Pade Approximants Method and Its Applications to Mechanics (Lecture Notes in Physics, Volume 47 1976)

Lecture Notes in Physics Edited by J. Ehlers, MLinchen, K. Hepp, Zerich, H. A. Weidenm~ller, Heidelberg, and J. Zitta...

Author: H. Cabannes

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Lecture Notes in

Physics

Edited by J. Ehlers, MLinchen, K. Hepp, Zerich, H. A. Weidenm~ller, Heidelberg, and J. Zittartz, K61n Managing Editor: W. BeiglbSck, Heidelberg

47 Pade Approximants Method and Its Applications to Mechanics

Edited by H. Cabannes

Springer-Verlag Berlin. Heidelberg. New York 197 6

Editor Prof. Henri Cabannes Universit@ Pierre et Marie Curie Mecanique Theorique Tour 66, 4, Place Jussieu 7 5 0 0 5 Paris/France

Library of Congress Cataloging in Publication Data

Main entry under title: Fade approximants method and its applications .to mechanics. (Lecture notes in physics ; 47) Bibliography: p. Includes index. I. Pade approximant. 2. Fluid mechanics. I. Cabannes, Henri. II. Series. QC20.7.P3P35 532' .01' 515 75-46504

ISBN 3-540-07614-X Springer-Verlag Berlin - Heidelberg • New York ISBN 0-387-07614-X Springer-Verlag New York • Heidelberg • Berlin This ,,york is subject to copyright, All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, re* printing, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1976 Printed in Germany Printing and binding: Offsetdruckerei Beltz, Hemsbach/Bergstr.

Henri Pad~

(1863 - 1953)

Henri Euggne Pad~ was born in Abbeville

(France) on Dec.

17th 1863.

Admitted in 1883 to the Ecole Normale Supgrieure, he left it in 1886 with the highest teacher's degree

(Agr~gation)

in Mathematics.

After teaching at the classi-

cal secondary school in Limoges, Carcassonne and Montpellier, in 1889 in order to study in Germany,

first in Leipzig,

he was granted a leave

then in G~ttingen.

On June

21st 1892, before the University of Paris, he defended his doctorate thesis on the approximate representation of a function by rational fractions. Henri Pad~ was appointed lecturer at the Faculty of Sciences of Lille in 1897, Professor of Rational and Applied Mechanics at the Faculty of Sciences of Poitiers

in 1902 and Professor of Mechanics at the Faculty of Bordeaux in 1903. In

1906, he was elected Dean of the Faculty of Sciences of Bordeaux and became Laureate with the major prize for mathematical on the report of Emile Picard.

sciences awarted by the Academy of Sciences

In 1908 he was named Rector of the Academy of

Besan~on, he was then the youngest rector in France.

In 1917 he became rector of the

Academy of Dijon and in 1923 rector of the Academy of Aix-Marseilles, kept until he retired in 1934. Henri Pad~ died in 1953 at the age of 89.

an office he

Henri

Pad~

(1863

-

1953)

/

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.

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de l'Enseiynement sup@rleur.

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d ht

d@dche en

Besanfo~4 le

,

j

juillet lglO.

tl~tte

du OBJET :

~emande dtautorisatlon -d'&bsence. J'al l'hormeur de vou~ informer que je compte pr~m/ dre quelque reDos p e n d a r ~ les va.cances. ~on absence durera!5

~

ma~xlmunk six semalnes,du G

/

aO~b au 15 s e p t e m b r e , p e n d ~ lesquelles je ne quibbei~als ) pas le ressor% et me t l e n < ~ i s Dr~t ~ reprendre mort set/ vice sl ies clrconsts/%ces~'exlgealent.J'aural d'ailleurs 2 soln de vous faire c o n n a ~ r e mort adress.e exac!be pendant mon dep i ac ement. Le servlee serait, con~ne d'habltude,assure

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v c . ' 2 ~"

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~iouvcznnon

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pay mes

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proJet.

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Editor's Preface

In 1892, in the Scientific Paris,

the french mathematician

proximate

representation

century later, rious methods

of representing

a table of rational

tion of functions

computers especially

led scientists

of the functions

algorithms.

fractions,

45

47

functions.

of pad~'smethod, series of powers

by these series.

have studied

the representa-

the main object being to obtain

The most celebrated

the sum of the following

4- + S = 4 - --3

or diverging

and applied mathematicians

by means of rational

of a

to consider va-

rapidly converging

represented

in

the ap-

Three-quarters

in 1955, D. Shanks showed the advantages

approximations

ever faster computing for calculating

functions,

Sup~rieure

an article concerning

fractions.

to deduce from any converging

Since then many physicists

(I)

Henri Pad~ published

the advent of arithmetical

it possible

of the Ecole Normale

of a function by rational

In a paper published which makes

Transactions

and now classic example

is

series:

+ (_)n

"'"

The sum S is equal to 7. But to obtain from series

2 n 4+ I

+ . ""

(I) -- which is a very poor al-

gorithm -- the value o f ~ c o r r e c t to eight digits, the first term which is neglected -8 in absolute value, which means t h a t w e must consider 20 mil-

must be less than I0 lion terms. sum S

If however one applies Padg's

of the first n terms,

n sequence

then terms

- S 2 Sn-I Sn+l n S n - 1 + Sn+ I - 2 S n

v Ln =

(2) to of

obtain series

the

same accuracy

(1)

and

The underlying

to

D,I~

transformation

that is if one associates

it

apply

principle

the

will

repeatedly

with the sequence

to the S

n

the new

' suffice

transformation

of Pad~'s method

to

consider

only

(2)

times.

five

is particularly

the

first

simple.

nine

Given a

power series co (3)

Pad~ proposed fraction Pm(X)

~ k=0

ck

finding / Qn(X)

xk

,

the closest approximation , with

to the sum, by defining a rational

(4)

Pm(X) =

m ~ ak k=0

xk ,

are polynomials

Ck

xk

-

This identity leads to m+n+l bk

(b

(6) A Pm(X)

0

=

=

1)

from the identity

co

Qn (x) k=0~

a k and

xk ,

of degrees m and n res-

These polynomials Pm(X) and Qn(X) are determinated oo

(5)

bk

k=l

in which the numerator and the denominator pectively.

n ~

Qn(X) = I +

can

be

Pm(X)

= xm+n+I k=O~

linear equations

determined.

Pad@

Yk

xk

from which the unknown m+n+l values

obtained

the

following

em-n+ 1

Cm_n+ 2

-..

Cm+ l

c

Cm+ 1

..-

Cm+ n

...

~ c. j=0 j

"" "

Cm+ 1

m

result:

m xj j!O ej-n

(7) A Qn(X)

m Cj-n+ 1 j=O

Cm-n+ 1

Cm-n+2

cm

Cm+ 1

x

n

In these two formulae,

x

xj

xj

Cm+n

n-I

it is necessary to take c. = 0 when j is negative, J the last line and the last column.

and A is

the minor obtained by eliminating

Pad@'s rational approximations

are widely used in computer calculations,because

they are generally more efficient than polynomial the number of operations required•

approximations.

These approximations

They almost halve

are particularly

convenient

when one takes m = n ( defining the diagonal of the Pad@ table ). For in this case the coefficients Yk of the right-hand

side of (5) usually decrease so quickly that

the first term Y0 x 2n+I , divided by Qn(X), constitues an excellent approximate value for the absolute error introduced

if one uses

Pm(X) / Qn(X)

instead

of

XI co

ck

x k. For

k=0

Ixl < 1 , the value of Qn(X) differs v e r y little from u n i t y because

the b k coefficients usually d e c r e a s e very rapidly.

to calculate

y

It will therefore suffice

, whence 0

& n

(8)

T 0

n

c1

c2

e

c2

c3

Cn+ 2

Cn+ 1

Cn+ 2

C2n+l

n+ 1

A

n

6n is the minor obtained by eliminating the last line

and the last column.

In an excellent article w r i t t e n for the collective w o r k entitled "Mathematical Methods for Digital Computers"

(published by John Wiley), K o g b e t l i a n t z

irrespective of accuracy required,

the Pad6 m e t h o d

shows that,

is the best m e t h o d for construc-

ting programs for calculating sin x and cos x:

(9)

cos x =

~ k=0

(_)k (2k) !

zk

with z = x 2

Taking m = n, we have

PI

12 - 5 x 2

Q1

12 +

P2

15120 - 6900 x 2 + 313 x 4

Q2

15120 +

x2

'

Y0

,

As we can see on the figure,

660 x 2 +

13 x 4

Y

0

3

1

2

6!

59

1

42 I0!

the a p p r o x i m a t i o n P2 / Q2 w h i c h is c o n s t r u c t e d w i t h the

XII five

first

terms

of

the

series

(I)

is b e t t e r

than

the s u m S 5 of

those

terms.

Sum S s

/ / !

!

I

P2

Q2

/

Exact

I

IX 5

-I

Approximations

of cos x

For m = n = 3, w e o b t a i n

with

P3

I +

a I x

+

a 2

x 2

+

a 3 x 3

Q3

]

bl

+

b 2

x 2

+

b 3 x 3

al

+

x

3665 7788

bl

229 7788

b2

2360

b3

39 251

71| a2

a3 = -

25960

2 923 7 850 304

value

I

127 520

XIII

and

1407 2596

Y0 =

Practically

I 14 ~ <

-12 7 . I0

the approximation

P3 / Q3 is put into the form of a continued

P3

fraction

C1

(I0)

=

C0

+

C2

Q3

x 2 + Bl + C3 -

x 2 + B2 + -

x 2 + B3

=

23535.603

B2 = - 3.50476

C2 =

4017.0378

B3 =

Ca =

1670.6950

B

C

92.70474 i

Since

i

Co =

rence between

41.76065

14 615 12-------~- one is led to calculate

-

two large numbers,

can be circumwented that C0(G)

resulting

by substituting

is small. Finally

(0, -~) to ten significant

and

using

from the diffe-

result.

~ z for z, the parameter

This drawback

~ being chosen so

the value of cos x is obtained over the interval

digits from this approximation;

tion put into the form of a continued tions

a small number

in an inaccurate

seven constants.

fraction,

we use a rational

performing

Calculating

P~(z)

frac-

only four m u l t i p l i c a -

/ Q3(z)

as a classical

frac-

tion would require eight operations.

This

short exposition

of the Padg method

dely used since the advent of computers. and used by three categories tical physicists on Mechanics

organized

it might be useful damentals

at the Toulon University

of the method

having kindly

included

numerical

in fluid mechanics.

to gather together

sent volume was born,

The Pad~ method

of scientists:

and specialists

explains why it has come to be wi-

several

or to its applications

and I am particularly

Verlag for having published

After

Center

articles

specialists,

the E u r o p e a n

in 1975, devoted

in mechanics.

indebted

it in the Lecture Notes

is currently being

analysis

it was felt that

This

series,

is how

1976

the pre-

Beiglb~ck

for

and to Springer-

it so quickly.

Henri Cabannes January

theore-

symposium

either to the fun-

to Professor

in Physics

studied

Contents

Part I. M a t h e m a t i c a l Theory of the Pad~ A p p r o x i m a n t s Method.

BACKER, G.A. : The Linear Functional E q u a t i o n A p p r o a c h to the P r o b l e m of the C o n v e r g e n c e of Padg A p p r o x i m a n t s . . . . . . . . . . . . . . . . .

