orderODEs. Knowing how to implementthe shootingmethod1brlinear ODEs by usinglinear ''shots." interpolationto generateaccurate Understandinghow derivativeboundaryconditionsare incorporatedinto the shootingmethod. Knowing how to solve nonlinearODEs with the shootingmethodby usingroot locationto generateaccurate"shots." Knowing how to implementthe finite-differencemethod. Understandinghow derivativeboundaryconditionsare incorporatedinto the finite-differencemethod. Knowing how to solve nonlinearODEs with the finite-differencernethodby using root locationmethodsfor systens of nonlinearalgebraicequations.
YOU'VEGOT A PROBTEM o this point, we havebeencomputingthe velocity of a free-fallingbungeejumperby integratinga singleODE: tlu -; :8 atm
-
c,t t --:u-
tll.lt
Supposethat ratherthan velocity,you are askedto determinethe positionof thejumperas a firncrinrr
nf time
Onp
urev tn
22.'I INTRODUCTION ANDBACKGROUND
541
of distance: dr dr
(22.2)
Thus, by solvingthe systemof two ODEs representedbyEqs. (22.1) and(22.2),we can simultaneously deterrnine both the velocityand the position. However, becausewe are now integratingtwo ODEs, we require two conditionsto obtain the solution.We are aheadytamiliar rvith one way to do this for the casewhere we have varluesfor both positionand velocity at the initial time; x(r:0):r:, tr(l:0):ur Given such conditions,we can easily integratethe ODEs using the numericaltechniques describedin Chaps.20 and 2l . This is refened to ils an initial-valueproblem. But what if we do not know valueslbr both positionand velocity at / : 0? Let's say that we know the initial position but rather than having the initial velocity, we want the .jumperto be at a specifiedpositionat a later time. In other words: .t(l : 0) :.v; x(t:t1):ry Becausethe two conditionsare given at ditferentvaluesof the independentvariable,this is called a boundary-valueproblem. Such problemsrequire speciaisolution techniques.Son.reof theseare relatedto the methodsfor initial value problemsthat were describedin the previoustwo chapters.However, others emplov entirely different strategiesto obtain solutions.This chapteris designedto introduceyou to the more comnlon of thesemethods.
22.1
INTRODUCTION AND BACKGROUND 22.1.1 Whot Are Boundory-VolueProblems? An ordinary differentialequationis acconrpaniedby auxiliary conditions,which are used to evaluatethe constantsof integrationthat result during the solutionof the equation.For an rzth-orderequation,n conditionsare required.If all the conditionsare specifiedat the samevalue of the independentvariable,then we are dealingwith an ilritial-vaLueproblem (Fig.22.1a).To this point,the materialin PartSix (Chaps.20 and 21) hasbeendevotedto this type of problem. ln contrast.there are often caseswhen the conditionsare not knor.vnat a singlepoint but rather are given at different valuesof the independentvariabie.Becausethesevalues are often specifiedat the extremepoints or boundariesof a system,they are customarily pntblerns(Fig.22.lb). A varietyof significantengineeling referredIo as bounrlary--value applicationstall within this class.In this chapter,we discusssomeof the basicapproaches
542
BOUNDARY-VALUE PROBLEMS
4't : , I r ( r .] r ' , r ' : l [ dy. V
= lzQ':"v):)
w h e r e a t r - 0 . r I . - J r . oa n d \ ' 2 - ) . r ,
I n i t i a lc o n d i t i o n s
0 {al a-\
ox-
= lQ'Y)
w h e r e a t r : 0 , . y- _ ) o x:1")"=YL
)
Boundary condition
\ YL
Boundary *\l,o condition
(b)
FIGURE 22.I lnitiolvolue versus problems boundoryvolue problem where o llheconditions {o)Aninrtiol-volue
o r e s p e cl i e d c ' l I e s o r n ev o l u eo f t h e i n o e p e n o e nv' o ' i o o l e .( b )A o o r ^ d c . y - v o l upe' o b l e r where lhe conditionsore specifiedoi differentvoluesof the independentvorioble.
22.1.2 Boundory-VolueProblemsin Engineeringond Science At the beginningof this chapter,we showedhow the determinationof the positionandvelocity of a falling objectcould be formulatedas a boundary-valueproblem.For thatexample, a pair of ODEs was integratedin time. Although other time-variableexamplescanbe developed,boundary-valueproblemsarisemore naturallywhen integratingin space.This occursbecauseauxiliary conditionsare often specifiedat differentpositionsin space. A casein point is the simulationof the steady-state tenperaturedistributionfor a long, thin rod positionedbetweentwo constant-temperature walls (Fig. 22.2).The rod's crosssectionaldimensionsare small enough so that radial temperaturegradientsare minimal and, consequently,temperatureis a function exclusivelyof the axial coordinatex. Heatis transfeffedalong the rod's longitudinal axis by conductionand betweenthe rod andthe surroundinggas by convection.For this example,radiationis assumedto be negligible.r
lWe incornorate radiation into this nrohlem
22.I INTRODUCTION ANDBACKGROUND
s43
A
i++l
-r
,r + A-r
FIGURE22.2
A h e o to o o ' c e I o o d i f f e ' e n ' oe e ' n e 1 o f o h e o ' e d ' o ds - b ; e . t - oc o ' r dr c l i o ' o ^ o . o n \ e c i o ^
As depictedin Fig. 22.2, a heatbalancecan be taken arounda differentialelementof thicknessA,t as
Q: q(r')A,.- q(x + Ar)A,.+ hA,(Tn - T)
(22.3)
whereq(.r) : flux into the elementdue to conduction[J/tm:'s)]: q(r * Ax) : flux out area[m2] : 7T12, r : the of the elementdueto conductionU/(m2' s)]; A,. : cross-sectional ' K radius[m]; h : theconvectionheattransfercoefTicient [J/1m2 ' s)l: A' : theelement's surfacearea [m2] - 2n r Lr Z- : the temperatureof the surroundinggas [K]; and Z : the rod's temperature[Kl. Equation(22.3)canbe dividedby the element'svolume (1rr'A,x) to yield ./(r)-./(r + A'{) 0_ + 2 h t T __ T l r A-r Taking the limit Ax -> 0 gives da
2h
0:__;+_(T__T)
(22.4)
arr
The flux can be relatedto the temperaturegradientby Fourier's law: q:-K,
dT
(22.s)
ax y / ( s . m ' K ) l . E q u a t i o n ( 2 2 . 5 )c a n b e d i f w h e r e , t: t h e c o e f f i c i e n t o ft h e r m acl o n d u c t i v i tU (22.q, and the resultdivided by k to yield, ferentiaredwirh respectro x, substitutedinto Eq. ,!27
0:T+h'tT--Tl
(22.6)
dx-
whereft' : a bulk heat-transferparameterreflectingthe relativeimpactsof convectionand 2] : 2hl?k). conduction[Equation(22.6)representsa mathematicalmodel that can be usedto computethe temODE, two conditions
544
BOUNDARY.VALUE PROBLEMS arerequiredto obtain a solution.As depictedinFig.22.2, a comnroncaseis wherethetemperaturesat the endsofthe rod areheld at fixed values.Thesecan be expressed mathematically as T (0) : 7,, T(L) : T|' The fact that they physicallyrepresentthe conditionsat the rod's "boundaries"is theorigin of the terminology:boundaryconditions. Given theseconditions,the model representedbyEq. (22.6) can be solved.Because this particularODE is linear,an analyticalsolution is possibleas illustratedin thefollowing example.
EXAMP2 L2 E. I Anolyticol Solutionfor o Heqted Rod Problem Stotement. Use calculus to solve Eq. (22.6) fbr a 10-m rod with h'= 0 . 0 5 m - 2 [ / z : l J l ( m 2' K ' s ) , r : 0 . 2 m , k : 200 J/(s . m ' K)1, T* = 200K, andthe boundaryconditions:
z(0) : 300K
z(10): 4s0K
Solution. This ODE can be solvedin a numberof ways.A straightforward isto approach first expressthe equationas d2T --ltT:-17'ln Becausethis is a linear ODE with constantcoefflcients,the generalsolutioncanbereadily obtained by setting the right-hand side to zero and assuminga solutionof the form T : eL' . Substitutingthis solutionalong with its secondderivativeinto thehomogeneous form of the ODE yields 7 2 n L_xh , d , r : 0 which can be solvedfor L : +JA
Thus. the gener-alsolutionis
T:Ae^'+Be-i' whereA and B are constantsof integration.Using the methodof undetermined coefficients we can derive the particularsolution T : T-. Therefore,the total solutionis T :To
* Ae,' 1 Br' "
The constantscan be evaluatedby applying the boundaryconditions 7,,:Tn+A+B Tr,: Tn I Ad'L I Be-t'r Thesetwo equationscan be solvedsimultaneouslyfor
A-
(7,-Tn)e-^t
-(7,,-Tn\
e,Lr, _ d,L (To-T*)-(7.-Tn)/'L
22.2 IHESHOOTING METHOD
545
I.K
400
10 "r,m
FIGURE22.3 fortheheoted rod Anolytico solulion
Substitutingthe parametervalues from this problem gives A:20.461 1 and B: 19.5329.Therefbre.the final solutionis (22.1) r : 100 + 20.4611n. ttos' + lg .532ge ffi5' As can be seeninFig.223, the solutionis a smoothcurve connectingthe two boundary temperatures.The temperaturein the middle is depresseddue to the convectiveheat lossto the coolersunoundingsas. for solving the same In the following sections,we will illustratenumericalapproaclres problem we just solvedanalyticallyin Example22.l.The exactanalyticalsolutionwill be useful in assessingthe accuracyof the solutionsobtainedwith the approximate,numerical methods.
22.2
THESHOOTINGMETHOD The shootingmethod is basedon conveftingthe boundary-valueproblem into an equivalent initial-valueproblem.A trial-and-errorapproachis then implementedto developa solution for the initial-valueversiorlthat satisfiesthe given boundaryconditions. and nonlinearequations.it is Although the methodcan be employedfor hi-eher-order nicely illustratedfbr a second-order,linear ODE such as the heatedrod describedin the previoussection: 0:
d)T a_*.
alttTo-T)
(22.8\
subjectto the boundaryconditions
r (0) - f,, T(L):Tb We convertthis boundary-valueproblem into an initial-valueproblem by defining the rate of changeof temperature,or gradient,as dT
546
BOUNDARY-VALUE PROBLEMS andreexpressing Eq.(22.8)as )tt . ( 2 2l.0 )
dx
Thus, we have convertedthe singlesecond-order equation(Eq. 22.8) into a pair of firstorder ODEs (F.qs.22.9and22.10). If we had initial conditionsfor both Z and:. we could solvetheseequationsasan initialvalue problemwith the methodsdescribedin Chaps.20 and 21. However,because we only havean initial vafi"re tbr one of the variablesI (0) : 7.,we simply mtrkea guessfor theother :(0) : {.r ord then perforrnthe integration. After perfbrmingthe integration,we will have generateda value of I at theendof the interval,which we will call 4r. Unlesswe areincrediblylucky,this resultwill differfrom the desiredresult Zr,. Now, let's say that the value of f,1 is too high (Tr,t > Tt) , it would makesensethara lower value of the initial slope z(0) - ;(2 might result in a better prediction.Usingthis new guess,we can integrateagainto generatea secondresultat the end ofthe intervalf,,2. We could then continueguessingin a trial-and-errorfashionuntil we arrivedat a guessfor :(0) thatresultedin the corect valueof T (L) : 7r. At this point, the origin of the name.shootirtg methodshouldbe prettyclear.Justasyou would adjustthe angleof a cannonin orclerto hit a target,we are adjustingthe trajectoryof our solution by guessingvaluesof z(0) until we hit our targetT(L) : Ty. Although we could cefiainly keep guessing.a more efficient strategyis possiblefor linear ODEs. In suchcases,the trajectoryof the perf-ectshot 1, is linearly relatedto theresultsof our two erroneousshots(s,,1,4r) and (zoz,Tnz).Consequently, linearinterpolation can be ernployedto arrive at the requiredtrajectory: i,, : l,,r + ;f -I bal r 7 1 , T 1 ' 1 ) 1b2-
(22.1r)
The approachcan be illustratedby an example.
EXAMPLE 22.2
The Shooting Method for o LineorODE Problem Stotement. use the shootingmethod to solve Eq. (22.6) for the samecondit i o n s a s E x a r n p l e2 2 . 1 : L : l 0 m , h ' : 0 . 0 5 m - 2 . T - : 2 0 0 K , f(0):300K, and 7(10) : 400K. Solution. Equation(22.6)is first expressed as a pair of first-orderODEs: .IT
E
:,.
.tat^
_ : _0.05(20tJ_ T) ax Along with the initial value for temperarureZ(0) : 300 K, we arbitrarilyguessa valueof ia | : -5 K/m for the initial value for r (0). The solutionis thenobtainedby integrating the
:"9,:""11;Y::il,f^'Li:.y':l.r*LAB's ocre45 ru'-ion bvfirst l:.11-":_10u-'^';".1'l^:
22.2 IHE SHOOTING METHOD
547
L u - ] e r. o r . . i y = E x 2 l 0 2 rx,y I dy= ly (2) ; -0.05" (2oo-y (t) )l ; We can then generatethe solutionas >>
Tbl=Y (lenSth
(Y) )
569.1539
Thrrs.we obtaina value at the end of the intervalof 761: 569.7539(Fig. 22.4a), which differs fronl the desired boundary condition of T6:400. Therefore,we make another gu€ss in2 : -20 and perform the computation again. This time, the result of T t z : 2 5 9 . 5 1 3 1 i s o b t a i n e d( F i g .2 2 . 4 b ) .
FIGURE22.4 Temperofure cistonceim.)compuied wrththeshoctingmethod;io) thefirsr"shot," {K)versus {b)theseconC"shoi,"ond lc) thefinolexoci"hit."
548
PROBLEMS BOUNDARY-VALUE Now, becausethe original ODE is linear,we can useEq. (22.11)to determinethecorrect trajectoryto yield the perfectshot:
-20 - (-5) (4oo- 569.7539) : -13.2075 2 5 9 . 5 1 3- 1569.1539 This value can then be usedin conjunctionwith ode45 to generatethe conect solution,as depictedinFig.22.4c. Although it is not obvious from the graph, the analyticalsolution is also plottedon Fig.22.4c. Thus, the shootingmethodyields a solution that is virtually indistinguishable from the exactresult.
22.2.1 Derivqtive Boundory Conditions The fixed or Dirichlet boundaryconditiondiscussedto this point is but oneof severaltypes that are used in engineeringand science.A common alternativeis the casewherethederivative is given. This is commonly referredto as a Neumannboundarycondition. Becauseit is alreadysetup to computeboth the dependentvariableandits derivative, incorporating derivative boundary conditions into the shooting method is relatively straightforward. ODE Justas with the fixed-boundaryconditioncase,we first expressthe second-order as a pair of first-orderODEs. At this point, one of the requiredinitial conditions,whether the dependentvariableor its derivative,will be unknown. Basedon guessesfor themissing initial condition,we generatesolutionsto computethe given end condition.As withthe initial condition, this end condition can either be for the dependentvariableor its derivative. For linear ODEs, interpolationcan then be usedto determinethe valueof themissing initial condition requiredto generatethe final, perfect "shot" that hits the end condition.
EXAMPLE 22.3
T h e S h o o t i n gM e t h o d w i t h D e r i v o t i v eB o u n d o r yC o n d i t i o n s Problem Stotement. Use the shooting method to solve Eq. (22.6) for the rod in E x a m p l e2 2 . 1 :L : l 0 m . f t ' : 0 . 0 5 m - 2 [ f t : I J / ( m 2. K . s ) . r : 0 . 2 m . k : 2 0 0 J l (s ' m ' K)1, f- : 200 K, and 7(10) : 400 K. However,for this case,ratherthanhaving a fixed temperatureof 300 K, the left end is subjectto convectionas in Fig.22.5.For
FIGURE 22.5 oi theolher. A rod wiih o conveciiveboundoryconditionoi one end ond o fixed temperoture
Convection T-
22.2 TIIESHOOTING METHOD I
549
simplicity, we will assumethat the convectionheattransfercoefficientfor the end areais the sameas for the rod's surface. Solution. As in Example22.2,Eq. (22.6)is firsr expressed as dT dx n7
ia x : -0.0sQ00- 71 Although it might not be obvious,convectionthroughthe end is equivalentto specifying a gradientboundarycondition.In order to seethis, we must recognizethat because the systemis at steadystate,convectionmust equal conductionat the rod's left boundary (-r : 0). Using Fourier'slaw (8q.22.5) to representconduction,the heatbalanceat the end can be formulatedas
h A , ( T o - f ( 0 l): _ - f e , # 0 1
(22.12)
This equationcan be solvedfor the gradient dTh - 7 *) E (0 ): t(7 (0 )
(22.13)
If we guessa value for temperature,we can seethat this equationspecifiesthe gradient. The shootingmethod is implementedby arbitrarily guessinga value for z(0). If we choosea value of Z(0) :Tnt:300K, Eq. (22.13)then yields the initial value for the gradient z n r:
dT.^ ;(0)
1 : -(300
- 2 0 0 ): 0 . 5
The solution is obtainedby integratingthe pair of oDEs from r : 0 to 10.We can do this with MATLAB's ode45 function by first settingup an M-file to hold the differentialequations in the samefashionas in Example22.2.we can then generatethe solutionas > > [ t , y ] = o d e 4 5 ( @ E x 2 3 A 2 l, 0 1 0 1 , t 3 0 O , O . 5 l ) ; >> Tb1=y(lengrh(y) ) 6 B 3 .s 0 B B As expected,the value at the end of the interval of 71,1:683.5088K differs from the desiredboundaryconditionof T6 - 400. Therefore,we make anotherguessf,2 : 150 K, which correspondsto Znz: -O.25, and perfbrm the computationagain. >>
Tb2 =y (length
-47.1544
(y ) )
550
BOUNDARY.VALUE PROBLEMS
7'K 400
300
200
10 r,m
FIGURE 22.6 The solutionof o second-order CDE with o conveciiveboundorycondltionot one end ond o fixed lemoerolureot the other.