3

BESSIS, D. : C o n s t r u c t i o n of V a r i a t i o n a l Bounds for the N - Body E i g e n s t a t e P r o b l e m by the Method of Pad~ A p p r o x i m a t i o n s . . . . . . . . . CHISHOLM, J. S.R.

: Rational P o l y n o m i a l A p p r o x i m a n t s

GRAVES - MORRIS, P.R.

in N v a r i a b l e s

: C o n v e r g e n c e of Rows of the Pad~ Table

WUYTACK, L / : The Use of Pad~ A p p r o x i m a t i o n

in Numerical

]7

. . . 33

......

55

Integration.

. . 69

Part II. A p p l i c a t i o n s of Pad~ A p p r o x i m a n t s Method to P r o b l e m s of Fluid M e c h a n i c s

BAUSSET, M.

: D e t e r m i n a t i o n of Shock Waves by C o n v e r g e n c e Acceleration.

CHENG, H.K. and HAFEZ, M.M. : Cyclic Iterative Method A p p l i e d to Transonic F l o w Analyses . . . . . . . . . . . . . . . . . . . . . . . . . DALE ~ R T I N , E. : A Technique for A c c e l e r a t i n g Iterative C o n v e r g e n c e in Numerical Integration, w i t h A p p l i c a t i o n in T r a n s o n i c A e r o d y n a m i c s . GAMMEL, J.L.

: The Rise of a Bubble in a Fluid . . . . . . . . . . . . . .

. 8] ]0]

. .]23 ]4]

SCHWARTZ, L.W. : Rational A p p r o x i m a t i o n s to the S o l u t i o n of the B l u n t - B o d y and Related P r o b l e m s . . . . . . . . . . . . . . . . . . . . .

]65

TURCHETTI, G. and MAINARDI, F. : Wave Front E x p a n s i o n s and Pad~ A p p r o x i m a n t s for Transient W a v e s in Linear D i s p e r s i v e M e d i a . . . . . . .

]89

VAN TUYL, A.H. : A p p l i c a t i o n of M e t h o d s for A c c e l e r a t i o n of C o n v e r g e n c e to the C a l c u l a t i o n of Singularities of Transonic F l o w s . . . .

209

~ . : The Use of Pad~ F r a c t i o n s in the C a l c u l a t i o n of N o z z l e Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

225

Part III. General B i b l i o g r a p h y

BREZINSKI, C. : A B i b l i o g r a p h y of Pad~ A p p r o x i m a t i o n and some Related Matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245

THE LINEAR a F U N C T I O N A L E Q U A T I O N A P P R O A C H TO THE S

P R O B L E M OF THE C O N V E R G E N C E OF PADE A P P R O X I M A N T S *

GEORGE A. BAKER, JR. Applied Mathematics Department Brookhaven N a t i o n a l L a b o r a t o r y Upton, N.Y. 11973

ABSTRACT

The Pade approximant problem is related to a orthogonal) type.

(not n e c e s s a r i l y

p r o j e c t i o n of a linear functional equation of the F r e d h o l m

If the kernel is of trace class and its upper H e s s e n b e r g form

is t r i d i a g o n a l

(this class includes H e r m i t i a n operators),

that not only do the diagonal P a d e a p p r o x i m a n t s converge, their n u m e r a t o r s and d e n o m i n a t o r s these results to C

separately.

then we prove but so do

The g e n e r a l i z a t i o n of

classes of compact o p e r a t o r s is given.

For kernels

P w h i c h are not only compact, striction,

but also satisfy an additional mild re-

a p o i n t w i s e c o n v e r g e n c e theorem is proven.

The a p p l i c a t i o n

of these results to quantum scattering theory is indicated.

*Work p e r f o r m e d under the auspices of the U.S0 ERDA.

4

Considerable progress a p p r o x i m a n t s can be made, functional equations. ginning,

in the study of the convergence of P a d e I think, by the use of the t e c h n i q u e s of

What I will report here is p r o b a b l y just a be-

and is drawn in part from a p r e v i o u s paper.~l]

First we w i l l

review the known relation of Pade approximants to linear functional equations,

then we review the p r o p e r t i e s of some special classes of

compact operators,

and give c o n v e r g e n c e results for these classes.

F i n a l l y we indicate how these results

lead to c o n v e r g e n c e of P a d e ap-

p r o x i m a n t s to the p a r t i a l wave scattering amplitudes

in certain quantum

m e c h a n i c a l scattering problems.

P R O J E C T I O N S IN THE C I N I - F U B I N I SUBSPACE Suppose we consider the f u n c t i o n a l equation f = g + k A f

(i)

w h e r e f, g, and h belong to some Hilbert space Z, and A is a linear o p e r a t o r whose p r o p e r t i e s are yet to be defined.

We also introduce

the a s s o c i a t e d sets of elements

~i =

Ai-i

'

g '

~i =

A %

(

)

i-I

h

i = i, 2 .....

w h e r e A % is the H e r m i t i a n conjugate operator to A.

(2)

We need as well the

N × N matrix !

Ri,j = (~i' ~j) =

(h, A i+j-2 g)

(3) !

d e f i n e d in terms of the inner p r o d u c t s of the ~j and ~i" in

a position

to

define

our

projection

operator

onto

the

We are now Cini-Fubini

subspace ~2] N

PN =

E ~i (R-l) i,j=l ij

( ' ~j'

(4) )'

provided detIR I ~ 0.

(It can be shown t3~ that there exists an in-

finite number of such N's.)

The operator PN is a projection on SN

I

from S N.

I

(The spaces SN and SN are respectively those spaces spanned I

by ~i and ~i for i = 1 ..... N.)

It has the properties

(5)

P N P M = P M P N = PM' M ~ N.

However,

it may not be an orthogonal projection.

its norm IIPNII will be greater than unity! jection on non-orthogonal directions. the "length"

If it is not, then

We show in fig. 1 the pro-

It is clear in this figure that

(a2 + b2) ½ of the projection can be greater than the

length of the original vector.

i

/

/

/

~

/ Fig. 1 Projection on non-orthogonal directions (heavy lines) of a vector (arrow)

Wit~ this machinery,

let us consider the truncated equation

fN = g + k PN A PN fN"

(6)

By the properties of PN' we expect a solution of the form N

fN =

~ a3 ~J" j=l

The substitution of (7) leads to the solution

(7)

N

fN =

N

( E E $i V. wj i=l j=l lj -

l)/detl

I

uij

(8)

'

where Uij

= wi+j_ 2 - I wi+j_ I

Vij

= i, jth m i n o r

of

(Uk~) .

(9)

Then N

(h,f N)

However

this

=

formula

Pad~ approximant

N

E i=l

E V wj i/det I j=l wi-I ij IUij "

is N u t t a l l ' s

to h(l)

defined

compact

form

C4,5]

(i0)

for the

by the L i o u v i l l e - N e u m a n n

f = g + k A g + 12 A 2 g

+ 13 A 3 g +

...

IN-I/N]

series

, (il)

h(l)

Thus

b y use

Pad~

approximants

RESULTS

as the

FOR THE TRACE

striction

a n d the of the

(h,f)

= w0 + k wI + k2 w2 +

of t h e p r o j e c t i o n

H e r e we w i l l

we use

=

imply

a basis first

solutions

CLASS

assume

about

truncated

operator

A?

Using

the

equations.

this

What

does

To e x a m i n e

b y the ~_ so t h a t l

span SN"

directly

OPERATORS

PN is o r t h o g o n a l .

linear

e. d e t e r m i n e d 1

(4) w e g e n e r a t e

of the

OF C O M P A C T

that

the

N of t h e m

operator

...

this

this

re-

question

the e. are o r t h o n o r m a l 1

basis,

we

see that A

is

form x x

x x x 0 0

x x x x 0

x x x x x

I 'I 0 0 0

,

(12)

7

that

is, upper

right

triangular

called the upper H e s s e n b e r g condition

implies

plus one subdiagonal.

form.

Now

This

form is

if A = A t, this H e r m i t i c i t y

at once that A is tridiagonal.

Therefore

if we

i

choose

h of eq.

the ~i'

(2) equal to g, then the ~i define

and PN is orthogonal.

tri-diagonality

of A w h i c h

But this c o n c l u s i o n

is more g e n e r a l

our results

Now

turns

trace class

and A is t r i d i a g o n a l

out that the n u m e r a t o r

definite,

operators.

only on the The re-

of Stieltjes,

If the o p e r a t o r

in its upper H e s s e n b e r g

and d e n o m i n a t o r

order to define the trace class non-negative,

series

as

extend them.

let us define

trace class

depends

spaces

than Hermiticity.

striction A = A t w o u l d yield only g e n e r a l i z e d however

the same

of compact

Hermitian

converge

operators

is

form,

then

separately. we

introduce

In the

operator

T = A T A,

w h i c h has the e i g e n v a l u e s

it

(13)

and e i g e n v e c t o r s 2 T ~i = ~i ~i"

(14)

11ATT1 = z ~l ,l

(15)

The trace norm of A is then

which

is something

operators

with

for an o p e r a t o r

like Tr(IAI),

and the trace

a finite trace norm.

S t a n da r d

class theory

consists [6]

of those

insures

that

of trace class we can d e f i n e

D(k)

= detlI + kA I = lim DN(k),

(16)

N~

where

the D

sulting

N

D(k)

are the d e t e r m i n a n t s is an entire

function

in a sequence of k.

of subspaces.

We note

at this point

The rethat

some condition related to the trace of A is required as the Pade denominator must go to

I - kEk? I + O(k2), l i and,

(17)

if -i El. = Tr(A) = ~,

(18)

1

then the denominator can't possibly converge In order to see the convergence

separately.

in this case,

it is convenient to

construct the Fredholm solution to the truncated equations.

To this !

end we introduce an orthonormal basis Xi,i=l ..... N spanning ~N = ~N" Then (19)

Aij = (ki' AXj) , fN = E b.ik.l,

N (20)

b. = (X., g) + k E Ajkb k. ] 3 k=l If DN(X) = detNI~ij

(21)

- XAij I ~ 0,

then the Fredholm solution is given by (22)

fN = g + kEXjDN, jk(k) (Xk'g)/DN(k)' where

DN,jk(X)

N 1A.3k Aj~ I k 2 = Ajk - ~1. ~=llA~k A I + ~ ' EEl

and are automatically polynomials

I +

....

of degree at most N - 1 in k.

we choose the X. to be eigenfunctions l

of

(23)

If

and use Hadamard's

determinant

all

•. .

inequality

aln n

Idet

I < •

.

.

N

~ (Z i=l j=l

(25)

laijl 2)½

a

anl

nn

then we can show ~I~ for any z in ~, of unit length N

N

{ Z I Z DN, jk(k) j=l k=l

+ Thus the operators

Zkl2}½ ~ 11AII2 (i

i.

22/2 11A111 + ~

.

33/2 IIA112 1 +

that give the numerator

in the Fredholm

are a sequence of uniformly b o u n d e d operators the series

in (26) converges.

IIAII1 finite makes

of the denominators. numerator

CLASSES

solution

in N for all k p r o v i d e d

finite also.

of the numerators

Standard arguments

and eq.

(16) gives that

Hence we have the separate convergence

and denominator

of the P a d ~ approximants

the two entire functions given by the Fredholm

C

(26)

As IIAII1 is finite the series does and

IIAII2 a u t o m a t i c a l l y

then insure the convergence

"'" )"

of the

to the ratio of

solution.