Linear interpolationcan then be employedto computethe correctinitial temperature 1 5 0- 3 0 0
4,:300f
- 683.5088) : 241.3643 -- 4 1 . 7 5 4 4 K - 6 8 3 . s 0 8@00 8'
which correspondsto a gradientof Zu - 0.2068.Using theseinitial conditions,ode45can be employedto generatethe conect solution,as depictedin Ftg. 22.6. Note that we can verify that our boundaryconditionhasbeensatisfiedby substituting the initial conditionsinto Eq. (22.12)to give |
JJK ir x(0.2m)'x(200K-241.3643K):-200 ---. rr xt0.2m)'x0.2068* mrKs mKs m
which can be evaluatedto yield *5.1980J/s: -5.1980J/s. Thus, conductionandconvectionareequalandtranSferheatoutoftheleftendoftherodatarateof
22.2.2 The Shooting Merhod for Nonlineor ODEs For nonlinearboundary-valueproblems,linear interpolationor extrapolationthroughtwo solutionpoints will not necessarilyresult in an accurateestimateof the requiredboundary condition to attain an exact solution.An alternativeis to perform threeapplications ofthe shooting method and use a quadratic interpolating polynomial to estimatethe proper boundarycondition. However,it is unlikely that such an approachwould yield theexact answer,and additionaliterationswould be necessaryto home in on the solution. Another approachfor a nonlinearprobleminvolvesrecastingit as a rootsproblem.Recall that the generalgoal of a rootsproblemis to find the value of r that makesthefunction f (x) :0. Now, let us usethe heatedrod problemto understandhow the shootingmethod can be recastin this form. First, recognizethat the solution of the pair of differentialequationsis alsoa "function" in the sensethat we guessa conditionat the left-handend of the rod zn, andtheintegration yields a predictionof the temperatureat the right-hand end 76.Thus, we canthink of the integrationas
Tu: f k,)
METHOD 22.2 THESHOOTING
551
That is, it representsa processwherebya guessof 2,, yields a predictionof 16. Viewed in this way, we can seethat what we desireis the value of ;,, that yields a specificvalue of 71,. I1',as in the example,we desireTo : 400. the problem can be posedas 400 : .l k.) By bringing the goal of 400 over to the right-handside of the equation,we generatea new function res(zn)that representsthe difference,or residual,betweenwhat we have, l (2,,), and what we want. 400. r e s ( 7 . , , ) J: ' k a ) - 4 0 0 If we drive this new function to zero, we will obtain the solution.The next exampleillustratesthe approach. The Shooting Method for Nonlineor ODEs
EXAMPLE22.4
Problem Stotement. Although it served our pu{posesfor illustratingthe shooting rnethod,F,q.(22.6) was not a completelyrealisticmodel for a heatedrod. For one thing, such a rod would lose heatby mechanismssuchas radiationthat are nonlinear. Supposethat the following nonlinearODE is usedto simulatethe temperatureof the heatedrod: (1:T
0 : = + h,(T&_ Tt + o,'q! _ ra1 dx' ' r ' ,
, ,
where o' : a bulk heat-transferparameterreflectingthe relative impactsof radiation and conduction:2.J x l0 e K 3 m-2. This equationcan serveto illustratehow the shooting method is used to solve a two-point nonlinear boundary-valueproblem. The remaining 2, problem conditions are as specified in Example 22.2: L : 10 r7t,h':0.05 m 7 - : 2 0 0 K , f ( 0 ) : 3 0 0 K , a n d7 ( 1 0 ) : 4 0 0 K . Solution. Just as with the linear ODE, the nonlinear second-orderequationis first exoressedas two first-orderODEs: tlT d: . ' : - 0 . 0 5 t 2 0 0 T ) - 2 . 1x l 0 u ( 1 . 6x l O q- 1 4 . ) tlx
An M-file can be developedto computethe right-handsidesof theseequations: dy=flydxn (x, Y) function (200-y(1,) (2) ;-0.05*
dy=ty
)
2.1e-9*
(1. 6e9 y(1)^
) I ;
Next, we can build a function to hold the residual that we will try to drive to zero as r=res (za) function fx,yl=ode45(@dydxn, t0 101, l3Aa zal); r=y (length (r), 1) -400 ;
BOUNDARYVALUE PROBLEMS
552
lK 400 300
10 r. rn
FIGURE22,7 Theresult to solve of usingfie shooting method a nonineorprobem
Notice how we use the ode45 function to solvethe two ODEs to generatethe temperature at the rod's end:y (length (x) , 1) . We can then find the root with the f zero function: >>
fzero
A1
(@res, -50)
?.41,1
T h u s , w e s e e t h a t iw f e s e t t h e i n i t i atlr a j e c t o r y z ( 0:) - 4 1 . 1 4 3 1 .t h e r e s i d u a l f u n c t i o n w i l l be driven to zero and the temperatureboundarycondition Z(10) :400 at the endof the rod should be satisfied.This can be verified by generatingthe entire solutionandplotting the temperaturesversusx: > > l x , y l = o d e 4 5 ( @ d y d x n ,t 0 1 0 1 , 1 3 0 0 f z e r o ( @ r e s , - 5 t l ) l ) ; >> plot(x,t'(:,1))
The resultis showninFig.22.7 alongwith the originallinearcasefionr Example22.2. As expected,the noniinearcaseis depressedlower than the linear model due to theadditional heatlost to the surroundinggas by radiation.
22.3
FINITE.DIFFERENCE METHODS The most colrmon altemativesto the shootingmethodarefinite-ditlerenceapproaches. In these techniques,tinite differences(Chap. l9) are substitutedfor the derivativesin the original equation.Tlrus, a linear differentialequationis transformedinto a setof simultaneousalgebraicequationsthat can be solvedusing the methodsfrorn PartThree. We can illustratethe approachfbr the heatedrod model (Eq.22.6): .l)r
o:;+h'(T--T)
(22.t4)
The solutiondomain is first divided into a seriesof nodes(Fig. 22.8).At eachnode,finitedifferenceapproximationscan be written fbr the derivativesin the equation.For example,
22.3 FINITE-DIFFERENCE METHODS
5s3
a-{
FIGURE22.8 Inorderto implement thefinitedifference opprooch, theheoted rodisdivided intoo series o{ nodes
at nodel. the secondderivativecan be represented by (Eq. 19.15):
d)T
Ti-t -2Ti *Ti+t
dx2
Ax2
(22.1s)
This approximationcan be substitutedinto Eq. (22.14)to give Ti r-2Ti aTi*1 A.rl
+ h ' ( T n* 4 ) : 0
Thus, the differential equationhas been convertedinto an algebraicequation.Collecting termsgives -Ti
t I Q + h' Lx2)Ti - Ti+r : h' L..r2Tn
(22.16)
This equationcan be written for eachof the rr - I interior nodesof the rod. The first and last nodesTs and 7,, respectively,are specifiedby the boundaryconditions.Therefore,the problem reducesto solving n - 1 simultaneouslinear algebraicequationsfor the n - I unknowns. Before providing an example,we should mention two nice featuresof Eq. (22.16). First, observethat since the nodes are numberedconsecutively,and since each equation consistsof a node(i) and its adjoiningneighbors(i - I andi + 1), the resultingsetoflinear algebraicequationswill be tridiagonal.As such.they can be solved with the efficient algorithmsthat areavailablefor suchsystems(recallSec.9.4). Ftrrtl.rer, inspectionof the coefficientson the left-handsideof Eq. (22.16)indicatesthat the systemof linearequationswill alsobe diagonallydominant.Hence,convergentsolutions can also be generated with iterativetechniques like the Gauss-Seidel method(Sec.12.l). E X A M P L E2 2 . 5
F i n i t e - D i f f e r e n cAep p r o x i m o t i o no f B o u n d o r y - V o l uP er o b l e m s Problem Stotement. Use the finite-differenceapproachto solve the sameproblem as in Examples22.1and22.2.Usefour interiornodeswith a segmentlengthof Lx : ) 111. Solution. Employing the parametersin Example 22.1 and Ax:2m, Eq.(ZZ. | 6) fbr eachof the rod's interior nodes.For example,for node l: -To l2.2Tr - Tz: 40 Substitutingthe boundaryconditionIt : 300 gives 2 . 2 7 1- T z : 3 4 0
we can write
554
BOUNDARY-VALUE PROBLEMS After writing Eq. (22.16) for the other interior nodes,the equationscan be assembled irr matrix form as
-r f2.2 .2 l-1 2 -r 0 | 0 L0
0 -r 2.2 -1
: :'lli{it}
Notice that the matrix is both tridiagonaland dia-qonallydominant. MATLAB can be usedto seneratethe solution:
4 0 4 4 r ) l' ; T= 283.265t) 283.1853 299 .14r6 336.2462
Table 22.1 provides a comparisonbetweenthe analyticalsolution (8q.22.1) andthe numericalsolutionsobtainedwith the shootingrnethod(Example22.2) andthe finitedifferencemethod (Example 22.5). Note that althoughthere are some discrcpancies, the numericalapproaches agreereasonablywell with the analyticalsolution.Further,thebiggest discrepancyoccursfor the finite-differencemethoddue to the coarsenode spacingwe used in Example22.5.Betteragreementwould occur if a f iner nodal spacinghad beenused. TABTE22. 1
Comporison of the exoctonolyticolsolutionfor temperoture with theresults
.:13:::-{"yl-f::k-:ns".::"ji stk*;"** Anolyticol Solution
(._) 2 4 6 IJ t0
300 2 8 28 6 3 4 2 8 25 7 7 5 '299 AB43 335 74A4 4AO
Shooting Method
Finite Difference
300 2 B 2B B B 9 2 8 25 t 5 8 2 9 91 2 5 4 J 3 5 . 7 )7B 404
300 283 2664 2 8 3I 8 5 3 299.7416 336 2462 4AA
22.3.1 Derivotive Boundory Conditions As mentionedin our discr,rssion of the shootingmethod,the fixed or Dirichlet boundan' cornlition is but one of severaltypes that are usedin ensineeringand science.A common
22.3 FINITE-DIFFERENCE METHODS
555
A-r
FIGURE22.9 A boundory nodeot theleftendof o heoied rod.Toopproxirnole thederivotive ot theboundory on imoginory nodeislocoted o drslonce Ax to theleftof therod'send.
altemative,called the Neumannboundary condition, is the case where the derivative is given. We can use the heatedrod introducedearlierin this chapterto demonstratehow a derivative boundaryconditioncan be incorporatedinto the finite-differenceapproach: 0:
d2T _ +/r'(f\-f) dr'
However, in contrastto our previousdiscussions,we will prescribea derivativeboundary condition at one end of the rod: dT
. (0): r:
dx T (L) : 7,,
Thus, we have a derivative boundarycondition at one end of the solution domain and a fixed boundarycondition at the other. Just as in the previous section,the rod is divided into a seriesof nodesand a finitedifferenceversionof the differentialequation(F,q.22.16)is appliedto eachinterior node. However.becauseits temperatureis not specified,the node at the left end rnustalso be included.Fig.22.9 depictsthe node (0) at the left edgeof a heatedplatefor which the derivative boundarycondition applies.Writing Eq. (22.16)for this node gives -T
r t (2 + h' L-r2)To* Tt : h' LxzTn
\22.11)
Notice that an imaginarynode(- I ) lying to the left of the rod's end is requiredfor this equation.Although this exterior point might seem to representa difficulty, it actually servesas the vehiclefor incorporatingthe derivativeboundaryconditioninto the problem. This is done by representingthe first derivativein the x dimensionat (0) by the centered difference(Eq. 4.25):
dT dr
Tr - T-t 2Lx
which can be solvedfor T-r:Tt
-2Ar',
dT dr
556
BOUNDARY-VALUE PROBLEMS Now we havea formula for I-1 that actuallyreflectsthe impact of the derivative.It canbe substituted into Eq. (.22.11) ro give
(2 + h' Lx2)7,* 2Tt: h' LxlTo - ro-#
(22.18)
Consequently,we have incorporatedthe derivativeinto the balance. A commot.texampleof a derivativeboundarycondition is the situationwheretheend of the rod is insulated.In this case,the derivativeis setto zero.This conclusionfollowsdirectly from Fourier's law (Eq. 22.5), becauseinsulatinga boundarymeansthat the heat flux (and consequentlythe gradient)must be zero.The tbllowing exampleillustrateshow the solutionis affectedby suchboundaryconditions. E X A M P L E2 2 . 6
I n c o r p o r o t i n gD e r i v o t i v eB o u n d o r yC o n d i t i o n s Problem Stotemeni. Generatethe finite-differencesolution fbr a 10-m rod with t.Tn:200K, Lx:2rl,h':0.05m a n d t h e b o u n d a r yc o n d i t i o n s 4: j : 0 a n d Tt : 400 K' Note that the first condition meansthat the slope of the solutionshouldapproach zero at the rod's left encl.Aside from this ca.se,also generatethe solutionfor dTldx: -20 at-v:0. Solufion. Equation(.22.1g) can be usedto represent node0 as 2 . 2 T 0 - 2 7 1: 4 9 We can write Eq. (22.16)for the interior nodes.For example.fbr node l. -To*2.2Tt-Tz:40 A similar approachcan be usedfor the remaininginterior nodes.The final systemof equations can be assembledin matrix form as
| / | - /t''| 1 2.2 l-r | -1 2.2 -r | -t 2.2 _r I | - 1 2 . 2IJ L Theseequationscan be solvedfor
lrtfit
To :243.0218 Tr : 241.3306 Tz:261.0991 Tt :281.0882 T+:330.4946 As displayedin Fig. 22.10, the solutionis flat at ,r : 0 due to rhe zeroderivativecondition and then curvesupward fo the fixed conditionof I : '100at x : 10.
22.3 FINITE-DIFFERENCE METHODS
557
200d FIGURE 22.1O Thesolution ofo second-order ODEwiiho derivotive boundorv condition ol oneendondo fixed boundory condilion Twocoses reflecting different ol fheother. oreshown derivotive volues ot-r: O For the casewherethe derivative&t,t : 0 is setto -20, the simultaneous equatibnsare 1t
a
-l
2.2 -t
To
,, 1 I
ll 1"{, T, rl
I
-1 .'t ^,
I
1
')a
whichcanbe soivedfor
T2 T3 T+
To: 328'2710 7r:301'0981 Tz:294'1448 Tt:306'0204 zr:339.100)
'
As in Fig.22.10,thesolutionatx : 0 nowcurvesdownwarddueto thenegative derivative we imposedat theboundary. 22.3.2 Finife-DiffercinceApprooches for Nonlineqr ODEs yieldsa systemof nonlinearsiFor nonlinearODEs,thesubstitution of finitedifferences multaneous equations. Thus,themostgeneralapproach to solvingsuchproblemsis to use root locationmethodsfor systems suchastheNewton-Raphson methoddeof equations is certainlyfeasible,an adaptation scribedin Sec.12.2.2.Althoughthis approach of sucprovidea simpleralternative. cessivesubstitution cansometimes Theheatedrod with convection andradiationintroduced in Example22.4providesa nicevehiclefor demonstrating thisapproach, ^
d:T
0 : ;; d
v a
., -.
+ h (I--
t .-t -, f) + o (T] -7")
.
558
BOUNDARYVALUE PROBLEMS We can convert this difTerentialequationinto algebraicform by writing it for a nodei and substitutingEq. tZZ.l5) for the seconcl derivarive: o-
T1 1-2TilTial
Ax2
+ h ' ( T n- 7 l )+ o " ( 7 1 - r : )
Collectingtermsgives -Ti-t I (2 + h'Lxz)Ti - Ti+) : lt' Lx2Tn o' tr2(r!. - r,a) + Noticethat althou-sh thereis a nonlineartermon the right-lrandside,tlreleti-handside is expressedin the form of a linear algebraicsystemthat is diagonallydominant.If we assumethat the unknownnonlinearterm on the right is eclualto its valuefrom the previous iteration,the equationcan be solvedfor , _
/ r ' A . r 2 r -+ o ' A r 2 ( {
-T,o)-tTi tlTi+t
(22.t9)
As in the Gauss-Seidel method,we can useEq. 122.19)to successively calculatethetemperatureof each node and iterateuntil the processconvergesto an acceptabletolerance. Although this approachwill not work fbr all cases,it convergesfbr many ODEs clerived from physicallybasedsystents.Hence,it can sometimesprove usefulfbr solvin-uproblems routinelyencountered in engineering and science. E X A M P L E2 2 . 7
T h e F i n i t e - D i f f e r e n cMee r h o d f o r N o n l i n e o rO D E s Problem Stqtement. Use the finite-difl'erenceapproachto simulatethe temperature of a heatedrod subjectto both convcctionanclradiation:
o: +
+ h , ( r n_ r ) + o ,e, l
14)
w h e r e o ': 2 . 1 x l O - e K - 3 l n - 2 ,L : l 0 m , l r , : 0 . 0 5 m . , Z _ : 2 0 0 K . I ( 0 ) : 3 0 0 K , and I ( l0) : 400 K. Usefour interiornodeswith a segmentlengthof Ar : 2 m. Recallthat we solvedtlre sanreproblemwith the shootingmethodin Example22.4. Solution. Using Eq. (22.11)) we can successively solvefor the temperatures of therod's interior nodes. As with the standardGauss-Seideltechnique,the initial values of the interior nodesarezero with the boundarynodes.setat the fixeclconclitionsof I, : 300and Z. :400. The resultsfor the first iterationare T.-
T.:
- 01)+ 300+ 0 0.05(2)2 200+ 2.7 x t}-e (2)2(2001 :159.2432 2 + 0.05Q)) :91.9614
559
PROBLEMS
Finitedifference
l0 x.m
22.11 FIGURE problem method lo solveo nonlineor Thef ledcircesoretheresu t of usingthefinite difference T h ei n e g e n e r o t e d w i t h t h e s h o o t i n g m e t h o d i n E x o m p e 2 2 . 4 i s s h o w n f o r c o m p c r i s o n - 01)+ 91.9614 200+ 2.1x t\-e'(2)2(2001 +0 'r- - 0.05(2)2 :70.4161 2 + 0.05(D2
Ta*
- 04)+ 10.4461 200+ 2.'/ x rc e'(2)2(2004 0.0-5(2)2 + 400: 2 2 6 . 8 1 0 4
2 + 0.0s(D2
until we converge on thefinal result: Theprocess canbe continued Zo: 300 Tr :250'4827 Tz:236'2962 Tt:215.7596 T',- 286.4921 I. :400 in Example22.4 in Fig. 22.11alongwith theresultgenerated Theseresultsaredisplayed with theshootingmethod.
PROBtEMS 22.1 A stcady-statehcat balance for a rod can bc rcprescntcdas
|a_-\'o . t 5 r : 0
22.2 RepeatProb. 22.1 but with the right end insulatedand the left end tempcratureflxed at 240. 22.3 Use the shootingmethodto solve ,t
, \.
d\-
a
I
v
-
ll
Y
: 8. and with the boundaryconditions.r.(0): 5 and,-v(20) Obtain a solution fbr a l0-m rod with 7(0):240 r(10) : 150(a) analytically,(b) with the shootingmethod, 22.4 Solve Prob. 22.3 with the finite-difl'crenceapproach usingAr : 2. and (c) usin-ethc finitc-ditl'erenceapproachwith Ax : l.