OF COMPACT OPERATORS

P We may define

larger classes of operators

and obtain special convergence operator belongs to class C

results

if, using

than the trace class

for them. (13) and

We say a compact (14),

P

IIAIIp= Czi ~p~i/p < ~. The trace class Cp_ I c

Cp.

(27)

is C 1 and these classes have the properties

We use the convenient

notation,

due to Nuttall,

that

10

P

i

~ F.x] p = ~ F.x 1 1 i=0 i=0 for any formal power

series.

are based on rewriting

IN-l/N]

By applying

DN,p(k)

the P a d d approximants

parallel

orthogonal

N

for operators

of class C

= detN(6ij

to those of Dunford and A in C

- kAij)

p

P

as

p (k) e x p { _ ~ n Q N ( k ) ]p-l} N _ PN (~) QN (k) QN(k) exp{-~ ~nON (k) ~P-I }

-

arguments

we can show for P

The results

(2s)

(29)

and Schwartz

[6],

that if we define

exp{-~ndetN(Sij

- kA..)]p-l] 13

(30)

then lim DN, p (k) = entire

function

of k

N-=

(31) exp (FIIAIIPkP), !

and in the N X N truncated -

~A

limllDN,p (k) {(6ij

space 8N

.) - I

13

-

~

.

8N -

~A

-

...

-

Ap-2]il

~p-2

13

p p-i

(32)

< exp[F(k p Allp + i)]

where F is a finite, (29)-(32)

we deduce

N-independent

for A in C

By combining

eq.

(ii),

that we have the form

IimFN-I/N]

where

constant.

=

p-2 ~ kjh. + entire j=0 3 entire

both entire

functions

in

function function

'

(33)

(33) satisfy

P lentire

functionl ~ k e x p ( B l p) .

(34)

11

The modified separately imant

to e n t i r e

does

general

numerator

too.

than

and d e n o m i n a t o r

functions,

The

result

defined

and thus,

of course,

for the P a d e

for the n u m e r a t o r

in this w a y the

approximant

and d e n o m i n a t o r

converge

Pade

itself

separately

I

approx-

is m o r e

as w e

shall

see b e l o w .

COMPACT

OPERATORS

First

let us c o n s i d e r

the

case

where

PN is an o r t h o g o n a l

projec-

!

tion

(%N = 8N) "

If we

subtract

f - fN = k A

Now

if IIfNlI < ~, then

goes

to zero

as A

eq.

(6) f r o m

(f - fN)

the

second

is compact.

+ k

term

Then

eq.

(I) we get

(A - P N A P N ) fN"

on the

eq.

right

(35)

hand

becomes

the

(35)

side

of

(35)

form

d ~ IAd. Thus,

if k is n o t a s i n g u l a r

by the F r e d h o l m

alternative

point

(36)

of eq.

theorem.

(I), w e m a y

conclude

d = 0

Hence

fN " f ' ~N-I/N]

If,

on t h e

other

hand,

=

(h,fN)

of unit

(h,f)

= h(k) .

(37)

I]fN1] - ~ for all N, w e can d e f i n e

dN = fN /

an e l e m e n t

-

norm.

Eq.

IlfNII '

(38)

(6) b e c o m e s

d N - k P ~ d N - 0.

As A

is compact,

over

the

there

subsequence

exists

exists

a subsequence

and has

the

(39)

of N ' s

value

such

that

the

limit

12

lim IAd

which

implies,

as A

N

(4o)

= d,

is compact, d - I A d = 0.

S i n c e w e are

assuming

the F r e d h o l m

alternative

tion. norm

Therefore tends

Therefore onal

to

k is not

there

infinity,

we conclude

projection

a singular

theorem

does but if A

operators

(41)

not

that

an

infinite

a finite

is a c o m p a c t that w h e n

of

(i), w e c o n c l u d e

]]dll = 0, w h i c h

exist

at m o s t

point

operator

k is n o t

is a c o n t r a d i c -

sequence

number and

by

of fN w h o s e

of s u c h the

a singular

equations.

PN are o r t h o g point

of eq.

(i) lim~N-i/N~ where

the

limit

is t a k e n

Pad~ approximants In the obtained.

case

over

the

= h(1), infinite

(42) number

of N ' s

for w h i c h

exist.~3~ PN is n o t

The problem

orthogonal,

here

trolled.

In p a r t i c u l a r ,

magnitude

of the

is t h a t

insofar

less

complete

the m a g n i t u d e

of

as w e are c o n c e r n e d

results

have

the

been

IIPNII is u n c o n it is o n l y

the

element, (43)

PN A e N ,

where

the

e. are d e f i n e d 3

before

eq.

(12),

that

is u n c o n t r o l l e d ,

as by

construction

PNAej

If w e m a k e tion

the a d d i t i o n a l

is m i s p r i n t e d

in ref. lim N-~

(44)

--- Ae.,3 J ~ N - i.

mild

assumption

that

(note

that

this

equa-

i) infIIPl~AP Ae

TI = 0,

(45)

13

where N

= N

Z e. (e., j=l 3 3

is the orthogonal p r o j e c t i o n on S N. Theorem:

)

(46)

Then we can prove:~l]

Let A be any closed b o u n d e d region in the complex k

plane not containing a singular p o i n t of eq.

(I) .

finite order Pad6 approximant to h(l)

(ii) is exact,

eq.

(42) holds,

or

(iii)

of eq.

for each I in A for w h i c h

Then either

(i) and

there exists an infinite subsequence of N's such that

or

(i) a (ii)

(ii) fail,

(ii) holds for

all other I in A.

A P P L I C A T I O N TO Q U A N T U M S C A T T E R I N G THEORY This a p p l i c a t i o n is a g e n e r a l i z a t i o n of that of G a r i b o t t i and Villani.~7~

We c o n s i d e r the p r o b l e m of p o t e n t i a l scattering in non-

r e l a t i v i s t i c q u a n t u m mechanics.

The f u n d a m e n t a l e q u a t i o n is the

S c h r o d i n g e r equation

-V20 (7) + kV(r)

0 (7) = k2% (7) .

We restrict the p o t e n t i a l function by a s s u m i n g that V(r) sign,

s p h e r i c a l l y symmetric, ~

(47) is of single

and satisfies

IV(r) lexp(21~Ir)

rdr < ~.

(48)

0 Then,

for the partial w a v e d e c o m p o s i t i o n of

(47) we can show:

(i) a

slightly recast version of the k e r n e l of the usual c o r r e s p o n d i n g tegral equation is of trace class. the k e r n e l is tridiagonal.

in-

( i i ) The upper H e s s e n b e r g form of

Thus the results we have r e p o r t e d show

that the numerator and d e n o m i n a t o r of the [M/M~ Pad~ a p p r o x i m a n t s

14

converge tion,

separately,

as o n e w o u l d In as m u c h

out

the

vergence large.

and the denominator

hope.

These

as F r e d h o l m

f i e l d of m e c h a n i c s , results

converges

conclusions

equations

appear

the potential

in the a r e a of t h i s

hold

if

to the Jost

IIm(k) I ~ ~.

very frequently

applications

conference

func-

seems

through-

of t h e s e

con-

to m e t o be v e r y

15 REFERENCES

i.

G. A. Baker, Jr., J. Math.

Phys. 16, 813

(1975).

2.

M. Cini and S. Fubini,

3.

G.A.

4.

J. Nuttall,

5.

G . A . Baker, Jr., "Essentials of Pade Approximants" Press, N. Y., 1975).

6.

N. Dunford and J. T. Schwartz, "Linear Operators, Part II: Spectral Theory" (Interscience, New York, 1963), Chap. ii, Sec. 9.

7.

C. R. Garibotti and M. Villani, A61, 747 (1969).

Nuovo Cimento ii, 142

Baker, Jr., J. Math. Anal. Appl. 43, 498 Phys. Rev. 157, 1312

(1954). (1973).

(1967).

Nuovo Cimento A59,

(Academic

107

(1969);

~CONSTRUCTION OF VARIATIONAL

BOUNDS FOR THE N-BODY EIGENSTATE

PROBLEM

BY THE METHOD OF PADE APPROXIMATIONS D. Bessis

CEN-Saclay,

Service de Physique Th4orique BP n°2 - 91190 Gif-sur-Yvette

We first recall how the Pad~ approximations N-body Hamiltonian

scheme,

applied to the resolvant of a

generate new improved variational

tal and the excited states, which generalize the Rayleigh-Ritz

we therefore analyse

principle

completely

rise to a variational

principle

- France

principles

the Rayleigh-Ritz

for the fundamenprinciple.

In this

is deduced from the lowest Pad~ Approximation,

the content of the next approximation, in which is embedded

the knowledge

which gives

coming from

more terms of the resolvant expansion. An application

to the case of N fermions

which is itself a sum of Gaussian potentials case, the reconstruction

interacting via a two-body potential is analysed. We show that in this

of discrete eigenstates

in a purely algebraic way without any multiple states being approximated

by a monotonously

and eigenfunctions

can be done

integral calculations

decreasing

sequence

: the eigen-

rapidly conver-

ging° In the conclusion we recall how the method extends to singular potentials (hard cores) and a very simple two-body problem tration.

is tested numerically

for illus-

18

INTRODUCTION In physics one is often faced with the problem of expanding a given quantity G in terms of an expansion parameter

G(~)

Go +

~GI + "'" + ~nGn + ...

(I-I)

Four cases can happen I) The series

(I-i) is, for the "physical" Value of ~

rapidly convergent. 2) The series is therefore,

One can therefore

Eq. (I-I)

useless for this precise problem:.

(I-I) is divergent for all ~ , but asymptotic.

value of ~ is sufficiently

If the physical

small, then again (I-I) can be considered as solving

the problem because the "effective 4) The series

consider that (I-I) solves the problem.

(I-I) is convergent but too slowly to be effective.

as it stands,

3) The series

, convergent and

convergence"

could be extremely

good.

(I-I) is divergent for all ~ , and has no asymptotic

properties

:

(I-I) is useless as it stands. Most of the time, one is left with cases 2 and 4, for which it is necessary to apply an "accelerator for deep reasons,

of convergence",

among which the Pad4 Approximation

one of the most powerful.

More generally one can ask the following question Given a finite number Go, GI, (I-I),

is,

:

..., G N. of coefficients

in the expansion

find "the best" upper and lower bound for G(a),

~(~;

Go,G I .... ,GN) 5

G(~) 5 ~+(~; G O ..... GN )~

How does one construct the unknown functionals ~+ ? Of course the problem makes sense only if, not only the Gi, O 5 i < N are known, of the function G(~) are given,

but also if additional

properties

in such a way that the bounds are achieved

for

G(~) in a precise class of functions. For instance, out subtractions, function

if G(~) fulfills a dispersion relation with respect to ~, withwith a positive discontinuity,

that is G(~) is a Stieltjes

: G(a)

=

So

~<x)dx I + x~

'

o(x)

~

0

'

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~W~en G(~) is complex,

one has to deal with inclusion domain see Ref~(1).

Ig

then for ~ real and positive by the

IN/N] and

[N-l/N]

the functionals

Pad4

~+ and ~

ApprQximants

are given

respectively 2)

built up from the coefficients

G. i

(0 < i < 2N) or (0 < i < 2N-I). Another

example

G(~) namely

if G(~)

is provided

=

~o

[N-l/N]

transform

~+

An interesting

fact~

for those

of the convergence

be divergent

@(x) ~ O,

of a positive

two examples,

for all ~ . The problem

derivatives

at

~ = O, what

Pad4 Approximants

[N/N]

G. 3) i

or divergence

: given the class to which

(1.3)

measure.

are given by Generalized

tion theory

for

,

built up on t~e coefficients

independently well

e "°cx p<x)dx

is the Laplace

Then the functionals and

by the case where

is that the functionals

of the series

is best expressed

G(~)

exist

(I-I) which may very in terms of informa-

belongs, given a finite

number of its

is the best upper and lower bound which

one can expect

G(~). Even more

where,

interesting

for some reasons,

for instance gularized thermore

is the case of variational

the G i themselves

in the so-called

into

Ga(~)

do not exist

case of singular

by means

bounds,

of a small

that is the case

and need

interactions

regularization

4)

regularization~

. Then if G(~)

parameter

is re-

a and if fur-

: G(~)

if finally we can derive

< Ga(~)

the functional

,

(I-4) ~+(~

; G ~ ;

o

GIe

;

E "''' GN)

for

Gg(~),

then

we find

G(~) This gives

Even though

value

<

inf

bound

~

of N will depend

:

(~ ; G o~ ; GIE ; "" "GN) e

the regularized

E -~ O, the functional

Taylor

contained

a scheme has been extensively

coefficients

gN(~)

used in ref.

of a Hermitian

of G(~) for some

is the best choice

in the Taylor

series

tend to infinity gN' which at that of the cut-off

regu-

stopped at order N. Such

(4).

Let us come now to a very well known case. of the resolvant

(1-6)

--~ will have a minimum

on ~ o This

lator for the information

(~-5)

G E E GE < ~+(~ ; o ; GI ; "''' N )"

_

the variational

G(~)

when

< GE(~)

Suppose we consider

operator H in Hilbert

space,

the mean value

having a negative

20

discrete

spectrum Eo, El,... , and a positive

body Hamiltonian

in Quantum

cp(z)

continuous

spectrum,

Mechanics. Then the resolvant

=

<~J l + zzH

reads

typically a

N-

:

j~>

(m-7)

and if we expand it formally, we get :

~(z)

=

z

The operator H being unbounded infinite,

(-z) n <~IHnlg>

n=O in general,

Then,

the moments <~IHnl~>

o

the Rayleigh-Ritz variational fulfills

principle asserts that the ground state

:

v

l~>~

,

(I-9)

E ° = inf

~

Therefore

the knowledge of only the first term of the expansion

;

(I-I0)

us to build up a variational

functional

important to remark that the series generic

are all

]~> in such a way that <~IHl~> exists.

Eo 5 i ~ i and

(I-8)

except if I~> belongs to the domain of all the powers of H. Let us suppo-

se first that we choose

energy E

~

l~>: this is so because

of H. It is

(1-8) has a zero radius of convergence

for a

R (z) is singular at z equal zero due to the fact

that the spectrum of H extends to ~ . the radius of convergence

for the lowest eigenstate

(1-8) allows

The point z = O being a singularity of R (z)

of (1-8) is of course zero. Nevertheless

the bound

(1-9)

holds. Now the interesting Given more moments of the expansion

(1-8),

derive variational E2,...

question is : : ~R = <~IHkl~>

(I-II)

can one improve the bound

principles

(1-9) and more generally%can

one

not only for the ground state E0, but also for El,

the so-called excited states ? The answer is yes, and it is the Pad4 Appro-

ximation technique which is the generating tool of these variational

improved

Rayleigh-Ritz principles

principles.

Furthermore,

: we shall call them the Pad~-Rayleigh-Ritz

these bounds can be shown te be the best possible 5) which,

construct from the knowledge of only II - THE PADE-RAYLEIGH-RITZ Given an even We construct Pad~ approximation

a finite number of moments

one can

~k :

PRINCIPLES.

number of moments

~o' ~i' "''' ~2N-I"

the following determinant built up on the resolvant)

(which is the denominator

of the

[N/N]

21

ko kl °'" ~N ~I ~2 "'" ~N+I A

..o...i.°.,.

(E,N)

(If-l)

~LN-I ~N " ' ° ~2N-I I

A

(E,N)

roots

is clearly a polynomial

of

E °.. E N

of degree N in Eo It can be shown 5"~ that the N

A (E~N) are all real and distinct

(N)< Eo Furthermore

_(N) KI

E(N) < "'" < p

E ("N"~ is an upper bound P

<

'

and therefore we can order them

.o< ~(N) ~N-I

(II-2)

to the pth excited E

<

:

state of H 5).

E (N)

O

O

E1

_< E~ N)

Ei

-< E!N)I

(II-3)

,.,,,o,.,,

EN The true eigenvalues

E

are of course

but clearly on

the bound ~(N) P I~> • Therefore

5

independent

is dependent

EL

_<

inf J~>

EL

=

inf

.(N) mN-I

on

of the test vector

I~>because

E~N)(~)

the moments

J~> we choose,

~k are dependent

.

(11-4)

More precisely we have 4)

~N)(q~)

(11-5)

I~> In that way, we generate interesting

variational

principles

fact is that if we now increase E (N+I) I % L ~)

In other words, nal principle

Ei

<

N by one unit,

states.

one proves

ELN)((%o).

by adding more moments,

for the i th excited

for the excited

A further

that 5)

:

(II- 6)

we are sure to improve

the variatio-

state°

(i+l) (~)

"<E (N)" .< °'° .5 E i(N+I)"L~)i ~ )"- < "'" -< E i

°

(II-7)

22

Even if we do not use the variational~.~ properties consider the sequence of the Ei~)(9)at fixed ly towards E L and in cases of practical

in

I~> of

the E~N)(~) bounds, and

I~> , the sequence decreases monotonous-

interest (l~kl j

(2k)!) the sequence con-

verges very rapidly towards E L . However here, we do not want to make use of the convergence of the bounds (11-7),

(because for the N-body problem, for instance,

it is not easy to compute

too high moments) but rather use them as new improved variational principles. For N=I, we have ~o ~I A

(E,I)

=

~0

=

I

E

~I ~

=

(II-8)

FoE - ~I

Then

E(1) o

=

(II- 9)

That is using (11-5)

E

=

o

inf

E (I)

I~>

o

=

inf

i~>

<~IRI~>

(1I- lO)

<~I~>

we find the Rayleigh-Ritz variational principle. III - THE CASE N=2, OR THE PADE-RAYLEIGH-RITZ

VARIATIONAL PRINCIPLE.

For N=2 we get

14]. ~2

~o A

(E, 2)

=

i Introducing the deviations

E

we get, solving

=

E2

:

<~I~>

83

(111-I)

~I ~2 ~3

~o

"

(I11-2)

(111-3)

<~ lq0>

Z~ (E,2) = 0

(iii-4) _(2) E1

=

~

/(83 + ~ 2 )2 + 62

63 + 26--~

28 In particular we obtain for the ground state E ° the improved variational principle : .63 .2

Eo "

inf j~>

we remark that,

- [2~'2)

~

82 being always positive,

E2(~ )

[

=

/ . 8 3 .2

63

+

82 - ~

] ]

(III-5)

the quantity :

8

,28~) + 82 - ~2

]

(III-6)

is always positive, and therefore comes subtractively to the Rayleigh-Ritz ordinary principle : it can be considered as the correction to it, induced by the extra information coming from the knowledge of the second and third moment. By changing the sign of the square root, we get the variational principle for the first excited state :

E1

=

inf [

~

+

(03 ) 2 ~2 + 62 +

~263 ]

(III-7)

J~> On (III-5), we see that if 62 is smell,that is if J~> is already near to the exact ground state eigenfunction; , the correction trary

82 is very large, the correction

e2(~) is small;if on the con-

a2(~) , being an increasing function

of

62, becomes large and that large correction will improve the bound significantly. Even if

82 is small, the precision on E ° ~s very much improved. As shown en

numerical examples, factors as large as Io

on the precision are gained by using

the Pad4 principle with respect to the usual Ritz principle° Finally we want to point out the important fact that g2(~) is also a monotonously increasing function of 83, as checked easily. Therefore if we replace 63 by a rough upper bound, (or equivalently ~3 by an upperbound), we still get a variational

principle for the ground state : 8~ E

where

67

o

=

inf

I~>

~ H \~J~/ ~

83 ~2 ] ]

)2 +

(III-8)

is an upper bound of 63 ~ obtained by using an upper bound for ~ . Eq.

(111-8) shows also clearly that, if only

62 is known, then it is not possible

to improve the ~ t z principle, because when knowledge of

83 ~

+~,

a2(~) ~ Oo Therefore the

63~ even through a very crude estimate, is fundamental in this me-

thod. (see a physical consequence of this last statement in section V) o

24

IV - THE EIGEN-FUNCTION APPROXIMATION. Let us suppose, from now on, that we have normalized the test vector

<~I~>

=

+I.

I~> :

(iv-l)

Then, following refo (5), the approximate ground state eigenfunction, for the N=2 case is : E2

I®~2)> = I~> - ~ (H-~)I~> It is easy to check that, if we use

(iv-z)

I ~ 2 ) > as a test vector in the Rayleigh-Ritz

variational principle, we get the Pad~-Ritz principle that is :

<®~Z) IHI%(Z)> = <~lnl~> <0o~2) io~')>

The reader will notice that

"l-4

)z+ 62 - ~-I

(IV-3)

'{ -l~'2)>and E (2) are the ground state eigenvalue O

O

and eigenvector of P2 H P2, where P2 is the projector onto the two-dimensional space ~(2) spanned by the vectors

I~> and HI~ > . This remark can be completely

generalized, and is linked to the fact that the Lanczos method fer matrices, and its generalization to Hilbert space the tri-diagonalisation Jacobi method, are deeply linked with the theory of Pad~ Approximation

6)

.

To end up this section we give the formulae generalizing (IV-2) for the N th approximation. Introducing the normalized polynomials

:

~O.....~N

~(E,N)

A(E,N)

>

0.

(iv-4)

~/GNGN_ I

~ . . . . . g2N The L th approximated eigenfunction at order N is given by : N-I

(Iv-5) L

j=O

The

I~-N$ are obtained by diagonalizing the Hermitian matrix ILl =

PN H PN

(IV-6)

25 where PN is the projector onto the N-dimensional space spanned by the vectors I~>, HI~>,..o HN'I]~>. While the

~L(N)\ar / e orthogonal but not normalized,the

of orthonormal vectors because it is known that the

A(H,j) I~> are a set

A(E,N) form a set of ortho-

gonalized polynomials with respect to the spectral measure of the operator H 5).

V - APPLICATION TO A SYSTEM OF N FERMIONS. To apply effectively the Pad4-Ritz principle (IV-5) it is necessary to find a way to compute easily the moments

~k

"

<%oI Hkl~0>~o

(v-l)

In the Z-body problem the moments are given Z-uple integrals which are difficult or even impossible to compute if k is large. However, there are cases in which the moments can be reduced to entirely algebraic expressions. An example is given in ref. (5)~ for the case of the most general d-dimensional anharmonic oscillator, by choosing a suitable variational test vector I~>

•

Here we want to consider the case of Z fermions interacting via a two body Z potential : Z ~ 2 ~ ~pi)2 HZ

"

~ i =I

P2 ~

+ i

I i<jsZ

~ ~ V( Iri'rj ])

i-I 2M

(V-2)

where we have subtracted the center of mass energy. We shall suppose that V, the two-body potential is a superposition of Gaussian potentials : .