PROBLEMS BOUNDARY.VALUE
560
22.5 The tbllowing nonlinear difl'erential equation was solved in Examples22.4 and 22.'7.
n:
o*4
+ h'(T* - D + o' (Tl
Tl)
(P22.5)
Such equationsarc sometimeslinearized to obtain an approximate solution.This is done by employing a flrst-order Taylor sericscxpansionto linearizc thc quartic term in the equationas o'71 : o'Ta + 4o'T3C -T) where 7 is a basetcmpcratureabout which the tcrm is linearized.Substitutcthis relationshipinlo Eq. (P22.5),and thcn solve the resulting lincar cquation with the finitedifl'erenceapproach.Employ 7 : .r00. Ar : I m. and the paramctcrsfiom Exarnple22.21to obtain your solution.Plot your results along with those obtaincd fbr the nonlinear versionsin Exarnples22.4 and 22.1. 22.6 Dcvelop an M-file to implernentthe shootingme(hod for a linear second-orderODE. Tcst the program by duplicatingExamplc22.2. 22.7 Dcvelop an M-file 1o irnplementthe flnite-differencc approach fbr solving a linear sccond-order ODE with Dirichletboundaryconditions.Testit by duplicatingExample 22.5. 22.8 An insulatedheatedrod with a unifolm hcat sourcecan b e m o d e l c dw i t h t h e P u i t t o t t( u t t ( t t i ( t L '
an( be
a,,.
uk: 1l=
wh len bo
FIGUREP22.IO of the cone wall. Thc equationhas the boundaryconditions: u(,r:0):
Q
a(,r: l):
l
Solvc this equation for the tcmperaturedistributionusing finite-diff'erencemethods.Usc second-orderaccuratefinitediff'crenccfbrmulas for the derivatives.Write a computer versus program to obtain thc solution and plot temperature (l- I axial distancefbr variousvaluesof p : I 0. 20, 50, and100. : .l(r) u.x22.11 CompoundA diffuses through a 4-cm-longtubeand The equationgoverningdiffusionwith Given a hcat source/(,1 ) : 25 "Clm2and the boundarycon- reactsas it diffr.rses. rcaction is : : 2 0 0 ' ' C . : 0 ) : , 1 0 ' C ( , t 10) solve and f d i t i o n s7 ( - t fbr the temperaturedistributionwith (a) the shootingmcthod J)A D kA:0 and (b) the finite-difI'crcncemethod(A-r : 2). ax' 22.9 RepeatProb. 22.8, but fbr the following spatiallyvaryAt one end of the tube (,1: 0), there is a largesourceof .4 i n g h e a ts o u r c e.:l { r 1 : 0 . l 2 . r r 2 . 4 1 2- 1 2 . r . 22.10 The temperaturc distribution in a tapered conical that resultsin a fixed concentrationof 0.1 M. At the other cooling fin (Fig. P22.10)is desclibedby thc fbllowing dif'- end of the tubc thcreis a materialthat quickly absorbsanyA, makingtheconccntration 0 M. If D : 1.5 x 10-6cm2/sand f-erentialeouaiion.which has been nondimcnsionalized: I, : x s is ft 5 l0-o what thc concentrationofA asa func0-u tion of distancein the tube') ,t.t 22.12 The tbllowing diffcrcntial equation describesthe whcrc a : temperature(0 < a < l). .r: axial distancc stcady-stateconcentrationof a substancethat reactswith (0 <,r < l). and p is a nondimensional parameterthat de- first-ordcrkineticsin an axially dispersedplug-flowreactor (Fig.P22.12): gcomctrv: scribcsthe heat transf-erand
.(:)(# P,):
NL
T
rJ U'- 2,,,]
where /z : a heat transfercocfficicnt, k: thermal conductivity, L : the lengthor height of thc cone, andm - the slopc
t a t
(1"(
A(
D__U__kt.:O drr d,r where 1) : the dispersioncoefficient 1m2/hr),c : concentration (mol/L), ,r : distance(m), U: the velocity(m/hr),
tra ter an ap Cc
wl
22 de
Ifr rea us
PROBTEMS
561
FIGUREP22.12 An oxiolly dlspersed plugflowreoctor andt : the reactionrate (/hr). The boundaryconditionscan beformr-rlatcd as
Usc thc finite-differenceapproachto solvc fbr the concentration of cach reactantas a tirnction of clistancegiven: D : 0 .I m 2 / n r i nU, : I m / r n i n ,t r : 3 / r n i n , k r : l l m i n , L : tlc : 0) U c i n* U c ( . r : 0 ) - D - ( r 0.5 m. c,,',,: l0 rnol/L. Enrploycenteredfinite-diltcrence approximirtionswith z\-r : 0.05 m to obtain your solutious dc -(r:/_):0 and assumeDanckwertsboundaryconditionsasdescribedin dI Prob. 22.12. Also, computr-the sum o1'the reactantsas a where.in: theconcentration in the inf]ow (mol/L), l : the tunctionof distance.Do your resultsmakesensc''J iengthof the reactor (m). Thcse are callcd Dant'kverts 22.14 Ahiolllnr with a thicknessL, (crn).grows on the surboundun'conditions. face of a solid {Fig. P22.14).After traversinga dillusion Usethe tinite-difl-erenceapproachto solve tbr concen- layer of thickness| (cm), a chemicalcompoundA diffuscs tration asa functionof distancegiven the tbllowingparame- into thc biofllrn where it is subjcctto an irrevcrsiblefirstrers: D: 5000mr/hr, U : 100rn/hr.li :2/hr. | : t00 rn, ordcr reactionthat convertsit to a productB. nndc'n : 100 mol/L. Employ centeredflnite-diffbrence Steady-statcmass balancescan be used to derive the approximations with A.r : l0 rr to obtainyour solutions. fbllowing ordinary diltbrcntial equationsfor cornpoundA: your numericalrcsultswith the analyticalsolution: Compale tl) c,, D +:0 0
DXt)).\e):t-
\U ,
D).)L.e) tl lL,t'":L/
'' -
,1). ),rr')'t r'i
,
where
,\:#(-'/;-#) 22,13A seriesof first-ordcr,liquid-phasereactionscreatea product(B) and an undcsirablcbyproduct(C): desirable
Dr!-frc.,,:0 ur'
< Ll
Lt
wherel) : the diftusion coefficientin the diffusion layer : 0.8 cm /d, D1 : ths ditTusioncoetTicient in the biofilm : 0.611 cmr/d, and A : thc flrst-ordcrlate for the conversionol A to B :0. l/d. The tbllowing boundalyconditionshold: (rt : Ll. ,
(ttl)
--r:0 tlx
n\s5c
l. <x
at -f :
0
ats=L-Lt
If thereactionstakeplacein an axially dispcrsedplug-flr>w where r',,e: the concentrationof A in the bulk liquid : 100rnol/L. Lr.sethc' finitc-difl-erencc nrethodfo cornp.rtethc reactor(Fig. P22.)2). .stead),-sIare nrass balance.scan bc : 0 to L * Lr. where steady-state fiom distribution of A .r usedto dcvelol'rthe following sL-cond-ordel'ODEs: f, : 0.008 cm and L, = 0.001 cm. Employ centeredfinite tl-L', al(, ditlerenceswith A.r : 0.001crn. D l1r'.,:() U; d-t' d-t 22.15 A cable is hangingfiom two supportsat A and B (Fig. P22.I5). The cable is loadedwith a distributedload c/lr'r dr'^ D lrr',, -(-ri,-0 U--J whosemagnitudevarieswith -r as ax' dx t ) 1
tl-a.
D -
d t-
tlt
-U---(lx
| {1r1,-9
-,t,(, ,t, f' *.'"(#)l
562
PROBLEMS BOUNDARY-VAIUE Solid surface
1
o
F tf I
-
II
I
Fl(
U
FIGUREP22.14 growing A biofilm ono soid surfoce ))
ela!
where ur,,: 450 N/m. The slope of the cable (d,yldx) : 0 ar .r : 0, which is the lowestpoint tbr the cable.It is also the point wherethe tensionin the cableis a minimumof 7,,.The difl-erential equationwhich governsthe cableis 1 )
1l- v
ux-
22.16 The basicdifl'erentialequationof theelasticcurvefor is a simply supported, unifbrmlyloadedbeam(Fig.P22.16) given as FI:
; ['.,'"(#)]
{1- I'
-
u'L-\
l|-T
)
dr)22
Solve this equation using a numerical method and plot the shapeof the cable (,r versus,t). For the numerical solution, the value of I,, is unknown, so the solution must use an iterative technique,sirnilarto the shootingmethod,to converge on a correct value ol ft, fbr variousvalueso1'I,,.