V( l~i-~j I)

"

OlS

2

Pp ([~i-~j ]2)

e- ~P(~i-~J)

(V-B)

p=l (where

P (x) is a polynomial in x~With such a superposition, we can build up a P large class of phenomenological potentials, The interesting case of hard cores will be postponed to a further section. We must now choose the test vector in such a way that I) The Z-uple integral reduces to an algebraic calculation. ..... ~ - - ~ - ~ - ~ 5 ~ z 9 - y ~ - ~ 5 5 ~ 5 ~ - ~ - ~ [ ~ 9 ~ 9 - ~ 5 - ~ - - 5 ~ z - f ~

illed.

~]~> is a given vector~ by using "m~trix" Pad4 Approxlment most of the results exposed here generalize to the case where ]~> is a set of vectors I~i>, ]~2> o.o l~R>, generating a "model" space.

2B We shall choose i~> to be an arbitrary sum of Slater determinants built up with the states of the harmonic oscillator. -+

-+

-~

-+

%0o(rl, t) o..go(rz, t) -+

P I

-+

<:I'~2 ..... rz> =

-+

~I (31':)"" 91 (rz' t) o~> 1

t=i

(V- 4)

. ° ° ° ° , . . , . ° . ° o . , ~ ° .

~-F.

-+

-+

-+

9z-l(rl' t)...~Z_l(rz-t~ -+

where the

-+

Qm(r,t) are a set of orthonormalized harmonic oscillator eigenfunctions :

-~-~ %°re(r, t)

=

%°ml(X, tl) o 9m2(Y,t2) . %°m3(z, t3 )

with

.

9ml(X,t I)

= L~

2 tlx2

2T M m! t: I] -1/2 Hml(tlx) e

and analogous expressions for Hermite polynomials. The parameters

(V-5)

Z

(V-6)

9m2(Y,t2) , 9~(z,t3). The Hm(P) are the usual

~ and : are the variational parameters to be used in the

calculation. Combining all these informations and taking into account the fact that the derivative of a polynomial times a Gaussian is again a polynomial times a Gaussian we see that the moment

~k will be of the form :

-Q:(71 ..... r~z) ~Ik = ~ S where

Q~

d:l'''d:z

is a quadratic positive

P~(:I .... :Z )

e

(v-7)

form of the ~i :

Z

(v- 8)

qij ri orj.

Q~ -= i=j=l and

P

is a polynomial

in the variables ~i,o. "'~Z"

Therefore, we are left with integrals of the form :

24~

~ I (xi2Xj2)~ 2..o (XilXj I)

am :xx im Jm) oxp

ZZ

qijxixj }

i,j~l

(v- 9)

27 which are equal to

~I+~2 + +~ ~la "''~m m

(-)~l+~2"''+~m a

~

a qilJl- • •

~

Z/2 1/2 ~

(v-lO)

(Det qij) q~mJm

In principle the calculation is straighforward and purely algebraic. In practice the enormous number of terms to consider limits the possibilities to the calculation of the first three moments. Even for the third moment one may have to consider one thousand terms contributing. This means that it is necessary to use the variational Pad6-Ritz principle with the upper bound on

63 . As a consequence

it is important to know how to get not too bad upper bounds for

63 = <~ [(T-}) + o

(V-If)

This problem will be considered in future work. VI - THE CASE OF AN INFINITE NUMBER OF BODIES Let us consider the case where the number Z of bodies tends to infinity. Let us suppose that the ground state energy per body, E(Z)/z" has a finite limit when Z o tends to infinity and that this limit E ° is such that

E

o

E (Z) o Z

<

for

Z > Z

(VI-I)

o"

This case seems to be not unlikely for nuclear physics. Then we can use the Rayleigh-Ritz principle and write :

Eo

<

E(Z)o Z

=

inf

inf

z

Iv>

<

<~IHz I~> -~<~ I~>

(VI- 2)

or

E°

<~ IHz I~>

(Vl-3)

z<~l~>

If we use the Pad6-Ritz principle we get

< IHgI > E

o

=

inf inf

z

l~>

E ~

i - ~ ~ I ~ 7

6
~ ]

28

It is interesting

to notice that while we can

E

o

=

inf

lim

I~> g ~

replace

(VI-3) by :

(VI-5)

<~J~>

because we expect the limit for not too bad

I~> to exist and be meaningfull,

in

general the limit of the correction term when Z ~ ~ will be zero 7), and so no improvement will be achieved by the rad6-Ritz principle limit Z ~ .

Instead,

if we take brutally the

if we look for the inf in Z, it is clear that this inf will

give a better upper bound to E

o

than the R~yleigh-Ritz

principle,

as best shown

on Fig. Io

Eo(Z} Z E@R

E(Z) o

Z

Eo

I

1 Zs2

I/z

Fig.l - E(Z)/z as a function of I/Z. and its approximations° o We see that

Eo, P (the Pad~-Ritz approximation)

the Rayleigh-Ritz

Approximation

will be always better than Eo, R

.

It will be for some value ZB2 (Z best) that we achieve the best upper bound for E ° This technique clearly uses the number of bodies itself as variational parameo tero If one takes the next Pad~ Approximation, one gets a new ZB3 which gives an improved upper bound for E o. It is possible mum converges

to E

o

in very general cases 4j. ~

A very analogous tials

to proove that this sequence of mini-

technique can be used to treat the case of hard core poten-

: one regularizes

nal parameter, see ref.

the potential and then uses the cut-off as a variatio(4) and (5) for a detailed discussion.and

forthcoming

paper. Before ending this section we want to point out the important fact, that, as explained at the end of section III the knowledge

of only

the calculation by no means, and that some knowledge necessary.

But up to third order,

the deviations

62 cannot improve

of 63 (upper bound)

(III-2),

is

(111-3) are identical

29 to the cumulants which occur in the Ursell-Mayer cluster expansion. This suggests that (in that scheme) some knowledge of the three-body correlations is necessary s together with that of two-body correlations

(Brueckner) to improve, in ~a variatio-

nal framework~ the result one can obtain from an independent particles approximation. VI - A NUMERICAL EXAMPLE. As numerical example, let us take the one dimensional harmonic oscillator :

H

=

p2 + x 2

;

p

=

d -i dx

(Vl-l)

The ground state of which is known exactly to be

E

=

o

I

(VI- 2)

As test vector we take

_ 2(x_ ~)2 <x]~>

=

e

(Vl-3)

The exact eigenfunction is obtained for The deviations

~=I and 8-O°

are easily computed for (VI~3) and one gets :

= ~ 1

I + 62 . = ½(~ + ~) 1)2 +

(vl-4)

2~ 2

(VI-5)

~2 -- ~(~ - 7 63 =

(~ + I)(~ _ 1)2 + 2 6 2 ( % _

We shall take ~as~variational

parameter and

I)

(Vl-6)

8 as a fixed parameter.

Then we get :

1 ERayleigh_Ritz

=

Epad~_Rit z - inf O<~

inf 0<~

{ ~(~ 1

!)

~ (~ +

+I) + 2

2 +

(Vl-7)

2+ %

-

-

53

(vl- 8)

For ~ - 0 we have : ERR

EpR

.

} I

f3

1 2

(vl- 9)

30

We see that while only the first derivative two derivatives

of ERR(S) vanishes at ~ = I, the first

of ERR(S) vanish at ~ = i. Of course in this case both minima

are equal and equal to the exact value. We give now a little sunmmry of the numerical

results for three typical values

of ~ .

2~ 2

Exact value

Rayleigh-Ritz

value

Pad~-Ritz value

Improvement

0

I

I. (exact)

I. (exact)

O ~ =

0.01

I

i. 005

I. 00000025

20 000

o;1

I

1.05

1.000195

250

I.

I

I. 5

1.0583

8

We have defined the improvement

I

=

factor.

factor by :

El~yleigh_Ritz

- E

Epad~-Ritz

- Eexac t

exact

~

__2

for ~ small.

(Vl-lO)

~2

We see, that~ even when the Rayleigh-Ritz method becomes meaningless the Pad~-Ritz method still gives no more than a

for 2~ 2 = i,

6% error.

CONCLUSION The method presented here has the great advantage fore completely

to be variational and there-

independent of how big would be the coupling constant in the

Hamiltonian H = H + k V o

for a

N-body system. %t can be thought as representing what would be "the next

corrections"

to the Rayleigh-Ritz

variational

up those new improved corrected variational perturbative

terms in the formal expansion

When we deal with N-fermions

principle.

principles,

large

to build

of the resolvant of H.

and a two body potential

Gaussian potentials we have seen that all calculations and a simple

It necessitates

the knowledge of more

superposition

of

can be done algebraically

numerical easily testable example as shown that improving factor as

as two to four order of magnitude

in the precision can be expected.

31

The problem when dealing with hard-core potentials can be looked in Refo 4 : it will be fully analyzed in a forthcoming paper. ACKNOWLEDGMENTS The author for fruitful Dr. G°

would like to thank Prof. B. Giraud, ~Schaeffer

and Ao Zuker

cormments and discussions° Vi¢hniac is thanked for his help in working out completely the nume-

rical example.

REFERENCE$o (I) - G. Baker Jr., in "Essentials of Pad~ Approximants".

Academic Press New York

1975. Chapter 17. (2) - See Ref. (I) Chapter 15o" (3) - J.O. Wheeler and R.@. Gordon~

Bounds for averages using moment constraint

in "The Pad~ Approximant in Theoretical Physics". editors~ Mathematics

Baker Jr. and Gammel

in Science and Engineering Academic Press 197Oo

(4) - D. Bessis~ L. Epele and M. Villani~

(5) - D. Bessis and I~o Villani,

J. Math. Phys. 15, p. 2071,

(1974).

J. Math. Phys. 16, p. 462 (1975).

See also Michael Barnsley, The bounding properties of the multipoint Pad~ approximant to a series of Stieltjes,

J.M.Po Vol,. 14 n°3 March 1975.

(6)

-

See Appendix H of ref. (5)°

(7)

-

D. Bessis~ P. Moussa and Mo Villani, Monotonous converging variational approximations

to functional

integrals in Quantum Statistical Mechanics°

Saclay preprint D.Ph.T/75-6. January 1975 to be published in Journal of Mathematical Physics.

RATIONAL

POLYNOMIAL

APPROXIMANTS

IN N VARIABLES

by

J. S. R. Chisholm, Mathematical Institute, The University of Kent at Canterbury, Canterbury, Kent, England.

34

I am reviewing work on many-variable by a group of applied mathematicians Canterbury,

since September,

1972.

approximants

at the University I reviewed

In writing

only those technical standing

method,

the work that we have done;

to realise

that,

every day throughout a considerable although

benefited

of the various

almost

in c o n ~ q u e n c %

in the development

made by each member

from some discussions

It is

problems

There was,

papers

a

of my initial paper 2,

and discussing

amount of give-and-take

of the contribution

recollections.

the winter of 1972-3.

the authorship

for a broad under-

this review is necessarily

after the production

a group of about six were working

give

and shall try to put into

view based to some extent on my personal important

part

and much of it

I shall therefore

details which are necessary

of the approximation

perspective

this account,

of Kent,

a considerable

of this work in a talk given in Marseille I in 1973, is now published.

carried out

of work,

gives a very fair idea

of the group.

with other members

We also

of the School of

Mathematics. During my visit to Texas times discussed generalisation

the possibility

between

Critchfield,

1966 and 1972.

proposed

another

to solve the problem.

possessed

of diagonal several

closely restricted

Pad& approximants,

later,

from

I made

of an

since these approximants

properties

generalisation, class.