where E : the modulusof elasticity,and1: themoment of incrtia.The boundaryconditionsare.y(0):.r(l) = 0. Solve lbr the def'lectionof thc beam using (a) the finite-difference approach(A-r : 0.6 m) and (b) thc shootingmethod. The f-ollowing parameter values apply: E : 200 GPa,1 = your 30,000 cma, ur : 15 kN/m, and L : 3 m. Compare
) ht= 15m
= w,,11 + sin (nxl2l I'r=60m
FIGURE P22.15
Nol fun, terr
For For
PROBTEMS
563 that thc end displacenrents are zcro.r'{0) :1,(L) : 0, and that the end momentsarezero,),"(0) : )-"(L) :0. Solvcfor the detlection of the beam using thc finite-ditferenceapproach (Al : 0.6 m). The following parametervaluesapply: E : 200 GPa,1: 30.000crra,ur : 15kN/rn,andL: 3 m. Compare your numerical results with the anall,tical solr,rtion g i v e ni n P r o b . 2 2 . l 6 . 22.lll Under a nunlber of sirnplitying assumptions,the steady-state heightof the \.r,atcr tablein a onc-dimensional. unconflnedgroundwateraquif'er(Fig. P22.18)can be modeledwith the fbllowing second-order ODE:
FIGURE P22.16
- (l- ll
Klt-+N:0 (IX-
numcricalresultsto the analyticalsolution:
'
u,L,t'l
u,,r4
lzEr
24Er
u,Llr
24Et
22.17ln Prob.22.I 6, the basicdifferentialequationof the clasticcurve for a unifbrmly loadedbeamwas fblmulated as FI:
tl2 t'
tLtLx
d -)(-
1a
u't2
Note that the right-hand side representsthe mornent as a functionof r. An equivalentapproachcan be fbrmulatedin termsof the fburth dcrivative of deflectionas EI
da t' .: _u, ar-
For this formulation. four boundaryconditionsare required. For the sr.rpports shown in Fig. P22.16,the conditionsare
wherex : distance(m), K: hydraulicconductivity(m/d), /r : height of the water table (rn), /r : the averageheight of the watertable(rn),andN: infiltrationrate(m/d). Solvc fbr the height of the water tablc for x : 0 to 1 0 0 0m * ' h e r ef t ( O ) : l 0 m a n d / t ( 1 0 0 0 ) : 5 m . U s e l h c following parametersfbr the calculation:K : I m/d and N : 0.0001m/d. Set the averagehcight of the watcr lable as Obtainyoul solution theavelageof theboLrndary conditions. with (a) the shooting mcthod and (b) the finite-difference mcthod(A,r: I00 m). 22.19 In Prob.22.18.a linearizedgroundwatcrmodel was usedto simulatethe height of the water table fbr an unconfined aquitcr. A more realistic result can bc obtained by usinsthc fbllowins nonlinearODE:
* (r,'#)+ N : 0
Ground surface
Water table
+
l n fi ltr a t i o n
+ I ++
qr"-
C o n f i n i n gb e d
F I G U RP E2 2 . 1 8 A,. ,'n.nnfined or "ohreolic"
:-:---
BOUNDARY-VALUE PROBLEMS
564
where x : distance(m), K : hydraulic conductivity (m/d), ft : height of the water table (m), and N : infiltration rate (m/d). Solve for the height of the water table for the same c a s ea s i n P r o b . 2 2 1 . 8 .T h a t i s , s o l v ef r o m x : 0 t o 1 0 0 0m with ft(0) : l0 m, ft(1000): 5 m, K: I m/d, and N : 0.0001 m/d. Obtain your solution with (a) the shooting method and (b) the finite-differencemethod (Ar: 100 m). 22.20 Justas Fourier's law and the heatbalancecan be employed to characterizetemperaturedistribution, analogous relationshipsare availableto model field problemsin other areasofengineering.For example,electricalengineersuse a similar approachwhen modeling electrostaticfields. Under a numberof simplifying assumptions,an analogof Fourier's law can be representedin one-dimensionalform as D:
-t-
dv dx
whereD is calledthe electricflux densityvector,s : permittivity of the material,and V : electrostaticpotential.Similarly, a Poisson equation (see Prob. 22.8) for electrostatic
fields can be represented in one dimensionas d2v
Pu
dx'
€
where pu : chargedensity.Use the finite-difference technique with Lx : 2 to determineV for a wire whereV(0)= 1000,Y(20): 0, e : 2, L:20, and pu : 30. 22.21 Supposethat the position of a falling objectis governed by the following differential equation: 11
d'x
'
c dx
__L___,'-il I t I
at'
r
tn dt
6
-v
where c : a first-order drag coefficient : 12.5kg/s, rz = : 9.81mA2. mass: 70 kg, andg : gravitationalacceleration Use the shooting method to solve this equationfor the boundaryconditions: r(0) : O .t(12):500
APPENDIX A EIGENVALUES Eigenvalue, or characteristic-value,problems are a special class of problems that are common in engineeringand scientificproblemcontextsinvolving vibrationsand elasticity. In addition, they are usedin a wide variety of other areasincluding the solution of linear differentialequationsand statistics. Before describingnumericalmethodsfor solving suchproblems,we will presentsome generalbackgroundinformation.This includesdiscussionof both the mathematicsand the engineeringand scientific significanceof eigenvalues.
A.l Morhemoticol Bockground Chapters8 through 12 dealt with methodsfor solving setsof linear algebraicequationsof the generalform [A]{r} : {b} Such systemsare callednonhontogeneoas becauseof the presenceof the vector { b } on the right-hand side of the equality. If the equationscomprising such a system are linearly independent(i.e., have a nonzerodeterminant),they will have a unique solution.In other words, thereis one set of x valuesthat will make the equationsbalance. In contrast,a homogeneorrs linear algebraicsystemhasthe generalform l A l { . t }: 0 Although nontrivialsolutions(i.e.,solutionsother than all x's:0) of suchsystemsare possible.they are generallynot unique.Rather,the simultaneousequationsestablishrelationshipsamongthe .r's that can be satisfiedby variouscombinationsof values. Eigenvalueproblemsassociatedwith engineeringare typically of the generalform ( a 1 1- ) " ) x 1 |
anxt*'.'*
{ t y x 1I ( a 2 2 - } , ) r 2+ . . . +
I q,1.1 I -T
alnxu:0 a2rFu:0
n r 3 x l * . - . - + ( e , , , ,- , 1 , ) x , , - 0
566
APPENDIX A EIGENVALUES where)' is an unknown parametercalledthe eigenvalue,or characteristicvalue.A solution {,r} tbr sucha systemis referredto as an eigenvector.The abovesetof equationsmay also be expressedconciselyas
l t a t - ^ t I ] ] { x :} o
(A.l)
The solutionof Eq. (A.l) hingeson determining.l..One way to accomplishthisis * ),[1]] must equalzerofor nonbasedon the fact that the determinantof the matrix ltal trivial solutionsto be possible.Expandingthe determinantyields a polynomialin )". which is called the characteristic'pol ,-nomial.Theroots of this polynornialare the solutionsfor the eigenvalues.An exampleof rhis approach,called the poll,yorrro,methocl,will be provided in SectionA.3. Beforedescribingthe method,we will first describehow eisenvalues a r i s ei n e n g i n e e r i nagn ds c i e n c e .
A.2 PhysicolBockground The mass-spring systemin Fig.A.lo is a simplecontextto illustratehow eigenvalues occur in physical ploblem settings.It also will help to illustrate some of the mathernatical conceptsintroducedin SectionA. l. ?l sintplity the analysis,assumethat eachrnasshasno externalor dampingfbrcesacting on it. ln addition, assumethat each spring has the samenaturallength / and the same springconstantt. Finally. assumethat the displacernent of eachspringis rneasuledrelative to its own local coordinatesystem with an origin at the spring's equilibrium position (Fig.A. la). Undertheseassumprions, Newton'ssecondlaw can be employedto develop a force balancelbr eachmass: d-Xr
mr-,i--krr
1k(x: --l 1.1
FIGURE A.I Positioning lhe mossesowoy from eqi,ilibriumcreotesforcesin the sprinoslhoi on rereoseieod to oscillctionsof lhe nrosses.The positionsof the mossescon be referenledfo locoJcoordinotes w i i h o r i g i n so t t h e i rr e s p e c t i veeq u i l i b r i u m posifions
000-l
tttt
7f---7t
f
000-l
nt2
?T-,7f
| - _0r 0 0
APPENDIX A EIGENVALUES
567
and 't) -'
tn:
u .\l -:-k(.r:-11)-(r3 dt-
where,t; is the displacement of massi away from its equilibriumposition(Fig.A.lb). By as collectingterms,theseequationscanbe expressed d-r'.
- ft(-2.r1*.r;1 : g
nr t'4
dt-
(4.2a)
t)
0--X1
n 1- : .-t r-: - - f t ( r t - 2 x ; ) : g
(A.2b)
From vibration theory,it is known that solutionsto Eq. (A.2) can take the form xi : Xi sin(a.rr)
(A.3)
where X; : the amplitudeof the vibration of massi anda: tion, which is equalto 2n
the frequencyof the vibra-
(A.4)
7,, where {, is the period.From Eq. (A.3) it follows that x'i : -Xia2 sin(a''t)
(A.5)
Equations(A.3) and (A.5) can be substitutedinto Eq. (A.2), which. after collection of terms,can be expressedas
(#--')",
-kX z - 0
(A.6ru)
m1
- !-x, . (# - ,')x, : o
(A.6b)
Comparisonof Eq. (4'.6) with Eq. (A. l) indicatesthat at this point, the solutionhas beenreducedto an eigenvalueproblem.That is, we can determinevaluesof the eigenvalue systemsuch as Fig. A.l, there ol2 Ihat satisfy the equations.For a two-degree-of-freedom will be two such values. Each of these eigenvaluesestablishesa unique relationship betweenthe unknownsX calledan eigenvector.SectionA.3 describesa simpleapproachto determine both the eigenvaluesand eigenvectors.lt also illustratesthe physical significanceof thesequantitiesfbr the mass-springsystem.
A.3 The Polynomiol Merhod As stated at the end of Section A.1, the polynomictlnrcthod consistsof expandingthe determinantto generatethe characteristicpolynomial.The roots of this polynomial are the solutions for the eigenvalues.The following example illustrateshow it can be used to determineboth the eigenvaluesand eigenvectors for the mass-sprinqsystem(Fie. A.1).
s68
APPENDIX A EIGENVALUES
EXAMPLE A.l
T h e P o l y n o m i oM l ethod I i
Problem Stotement. Evaluatethe eigenvalues anclthe eigenvectors of Eq. (A.6) for the casewhererl | : tt12: 40 kg and li : 200 N/m. Solution. Substitutingthe parametervaluesinto Eq. (A.6) yields (f0
@2)xt-
5 X z- 0
- 5 X 1 * ( 1 0 - r r . , r ; X 2- o The determinantof this systemis ( r ' ) t - 2 o o 2+ 7 5 : o which can be solved by the quadraticformula for o,f : l-5 ancl-5s 2. Therefore,thefrequenciesfor the vibrationsof the massesare .D: 3.873s I and 2.236 s-t, respectively. Thesevaluescan be usedto determinethe periodsfbr the vibrationswith Eq. (A.4). Forthe first mode"Tp : 1.62s. and fbr the second,7,, :2.81 s. As statedin SectionA.l, a unique set of valuescannot be clbtainedfor the unknown amplitudesX. However,their ratios can be specifiedby substitutingthe eigenvalues back
A.2 FIGURE
T h e p r i n c i p o lm o d e so f v l b r o t l o no f t w o e q u o l m o s s e sc o n n e c t e db y t h r e ei d e n l i c osl p rn g s betweenfixed wolls
T_
1.62
2.81
(a) First mode
(b) Secondmode
APPENDIX A EIGENVALUES For example.for the flrst mode(ro2: into the equations.
569 l5 s 2):
(10-15)xr -5X2:g - 5 X r * ( 1 0 - 1 5 ) X 2: 0 T h u s . w e c o n c l u d e t h a t X- -l X 2 . l n a s i n r i l a r f a s h i o n f b r t h e s e c o n d m o d e (:r5, .s,-1r ) . arethe eigenvectors. Xr : Xt. Theserelationships This exampleprovidesvaluableinformation regardingthe behaviorof the systentin Fig. A.L Aside from its period,we know that if the systemis vibratingin the first mode, the eigenvectortells us that the amplitucle of the secondmasswill be equalbut of opposite sign to the amplitudeof the first. As in Fig. A.2a, the massesvibrateapartand thentogether indefinitely. In the secondmode, the eigenvectorspecifiesthat the two mersses have equal amplitudesat all times.Thus,as in Fig. 4.2b, they vibratebackand forth in unison.We should note that the configurationof the amplitudesprovidesguidanceon how to set their initial valuesto attain pure motion in either of the two modes.Any other configurntionwill lead of tlrerrodes. to superposition
We shouldrecognizethatMATLAB hasbuilt-infunctionsto facilitatethe polynomial method.For ExarnpieA.l, the poly functioncan be usedto generatethe characteristic polynontial as in >> A = t10 -5;-5 -_.-> p
IO);
= pot\,(A)
r
-20
7a
Then,the roots functioncan be employedto computethe eigenvalues: >> roots (p) utt t=, 5
A.4 The Power Merhod The power method is an iterativeapproachthat can be employedto determinethe largest or dominanteig,envctlue. With slight modification,it can alsobe employedfo determinethe smallestvalue.It hasthe additionalbenefitthat the corlesponding ei-eenvector is obtained as a by-productof the method. To implementthe power method,the systembeing analyzedis expressedin the form
[ A l { . r }: ) " { r )
(4.7)
As illustratedby the following example,Eq. (A.7) forms the basisfbr an iterativesolr.rtion techniquethat eventuallyyields the highesteigenvalueand its associatedeigenvector.
APPENDIX A EIGENVALUES EXAMPLE A.2
PowerMethodfor HighestEigenvolue A.2,we canderivethefolasin Section ProblemStotement.Usingthesameapproach for a threemass-fourspringsystembetween two lowing homogeneous setof equations fixed walls:
( ' * _ - ' ) x ,\lnr
r r.
:0
htt
/
k --- v
/2k - (l?1r
,\ .')r,
-
^t, m)
X,:o
fr,, (* -')x,:o If all the massesm : I kg and all the spring constantsk : 20 N/m, the systemcan be expressedin the matrix format of Eq. (A. 1) as
[40 -20 | L 0
-20 0 I 40 -20 - rli l:0 - 2 0 4 0 )|
). is the squareof the angularfiequencyc,r2.Employthe power wherethe eigenvalue andits associated eigenvector. methodto determine thehighesteigenvalue is firstwrittenin thelbrm of Eq.(A.7): Solution. Thesystem 4 0 X r- 2 0 X 2
: ),Xt
-20X1+40X2-20X2:)"Xz -20x2 -l40Xt : )'Xs At this point, we can make initial valuesof the X's and usethe lefrhand sideto computean eigenvalueand eigenvector.A good first choiceis to assumethat all the X's on the left-hand side of the equationare equal to one: :20 4 0 ( 1 )- 2 0 ( l ) -20(1)+40(l)-20(1):0 -20(l) -l40(l) :20 Next, the right-handside is normalizedby 20 to make the largestelementequalto one:
[20] [tl { 0l:20{01
lzol
lrl
Thus, the normalizationfactor is our first estimateof the eigenvalue(20) and the conespondingeigenvectoris I I 0 I .lr . This iterationcan be expressedconciselyin matrix
form as 40 [ -20 | Inan
-20 40
APPENDIX A EIGENVALUES
571
T h e n e x ti t e r a t i o nc o n s i s l o s f m u l t i p l y i n gl h e m a t r i xb y I
0
-1ll:-' jl, -n] [1l' l?]:
1lz togive
I -l
1
Therefore.the eieenvalueestimatefor the seconditerationis 40. which can be emplovedto detenrinethe errorestimate: 'r,,,:l 140-l0l ^ lxlf)07: I +r, The processcan then be repeated Tltird iterutiott:
[ 4- 200 |
-20 40
0lll -l(l I I -l
L;" -; r;ll where le,,| :
60 -80 60
-
_RO
-0.75 I -0.75
l50c/o(which is high becauseof the signchange).
Fourtlt iteratiou:
: :;;n] rl:l [1$ { i,,}: "
'or'.;troo'"nul
where lc,,| : 2147o(wlrichis high becauseof the signchange).
FiJthiteratiort: [40 t-20
L o
-20 40
0l]-0.1142e -t0t{ |
:llfii'i^
_20 +ol[_.0.1142s
-0.70833 1 :68.57141 I t -0.70833
wherele,,| : 2.08c/c . Thus, the eigenvarlue is convergir.rg. Afier severalmore iterations,it stabilizeson a value of 68.28421with a correspondingeigenvectorof | -0.707 107 I -0.101 lU )r . Note that there are some instanceswhere the power method will convergeto the second-largest eigenvalueinsteadof to the largest.James.Snrith,andWolford( 1985)proin Fadeevand Fadeeva vide an illustrationof sucha case.Otherspecialcasesarediscussed (l 963). In addition, there are sometimescaseswhere we are interestedin determining the This can be doneby applyingthe powermethodto the matrixinverse smallesteigenvalue. other of [Al. For this case,the powermethodwill convergeon the largestvalueof l/l-in will be left as words,thesmallestvalueof i. An applicationto find the smallesteigenvalire a problemexercise.
572
APPENDIX A EIGENVALUES Finally, after finding the largesteigenvalue,it is possibleto deternine the next highest by replacingthe original matrix by one that includesonly the remainingeigenvalues.The processof removing the largestknown eigenvalueis calleddeflation. We shouidmentionthat aithoughthe power methodcan be usedto iocateintermediate values,bettermethodsare availablefor caseswherewe needto determineall the eigenvalues as describedin SectionA.5. Thus, the power methodis primarily usedwhen we want to locatethe largestor the smallesteigenvalue. A.5 MATLAB Function: ers As might be expected,MATLAB haspowerful and robustcapabilitiesfor evaluatingeigenvaluesand eigenvectors.The function e:ig, which is usedfor this purpose,can be usedto generatea vector of the eigenvaluesas in -
e
elgt'4t
where e is a vector containingthe eigenvaluesof a squarematrix a. Alternatively.it canbe invokedas
where Dis a diagonalrnatrix of the eigenvaluesand I'is a full matrix whose columnsare the correspondingeigenvectors. E X A M P L EA . 3
U s e o f M A T L A Bt o D e t e r m i n eE i g e n v o l u e os n d E i g e n v e c t o r s Problem Stotement. Use MAILAB to determineall the eigenvaluesand eigenvectors for the systemdescribedin ExampleA.2. Solution. Recall that the n.ratrixto be analyzedis
[ 4-20 0 | L 0
-20 0 I 40 -20 | -20 40 _.1
The matrix can be enteredas >> A = 140 -20 O;-2A 4A -2A;0
20 4al;
If we just desirethe eigenvalueswe can enter r
e
= eiq(Al
II
.1 I51
e= 40. OOOO 68.2,843
Notice that the highesteigenvalue(68.2843)is consistentwith the value previouslydetermined with the power methodin ExampleA.2.
APPENDIX A EIGENVALUES
573
If we want both the eigenvaluesand eigenvectors,we can enter
0.5000 a.7alI 0.5000
0.7411 -0.0000
LI.1 I51 0 0
0 40.0000 0
-0.5000 4.107r -0.5000
0.70?1
C1 -
0 0 68.2843
Again, although the results are scaled differently. the eigenvectorconespondingto the 0.707I -0.5]7 is consistentwith the value previously highest eigenvalue[-0.5 d e t e r r n i n ewd i t h t h ep o w e rm c t h o di n E x a m p l eA . 2 : | - 0 . 7 0 7 1 0 7 | - 0 . 1 0 7 1 0 7) ' .