A few weeks

seemed to me to be the definition

characteristic

by any two-variable

generalisation 3,

in certain basic respects

for a single variable.

analogue

In 1972, John Gammel,

a two-variable

the Pad& method

The core of the problem

a two-variable

and I looked at the problem on

but noted that their scheme differed

attempt

John Gammel and I several

of inventing

of Pad& Approximants,

a number of occasions with Charles

in 1965-6,

which should be shared

and were therefore

There were other properties

the most of two-

35

variable

approximants

which

there was no analogue

also seemed very desirable,

for a single variable.

for which

The properties

I

wished to satisfy were: (i)

Defining Property Using the notation

z = (Zl,Z2) , y -- (71,72)

and ~~ = (~,~),

a

double power series

f(z) is given.

The diagonal

:

~ c z~ y=o ~

approximant

function

is to be a rational

~ ~cp fm/m(Z)

=

~ ~cp

polynomial

a s ~z ~ ~

m m

b B z~ ~

(2)

'

where Pm is a finite set of lattice points with ar(r=l,2) values

on set {O,1,2,...}.

The ratios of the coefficients

are to be determined by a set of linear equations, equating

to zero the coefficients

E(z)

:

(ii)

values

Symmetry

corresponding

pairs (iii)

formed by

b~z

o ~

to all

-

-

~

lattice

a

-~-z ~

,

m

points

i n some s e t

symmetry between

of suffixes, Existence

for example,

between

and Uniqueness in the set Qm should be twice the number the correct number of equations.

determinant

should,

of coefficients

Homographic

the

I 1 and 12 .

in Pm' less one, to provide

Suppose

Qm"

z I and z 2 in the

the sets Pm and Qm should be symmetric between

The number of points

(iv)

(3)

Property

In order to preserve definitions,

a s and b B

of z I~ in the expression

m

} taking

taking

in general,

The

be non-zero.

Covariance

that the substitutions ArW r , Zr - I+B w r r

(r=l,2)

(4)

36 where A r are any non-zero complex numbers numbers,

are made in the series

expansions

and B r are any complex

(i), and formal term-by-term

give a series ~

gown-Then if gm/m(W)

d

o

is the approximant

2 I ArWr ] Yr

c

l 7 rWr j

defined from the series g(w), it

should be given by gm/m (w)

Iv)

Reciprocal

=

I AlWl A2w 2 ] fm/m ~l+~lWl , 1+~2w2 j

(6)

Covariance

The [m/m] approximant

formed from the reciprocal

of the series

(I) should equal Efm/m(Z)~ (vi)

Projection Property If z2~O , the series

z I.

-I

(i) becomes a series in the single variable

The Pad~ approximant

to this reduced series should equal

fm/m(Zl,O) • (vii)

Convergence

in Measure

There should be an analogue in measure of diagonal sequences It was not possible

of Nuttall's

of Pad6 approximants.

to consider convergence

defining the approximants,

theorems before

so the problem was one of choosing the

lattice regions Pm and Qm so that conditions The problem was like solving a jig-saw puzzle comprehensible

theorem 4 on convergence

(i)-(vi) were satisfied. : difficult

while one examined one or two pieces at a time, but

as the pieces began to fit together,

leading rapidly to a solution;

apart from a relatively minor problem over symmetry, of two-variable

and in-

diagonal approximants

after two or three days work, At this stage, though,

satisfying

(i)-(vi)

and was then submitted

existence

and uniqueness

the definition emerged

for publication.

of the approximants

37

had not been properly

studied.

The region Pm that denoted by S 1 in Figure 2(m+I)2-i

I eventually i.

The number

to satisfy property

could be satisfied along the "axes"

(iii);

if Qm contained

of the lattice.

1.

(2m+l,O)

These

(O,2m+l)

could not be used,

lead to overdetermination excluding

preserve

2m lattice symmetry,

coefficients

I thought

in E(z)

with X = m+l,...,2m; Geometrically, required

of equating

(2m+l-l,l)

this would provide

rectangle

R of Figure

in the set Qm" (excluding

axes.

This

(7)

m "symmetrised"

However,

conditions

equation,

symmetrised

points.

equations. on S 4 are

(less one:)

and

from

of the

running

does not of itself

point",

in the

from O to X, must be

an angle

points

~

contributing

in the rectangle

So all symmetrised

to

rule";

imply that the boundary

should be a line making

no other points

(v) reduced

the "rectangle

of Qm then all points

if h is a "symmetrised

symmetrised

(iv)

called

2, with diagonal

the axes)

of

sum

of points

half the points

on the lattice,

that if h is a point

To

two triangles.

that the covariance condition

equation

the line S 4 is a diagonal

between

it states

Qm

to f(Zl,O ) and

to zero the

(h,2m+l-~)

and is "shared"

a geometrical

they would usually

I had to select m.

to pairs

O~Xr~2m+l;

in

the points

the reason why only half the equations

square

I showed

however;

(O,2m)

suggested

line $4, with

corresponding ,

and

2m + 1 ,

out of which

is that we need to choose

the lattice square,

points,

=

(vi)

such as SI+$2+$3

since

the next

hl + X2 contained

up to (2m,O)

of the Pad~ approximants

these points,

property

two requirements

region

O~Xr~m ,

in Qm had to be

the projection

This left me short of m equations, and

f(O,z2);

of points

the points

that Qm might be a triangular-shaped Figure

chose was the square

must

of

with the to a

R may be

lie on a single

38

line making an angle are adopted, Figure

i.

(O,2m+l)

~

with the axes, and if symmetrised

one is forced to choose the triangular

configuration

The fact that Qm may not contain points means that the triangle

configuration

can be no larger,

equations

(2m+l,O)

and

and that the

is unique.

When I showed this solution to Peter Graves-Morris Dick Hughes Jones,

and

they quickly pointed out that the constants

in any transformation

(4) had to be equal if the form of the

symmetrised

(from $4) were to be preserved,

equations

change of scale of only one variable of the points

(7) in these equations.

was therefore

arbitrary;

studied,

of

Two additional

and that a

changed the relative weights My choice of equal weights

this problem of weighting

and I shall discuss

AI,A 2

has been fully

it later.

properties

of the approximants

I proposed were

established 5 by Alan Common and Peter Graves-Morris: (viii) Unitarity

Property

The (diagonal) This property

is derived directly

Pad~ approximants, scattering

approximants

of a unitary

function are unitary.

from reciprocal

and is of particular

importance

covariance,

as for

in quantum

theory.

(ix) Factorisation

Property

The [m/m] approximant

to a function

of the form f(zl)g(z2)

is

fm/m(Zl)gm/m(Z2 )John Gammel pointed out a further property6: (x) Addition Property The [m/m] approximant

to a function

of the form f(zl)+g(z2)

is

fm/m(Zl)+gm/m(Z2). The last two properties, and the projection approximants

behave

like reciprocal

property,

and homographic

are important

in many respects

in showing

covariance

that the

like the functions

they

39

approximate. In the paper that I published 2, I wrote out a list of possible generalisations

and further problems,

work on several

of these problems.

of acceleration

of convergence

suggested by the two-variable

and a group of staff began to

Alan Genz began to study methods

of double

sequences,

approximation,

some of his results

at this colloquium.

Peter Graves-Morris

and Gordon Makinson

and he is reporting

Dick Hughes Jones, (working on his own initiall~

began to study the algebraic properties

of the linear equations,

while John McEwan and I looked for a generalisation approximants geometrical

to three variables conditions,

and N variables,

notably

John McEwan became

the "rectangle

involved

in the work when

I explained

triple

In our search for the 3-dimensional

asked my daughter morning,

when she had done this,

rule was violated, region, model

of a rather complicated

Carol to make a model

which seemed to be right.

of this solid,

explaining

m 3 of the cube shown in Figure these models,

Figure

3.

symmetrisation

Figure 4, and of triple edges where becomes

configuration,

Again

next

that the rectangle

I asked Carol to make a

that its volume was twice the volume 3.

Next morning

John McEwan

she had made four of

symmetrisation

exactly

to

are shown in problem;

after

came up with the solution

over the three surfaces

these surfaces met.

I

of a much simpler volume

(2m) 3; two of these solids

and discussion,

for

of the volume proposed;

We still had to solve the symmetrisation

much thought of double

approximants

and pointed out that they fitted together

make a cube of volume

to him

configuration.

it was evident

and we then thought

the same

rule".

nature of the problem of diagonal

he and I first thought

of the diagonal

imposing

the geometrical series.

some of them

S 4 shown in

over points

on the three

The reason for this symmetrisation

clearer when one sees the four volume

elements

fitted

40 together between volume

to form the larger cube; two volume

elements.

in concise

elements,

while

algebraic notation,

is shared

each edge is shared between

three

and f o u n ~ that it was remarkably

I realised

between volumes,

that the partition

surfaces,

number N of variables. easy to generalise

of numbers

and edges followed

rule which could easily be generalised

geometrical

surface

We had to write down the sets of linear equations

simple to do; further, equations

each triangular

of

a very simple

to systems with an arbitrary

In this way it turned out to be relatively

the scheme

structures

to power series

of Figures

in N variables.

1 and 4 thus generalise

The

to N

dimensions. In Figure zero

4, if one looks at the points with one component

(on the rear vertical

the configuration

of Figure

plane,

say), we see that they form exactly

i, with the correct double

These points

correspond

is put zero,

so that the three-variable

approximants

satisfy the projection

approximants

thus forms an infinite-dimensional

projection mimics

property.

the behaviour

encourages functions diagonal period,

(and likewise

property.

The fact that the whole

approximants

space of approximants

Jones

structure

carry out computations.

They produced,

a series of three papers

containing

and Makinson,

my two-variable

equations;

During

so well

the same

and Peter Graves-Morris

of the two-variable

generalisations,

by Hughes Jones

series

This work on N-variable

Dick Hughes

the algebraic

off-diagonal

space with a natural

will often represent

was published 7 in 1974.

Gordon Makinson,

the N-variable)

The system of

of the original many-variable

well.

symmetrisatiom

obtained when one variable

our belief that the approximants

were studying defining

to the equations

of several variables

of X

equations,

and writing programmes

to

over a short period of time,

their results.

elucidated

this structure

8 The first paper,

the algebraic

structure

of

turned out to be remarkably

41 simple

and convenient.

symmetrised

Apart

equations,

from the complication

due to the

the matrix which had to be inverted was of

the lower block diagonal

form

D(1) m

a(1)

D(1)

m

m

. . . .

o

.

°

o

.

o

o

.

. . . .

R~I)

• •

_(1) u1

,

D(2) m R (2) m •

where D~l),...,Um_(1)

.

.

D (2) m

.

,

in order

of the single power

the inversion

of a matrix

to the calculation

o

.

.

.

.

• .

.

series

of dimension

of two sequences

inverse,

and that existence

could be established.

corresponding number

The structure

of "prongs";

as a block. to points

corresponding

of a prong

to points

inverted•

ensuring

inside

outside

was

inter-

in Figure

1

on the line of

the square

is directly

that the matrix

theorems

a set of equations

on the diagonal

up

in general

set of points

The fact that the number

fact that the block matrices square matrices,

it defines

redu~d

Pad~ approximants

of the equations

(p,p)

Thus

is essentially

and uniqueness

the L-shaped

region;

the first m Pad~

that the matrix

and is defined by the point

of the lattice

are considered

.

of diagonal

had a unique

symmetry

.

(m2+2m)

it was now clear

is a "prong",

.

f(Zl,O ) and f(O,z2).

Further,

in terms

.

to calculate

to order m.

preted

.

and D~ 2),...,D~ 2) are just the square matrices

which have to be inverted approximants

.

(8)

which

of equations is equal

related

of the matrix

(8) can generally

to the

to the (8) are be

42

In the second paper

9

, Graves-Morris,

Hughes Jones and Makinson

used the prong method to define symmetric (S.O.D's); generally

this method ensures soluble,

two variables diagonal

and uniquely defines

which have properties

also satisfy the projection

property.

approximating

and S.O.D's,

its singularities,

work provided welcome

approximants

analogous

including

approximants

reciprocal

covariance;

calculating

that the approximants

5 shows the lattice configuration

The third paper

Ii

, by Dick Hughes Jones,

extension of 2-variable

approximants

I; this meant that these approximants and (viii)-(x),

powers

off-diagonal

possible.

The G.O.D's

again satisfy

similar to those of the diagonal

dimensional

S.O.D's

satisfied

in both numerator

the G.O.D's were therefore

covariance.

corresponding

approximants,

series,

the

and defined

in which the maximum and denominator

the fundamental

approximants,

approximants

excluding

of course

The lattice point configurations

6, 7 and 8.

In Figure

needed to define

the system of equations prong for a G.O.D.

were

properties

and for two- and three-dimensional

2-dimensional

to

all the properties

the most general

shown in Figures

typical

along with this

defined by John McEwan and

(G.O.D's),

arbitrary;

homographic

did in fact

using this approach

to N-variable

approximants

of all N variables

the

and also had good algebraic properties.

The paper also extended S.O.D's general

and

extended the prong

for diagonal

turned out to be exactly the approximants

(i)-(vi)

using

and typical prongs.

method to series with N variables; natural

they

in this paper;

paper.

linear equations,

in

the beta-function

An ALGOL procedure IO was published

the S.O.D.

symmetric

The first computations

were reported

confirmation

are

to those of off-

approximate~ Figure

approximants

that the linear equations

Pad~ approximants,

diagonal

off-diagonal

for three-

G.O.D's

are

6, the two types of prong are exhibited,

and a

is shown in Figure

7.

43

In the early summer of 1973, Peter Graves-Morris began thinking

about convergence

in trying to generalise This

de Montessus'

theorems, and decided to collaborate

the theorem of de Montessus

involved simplifying

and generalising

theorem for simple poles,

enough starting point.

in order to provide

The work took nine months

we each contributed

several

in establishing

de Montessus'

crucial

and complexity

of functions

impose conditions appears

which are not needed

to sequences

properties

equations,

makes

scheme.

of

; the variety

it necessary

to There

cannot be generalised approximants.

arbitrary weights

since this arbitrariness

of the whole

12

for a single variable.

of N-variable

In this paper, we introduced symmetrised

and I do

the work alone.

approxlmants

to be no reason why our theorems

straightforwardly

ideas,

•

of two variables

and

and algebra of

some limited generalisations

theorem to two-variable

a simple

to complete,

analysis

not think that either of us would have completed We succeeded

de Ballore.

Gragg's proof of

we had to do a great deal of hard classical determinants;

and I both

into the

affected none of the

The problem of weighting

had been

with us from the beginning,

and in the autumn of 1974, while he was

working

Peter Graves-Morris

at Brookhaven N.L.,

a proposal

to choose

determinant

the weights

in solving

that this choice might under relative

providing

also ensure

of weights

However,

that covariance 14

full covariance

; we therefore

of diagonal

for any N.

disaster because

degenerate.

convariance

scale transformations.

(4) with r=l,2,...,N, numerical

in order to maximise

telling us of the d e n o m i n ~ o r

the linear equations 13", it occurred

and I quickly discovered inverse choice

wrote

to me

of the equations Dick Hughes Jones

was ensured by exactly had solved the problem

approximants

of

the group

nevertheless,

courts

it may help to make the equations

almost

Peter Graves-Morris

This choice,

under

the

and David Roberts have compared

44

numerically 15 the various

choices

of weighting

a great deal of difference,

but PETCH

give rather more consistent

results

choice).

In their paper

analysed possible

13

(Peter's

choice)

than SCINCH

, Graves-Morris

degeneracies

factors;

of G.O.D.

showing that new types of degeneracies

there

is not

appears

to

(scale invariant

and Hughes Jones have two-variable

arise,

approximants,

compared with Pad~

approximants. Two applications Peter Graves-Morris two-variable

to problems

in physics

and his student

approximants

Charles

University

approximants

was present, several

double

series

results

from Nottingham 17'18

solve some problems

encouraging double

results.

(and, later,

conference

Dr. David Wood

in critical indicate

not previously

in the study of a number

to double

field in which

series

in theory be applied, the investigation

The first

Through David Roberts,

for this work,

and he has collaborated

examples 20, which again give

power series

series.

this technique

is enormous,

Theoretical

chemistry

which give is another

arise naturally.

to solve all problems

to

and at this

in fluid mechanics

in several variables

can be expected

on

solved.

we have heard of problems

no technique

phenomena.

The scope for applying

rise naturally

(of Nottingham)

that the new approximants

of numerical

triple)

Society;

proposed using these approximants

arising

we supplied programmes IO'19

arose out of a lecture

but could not resist mentioning

at the end.

and immediately

later in the

Student Mathematical

talking on Pad~ approximants,

N-variable

in potential

plus one by John Gammel,

The other application

I gave to the Nottingham I was

at Marseille

My own view is that these examples,

are very encouraging.

Samwell have applied

to the study of problems

theory 16", this work is being reported week.

have been made.

to which

While it can

it seems that we are just at the beginning

of the applicability

of this N-variable

of

approxi-

4S

mation method to a wide variety of problems

arising

in many d i f f e r e ~

fields. There have been two other investigations this general

type, by Levin 21, and by Lutterodt

These studies were framed more generally Canterbury,

and were not directed

ing the very specific properties satisfy.

Two other approaches

mants have been studied

thesis 25, O'Donohoe

of continued

fractions,

approximants

differing

approxi-

generalisations

a power series but also

Both of these approaches

from those I have described;

continued

fractions

give

it may be that

for one-variable

series,

and the moment

for many variables

which are

different.

One factor which has been of great importance work has been the readiness Canterbury

sought to

moment problem 24, and in an excellent

has defined N-variable

have generalisations

satisfy-

and Butera have defined approxi-

problem.

namely Pad~ approximation,

essentially

approximants

to defining many-variable

three topics which overlap considerably

problem,

and John 22'23

which I originally

not only matching

solving the interpolation

of

than those carried out in

at defining

: Alabiso

mants through the two-variable doctoral

of approximants

to communicate

a major factor

for her consistently more complicated I am grateful

and share his ideas;

the unselfish

connected with the work.

this was undoubtedly

of the theory,

co-operation

and I would

of all those

I would also like to thank Sandra Bateman

excellent work in producing

papers,

this

of each member of our group in

in the rapid development

like to acknowledge

throughout

this and many other

often under serious pressure

of time.

to Dick Hughes Jones for allowing me to copy or adapt

a number of his very clear diagrams, drawing some of the diagrams.

and to my daughter Carol

for

46

REFERENCES i.

J.S.R. Chisholm, Proceedings of Third International Colloquium on Advanced Computing Methods in Theoretical Physics (Marseille), ed. A. Visconti (1974), C-XIV-I.

2.

J.S.R.

3.

J.L. Gammel, "Pad~ Approximants and Their Applications", P.R. Graves-Morris (Academic Press, 1973), 3.

Chisholm,

C. Critchfield, 4.

J. Nuttall,

5.

A.K.

Math.

Comp.

27 (1973),

Rocky Mountain J. Math.

J. Math. Anal. Applic.

ed.

4, 2 (1974).

31 (1970),

Common & P. R. Graves-Morris,

(1974),

841.

147.

J. Inst. Math. Applic.

13

229.

6.

J.L. Gammel

(Private Communication).

7.

J.S.R.

8.

R. Hughes Jones & G.J. Makinson, 299.

9.

P.R. Graves-Morris, R. Hughes Jones & G.J. Makinson, Math.Applic. 13 (1974), 311.

J.Inst.

iO.

P.R. Graves-Morris, 18 (1975), 81.

Comp.J

Ii.

R. Hughes Jones, Theory.

12.

J.S.R. Chisholm 421.

13.

P.R. Graves-Morris AMD 674 (1974).

14.

J.S.R. Chisholm A344, 365.

15.

P.R.

Graves-Morris

& D.E. Roberts,

Kent preprint

(1975).

16.

P.R. Graves-Morris

& C.J. Samwell,

Kent preprint

(1975).

17.

D.W. Wood & H.P.

18.

P. Fox & D.W. Wood, J.Phys.A

19.

P.R. Graves-Morris 46-50.

20.

D.E. Roberts, D.W. Wood & H.P. be published.

21.

D. Levin, Tel-Aviv preprint

Chisholm

& J. McEwan,

Proc.Roy. Soc. A336

(1974),

J.Inst.Math.Applic.

13 (1974),

R. Hughes Jones & G.J. Makinson,

Kent preprint

(1973),

& P.R. Graves-Morris, & R. Hughes Jones,

submitted

to J.Approx.

Proc.Roy. Soc. A336

J.Phys.A. (1975),

& D.E. Roberts,

(Letters)

(1975),

7(1974),

LIOI.

to be published.

Comp.Phys.Comm.,

Griffiths,

(1973).

(1974),

Brookhaven N.L. preprint

& R. Hughes Jones, Proc. Roy. Soc.

Griffiths,

421.

J.Phys.A

9(1975), (1975), to

47 22.

C. Lutterodt,

J.Phys.A,

7 (1974),

1027.

23.

G. John & C. Lutterodt,

submitted

to Nuovo

24.

C. Alabiso

J.Math.Phys.

25.

M.R.

& P. Butera,

O'Donohoe,

Ph.D.

thesis

(Brunel

Cimento

16, 4 (1975), University,

(1974). 840.

1974).

48

to.

Fig.l

'1. ~,.

Parameter regions : 2-variable diagonal

X~

X

Fig.2

Rectangle rule : 2-variable

-->

49

Fig. 3

Volume regions : 3-variable

50

$4

I III ".,{ 2~

S~

Fig.4

Parameter regions : 3-variable diagonal

51

T~

Y

I

:C L

©

-

-

-

n

Fig.5

Parameter regions : 2-variable S.O.D.

52

52 /

I

S~

I

I / A

Fig.6

Parameter regions

: 3-variable S.O.D.

53

t i

J /

/

/

S I .

I

Fig.7

Parameter Regions : 2-variable G.O.D.

|

54

~+~

x I

/ x I" z" i t,ll E

t',O. \',~

l

3

Fig.8

Parameter regions : 3-variable G.O.D.