AND EARTHAUAKES , . . EIGENVALUES Engineersand scientistsusemass-springmodelsto gain insight into the Bockground. dynamicsof structuresunderthe influenceof disturbancessuchas earthquakes. FigureA.3 shows such a representationfor a three-storybuilding. Each floor massis representedby nt,, and each floor stiffnessis representedby ft, for i : 1 to 3. For this case,the analysisis limited to horizontalmotion of the structureas it is subjected to horizontalbasemotion due to earthquakes. Using the sameapproachasdeveloped in SectionA.2. dvnamic force balancescan be developedfor this svstemas
-^t { ^ 'lrr - r l . l x , /t.
I
t.
\
\
K2
t.
___: mz
/kz*kz Xt I l - - u t , l \
tTlt
L
^3
*,
-n
x2
*t
/
'\ /
L-
X::0
X2mZ
*,*(h-oi)x.:o
where X, representshorizontal floor translations (rn), and @,,is the natura[ or rcsonant, frequency (r'adians/s).The resonantfiequency can be expressedin Hertz (cycles/s)by dividing itby 21r radians/cycle. Use MATLAB to determinethe eigenvaluesand eigenvectorsfor this system.Graphically representthe modesof vibration for the structureby displayingthe amplitudesversusheight for eachof the eigenvectors.Normalizethe amplitudesso that the translationof rhe third floor is one.
574
APPENDIX A EIGENVALUES
continued lar : 8,000 kg
| .....
-UUU-..--]
t,-
1,800kN/m
t,-
2,400kN/m
t.-
3.000 kN/m
rh : 10,000kg
l- *
uuu --l
'|1r: 12.000kg
[*-
FIGURE A.3 Solution.
The parameterscan be substitutedinto the force balancesto give
-o
2oox2 (+so-.|,)xr- 2 4 0 X t +( 4 2 0 - r T ) X ,-
180x,:s 225X2+(225-u'l)x.:g
A MAILAB
session can be conducted to evaluate the eigenvalues and eigenvectors as
>> A= f 450 -2llC 0; -?.40 42i) '>:>
lv,
clr.-ciq
0.5!ll!r O.rl0'i 0 . _ 14 1 1 ci= irq3.!982 0 {t
18C;0 -225
2251;
(;r)
-Lr.6l4i1 -0.3506 0.5890
0.2911 0.5725 0.1654
0 139.4t'i9 0
0 0 56.9239
Therefore, the eigenvalues are 698.6, 339.5, and 56.92, and the resonant frequencies in Hz are :> L/:r-s.irt {ci-iaq(d)) 'i 2ipi 4.2t65
2.9324
1.2008
The corresponding eigenvectors are (normalizing is one)
so that the amplitude for the third floor
0.38008e 1
I } 0.746eee {lt:'#:}{-3.33il;
575
A EIGENVALUES APPENDIX
continued
-1 Mode 1 la\:1''OOU^r,
0
1
Mode 2 ( a L : 2 . 9 t r Or r ,
-1
0
1
Mode 3 ko,, = 4.2066 Hzl
A.4 FIGUR.E A graph can be made showing the three modes (Fig. A.4). Note that we have orderedthem fiom the lowest to the highest natural frequency as is customary in structural engineering. Natural frequenciesand mode shapesare characteristicsof structuresin terms of their tendenciesto resonateat thesefrequencies.The frequencycontentof an eafihquaketypically has the most energy between0 to 20 Hz and is influenced by the earthquakemagnitude,the epicentral distance,and other factors. Rather than a single frequency,they contain a spectrum of all frequencieswith varying amplitudes.Buildings are more receptiveto vibration at their lower modes of vibrations due to their simpler deformed shapesand requiring less strain energy to deform in the lower modes.When theseamplitudescoincide with the natural frequenciesof buildings, large dynamic responsesare induced creating large stresses and strainsin the structure'sbeams,columns,and foundations.Basedon analyseslike the one in this case study, structural engineerscan more wisely design buildings to withstand earthquakeswith a good factor of safety.
1ilii;l ,' .i,ii
'', ,iii'
ii.il"
APPEND XB MATLABBUILT.IN FUNCTIONS abs, 30 acos, 30 asci i, 5l axis square,35 beep,62 b,-ssel j, 356 ce11,31, I 80 chol,245 c1abe1, 180,466 clear, 50 cond, 257, 338 contour, 180,466 con_,,,157 cumtrapz,4l4 db1quad,4l8 deconv,156 det,2l7 dtaq,574 dlff.4l3, 460,462 disp,47 double,5l eis,572 e1fun,30
fminsearch , 181,326 format bank,23 format f crrmat format formac format forma[
compact,2l 7onrJ,23 tong e,23 iong eng,23 toose,2l short,23
format short format short fp1ot,67 fprint f, 48 fzero, l-51 gradient,463 grrlo!
-1J
help, 30 help elfun.30 hist,29l nord oIt, J4 hold on.34 humps, 78,440
eps, 90 erf,445 error, -52 exp,30 e1'e, 202 factorial.40,59 fix,365
inline,67 input,47 inr-erp1,376 interp2,38l int-erp3, 3[32 inv, 201,203 isempcrr, 127 1 e _ q e n d3, 8 5 , 4 1 5 1engtl-r.32
f1oor,31,180 fminbnd, 178
linspace,26 1oad,50
e.23 eng,23
roq,30 1oql0,30 Ios2,30, 126 loslos, 40,106 logspace,26 Lookfor,36,44 )..
)L'.)
max,31,226,290 mearr,31,290 median, 290 mesh.62 m e s h s r i d , I 8 0 .4 6 6 min.31,290 mode,290 riargin, 57 norm,25J ode1l3,5l7 ode15s,529 ode23,5l6 ode23s,529 ode23L,529 ode23tb,529 ode45,517,546 odeset,519 ones,25 o p t i m s e t , 1 5 3 ,1 7 9 . 3 2 6 pause,62 pchip, 374,377 p1ot,33 p l o t 3 , 3 5 ,- 5 0 8 porv. 156,569
APPENDIX B MATLABBUILT-IN FUNCTIONS p o l y f i , r , 3 0 1 , 3 2 5 ,3 3 9 P o f y \ / a l . 1 5 6 ,3 0 7 , 339 prod,3l quad,439 qrLadi,439 quirrer,4(t5 reatmax,90 realmin,90 roots, 155,569 ror-rncl, 3 | save, 50 semriogy,40 set,385
sign,55 sin,30 size,202 sort,3l sp1rne,374 sqrt, 30 sqrtm,3l srd,290 S U r r p LO t ! - 1 4
surn,3 1, 246 surfc,180 tan, 30 l-anh, 7
577 rrc,62 rr-
o
11
toc,62 trapz,4l4,422 triplequad,4lS '",ar.290 vararqJin, T0 who, 25 whos, 25 )414De1,
-1J
ylabel, 33 zeros,25 :labe1.180
APPENDIX C MATLABM.FILEFUNCTIONS
M-file Nome
Descripfion
Poge
brsecL euLode GaussNaive
R o o tl o c o l i o nw i t h b i s e c t i o n n l e g r o l i o no f o s i n ge o r d i n o r yd l f f e r e n t i oe q u o t L o nw i l h E u l e r ' sm e t h o d S o v i n g l i n e c rs y s l e m sw i l h G o u s se i m i n o l i o nw f h o u lp i v o l r n g S o v i n g i n e c r s y s i e m sw i t h G o u s se m i n a l i o nw i t h p o r t i o lp i v o t i n g S o l v i n g i n e o r s y s l e m sw i t h t h - -G o u s s - S e i d eml e t h o d M i n l m u mo f o n e - d i n , e n s i c t nf o u in c t i o nw i t h g o l d e n - s e c t i osne o r c h R o o t o c o t i o n r , v i l ho n i n c r e m e n i csi e o r c h I n t e r p o l o l r ow n i t h t h e L o g r o n g ep o y n o m i o l l i r q o I ' og \ l l i r e r r ' r 'I ' r : o l e g - - , i o r
127 486 222 227 269 176 t2c 349 306 383 346 276 149 542 432 364 404 413 '2')9
t,allssPrvor
GaussSeidel goldrrLin incse.rr-.:h 1rnregr n-rf
cniina
N e w Lr n t nerrtmu f t newtraph rk4 sys romberg Tabl eLo.ik arap r rapuneLl Tridiag
C u b i cs p i n ew i t h n c l u r c e l nd conditions nlerpolotion with the Newion polynomiol R o o l o c o l i o n f o r n o n i n e o r s y s t e m so [ e q u o i r o n s Root locolion with lhe Ner,r,ion-Rophson method I n t e g r c l i o no l s y s t e mo l O D E s w l t h 4 t h o r d e r R K m e t h o d I n t e g r o i i o no f o [ u n c l i o nv " , i l hR o m b e r gi n l e g r o t i o n T o b l e o o k u p w i t h l i n e o ri n t e r p o J o t i o n lniegrolioncl c funclicn \\,rthlhe .')rlposit6 i16pa1616Jo 1ul-. Integrofion o f u n e q r i s p c c e dd c r t cw i t h t h e i r o p e z o i d c r u l e S o l v i n gl r i d i c g o n c ll i n e c rsr y s t e m s