CONVERGENCE OF ROWS OF THE PADE TABLE

P. R. Graves-Morris Mathematical Institute University of Kent Canterbury Kent England

Abstract Some of the convergence theorems of analytic functions de Montessus'

theorem.

are reviewed, The progress

third row, the "poles out" theorem, two variables

are explained.

about rows of the Pad~ table

especially Beardon's

theorem and

on convergence theorems and de Montessus'

Two conjectures

theorem for

about convergence

rows, one of which is a counterpart of the conjecture Gammel and Wills for diagonal sequences,

for the

of Baker,

are boldly made.

of

56 Introduction The ideal Approximants"

introduction Eli.

is Chapter

However,

the pertinent

the tale is not too complicated. Approximants, a polynomial

P.A's

at most L, B(z)

and the Maclaurin

that of f(z) up to and including But before

commencing

table.

it represents?

= A(z)/B(z)

the coefficients

that the sequence

converges,

then one is either

is the same as saying

the given power series practice,

convergence

One of the methods which

converges, convergence

and there

again the problem

convergence

of convergence

of

in principle.

In

the second

again by looking

a good answer

of using

functions

subsequence

of approximants

8 2 method,

row of the Pad~ If this Likewise,

at the third row,

In practice,

accelerated,

What has been lost?

holomorphic

is Aitken's

is solved in principle.

has been greatly

and the price

of EL/O] approximants

that the Taylor series

What have we gained?

far down the third row gives

nothing,

If the first

of EL/I] approximants.

can be accelerated

first row is slow.

the

may be too slow for the method to be of value.

is the sequence

is that convergence

sequences.

is no difficulty

of accelerating

fourth row et cetera.

of z

for using the Pad~

in or on the circle

turns out to be the same as using

table, which

agrees with M+N

is little else one can do except form

row converges,

which

at

how does one reconstruct

some suitable

converges,

of degree

and their justifications,

the Pad~ table and inspect which means

is

is a polynomial

some of the motives

There

and

where A(z)

expansion of fL/M(Z)

Given a formal power series,

function

facts may be selected

on the theorems

it is as well to reconsider

of Pad~

It starts by defining Pad~

for short, by fL/M(z)

of degree

most M, B(0)=I

II of "The Essentials

the answer

and going not too

if convergence

of the

You never get something

any row except

the first is that

are known to exist for which does not converge.

for

an infinite

The price

is the loss

57

of the guarantee

of convergence

of the Taylor series of holomorphic

functions. The natural

suggestion prompted by the foregoing

the diagonal

sequence.

the diagonal

sequence,

Various

reasons

approximants

expansion

is that you should normally use

and I entirely support

support the use of the sequence

I propose that the following One conjectures

of a meromorphic

meromorphic

function

These remarks

function

f(z).

is the

sequence

of P.A's to f(z)

(f(z)~ -I symmetrically.

There is also considerable numerical

to back the choice of diagonal

quote the theorem of Baker,

accuracy to give

approximants.

Gammel and Wills

E2~ as

to support the use of diagonal P.A's, but I think the

argument

is misleading

because

at infinit~.

so many functions

Of course,

example of a holomorphic

subsequence mentioning

of diagonal

approximants

the importance

the importance

of proving

of diagonal

have essential

the fly in the ointment

function

diverge E31.

The reason for

approximants

is to emphasise

the conjecture

of Baker,

Gammel

of discovering

functions

the class of Stieltjes

for which the diagonal

includes

sequence

not going to be easy to find,

sequences

of P.A's converges.

and, as a start,

easy, to consider row sequences. is to learn something

is

for which an infinite

and equally to state the importance (which certainly

then

numerical

evidence

Gammel's

is

are based on the theorem that if g(z) = (f(z)~ -I

results)

singularities

an

Thus (f(z)~ ~I is also a

(by authors who use sufficient

(Other authors

to suggest

that the given power series

and the diagonal

gL/L (z) = (fL/L(Z)~ -I

credible

of diagonal

reason is as strong as

the only simple sequence which treats f ( z ) a n d

experience

this popular movement.

in cases where there is no information

alternative. any other.

Pad~ folklore

is the use of

and Wills,

the class of functions)

The answers

are

it is easier, but not

So one motive for studying about how to treat diagonal

row sequences.

58 The first and second rows There is little to say about the first row except that it is the set of truncated Maclaurin

series.

The second row is much more interesting be typical of what is expected, general

row.

Everything

which have the following

is contained

in Beardon's

be at z=pL.

as this suggests,

approximants any p
R.

theorems

Let the poles

[4]

of the

PL

that an infinite

as the inequality

subsequence

do not converge

of [L/I]

to f(z) on IzI~p for

suggests,

that functions

in Izl
[L/I] approximants

for the

Then

exists which converge uniformly

It follows,

to

implications:-

lim PL ~ R ~ ~ It follows,

it appears

but not yet established,

Let f(z) have radius of convergence [L/ll approximants

because

subsequence

of

at any point in Izl
a point set which is dense in Izl
are quite simple

given by the ratio of successive f(z).

and follow from the fact that PL is terms of the Maclaurin

The final remark about divergence

from Perron's

example

series of

on a dense point set follows

[51.

de Montessu~ theorem Let f(z) be meromorphic ]zl
in Izl
on Izl

as L+~ uniformly

small open neighbourh~ds

of the poles. This means that f(z) has M poles count double,

in Izl
and pth order poles count p times.

is that, given R, we must know M. and the number of poles enclosed, theorem establishes

The important

thing

Once the radius of meromorphy, M, are known,

as much convergence

R,

de Montessus'

as one can possibly

expect,

59

and is a very powerful by Saff's

theorem.

elegant proof

[7].

The old proof

[6] is now superseded

From a constructive

viewpoint,

the

work of Gragg [8] and simplified by Chisholm and Graves-Morris has some advantages. dimensional

These last three authors

H~nkel determinants

show that the M

of the coefficients

approach

determined by the M poles of the given function nearest and their residues. of multiple

row, which

case.

Then de Montessus' is the sequence

Suppose f(z)

theorem only applies

of [L/O] approximants.

that [L/M] approximants

converge

is no, not necessarily.

Suppose the poles of f(z) from the origin,

M+I th poles are equidistant

from the origin.

centre z=O, which contains de Montessus Existence

precisely

Does it follow The answer

are ordered

and the M-I th, M th and Then there is no circle,

M poles of the function

and

theorem does not apply in this case.

of Approximants

It is worth mentioning

that the apparently harmless

of P.A's stated in the introduction Baker definition

[iO].

conditions,

important

idea that interpolatory circumstances

The

The best known

of a [i/i] approximant

rational

is well known,

consequences.

cannot be found to satisfy the

then they are declared not to exist.

is the non-existence

definition

is, in fact, the modern or

It has several

The first is that if approximants

example

to the first

in some domain as L+~?

to their distance

is

Or alternatively,

suppose that we know that f(z) has at least M poles.

according

the origin

of de Montessus' theorem and the need for other

are most simply seen by a few examples.

holomorphic.

limits

of Saff's proof is that the case

poles need not be treated as a special

The drawbacks results

The advantage

[9]

fractions

to l+z 2.

The

do not exist in certain

and one advantage

of the Baker

definition

is that it does not obscure the problem by introducing

deficiency

indices

or by cross-multiplying

by zero or by any other

6O subterfuge.

The statement

that an infinite

subsequence

approximants

of any row of the Pad~ table exists

of

is then a non-

trivial

theorem

[3], and is true for any formal power series.

Nothing

follows

about convergence

of the extant

approximants

to any

limit function. The Third Row The principal

new result

I wish to mention

theorem for the third row of the Pad~ table. established

that at least an infinite

approximants

of a holomorphic

uniformly

in any compact

Proof

The proof

the principal

function

Baker and I [ii] have

subsequence converge

Suppose

f(z)

that the given holomorphic

=

function

in the space C 2 , and

It is quite easy to show that,

X no matter how large,

there exists

of ratios R i = vi/vi+ 1 which are greater can show that f(z) has radius

is

[ c. z i i= 0 i

= vi = {Icil2+ICi+l [~}i

positive

to the function

but it is quite easy to outline

Set up an array of vectors -i v. = (ci,ci+l)

IZi[

of [L/2]

region of the complex plane.

is complicated,

ideas.

is a convergence

an infinite

than X.

for any

subsequence

For otherwise we

of convergence

equal to X, which

the denominators

of the [L/Z] Pad~

is

untrue by hypothesis. Second, we may examine approximants,

which

are QL(Z)

Provided that

that an infinite

[~el and

in the compact converge.

IBel

= i + ~L z + 8LZ2

subsequence

are sufficiently

region where

of values

small,

of L may be found so

then Qe(z) has no zeroes

the approximants

are expected to

61

Since

cL

CL+21

cL

CL+ 1

CL+ 2

cL

CL+ 1

and

we s e e t h e

condition

for

~L and ~L t o be s m a l l

is

1%-1 % I

that

CL+ 1 I

I cL is not

anomalously small.

were a l m o s t p a r a l l e l ,

T h i s w o u l d o n l y be t r u e

and i n t h a t

case,

the

[L-l/2]

i f g L - 1 a n d gL approximant

c

turns

o u t t o be a h o p e f u l

complicated, large,

the

candidate.

The p r o o f

but

runs along the

lines

that

[L/2]

approximant is

superficially

to f(z).

The o n l y p r o b l e m o c c u r s when t h e

is

in which case the

small,

good a p p r o x i m a n t . approximant small,

is

found,

or else

infinite

the

function

of convergent

holomorphic

in Baker's

is

Either

determinant

superficially

closely

a

a good

a whole sequence of determinants approximates

are

a geometric

Thus there must exist an [L/23 approximants.

This is only the outline of the proof. the condition

sufficiently

denominator

approximant

which is not holomorphic. subsequence

RL i s

a good a p p r o x i m a t i o n

We p r o c e e d b y i n d u c t i o n .

w h i c h means t h a t

function,

[L-l/2]

if

now b e c o m e s more

earlier result

[3] that f(z) had to be a

function of order less than one.

does not go as far as we would like.

The actual proof removes

However,

our theorem

The best theorem is expected

to require only that f(z) have a circle of convergence,

and to

prove that an infinite

converge

subsequence

f(z) within that circle. A Conjecture

conjecture

That remains

about Convergence

The foregoing

[i13:

remarks

of [L/23 approximants to be established.

of Rows

led Baker and me to make the following

to

62

"At least an infinite function converges

subsequence

of EL/M] approximants

to the function within the largest

of a

circle,

centred

on the origin which contains not more than M poles of the given function and within which the function We can establish credibility

this result in various special cases which give

to the conjecture.

precisely M poles,

For a start,

the theorem follows

If the given function

is holomorphic,

theorem proves the conjecture analytic within the circle, conjecture

and with precisely

from de Montessus'

theorem.

the result of the previous

then Beardon's

There remains

rows and also the question

if the circle contains

for the third row.

for the second row.

the first row.

is meromorphic".

If the function is

theorem establishes

Our conjecture

is trivially

our

true for

the question about fourth and lower

of functions

meromorphic

one pole in the circle.

in the circle

The latter possibility

we deal wilth next. A "~ole out" theorem Suppose the function

c(z)

is given, which is meromorphic

circle PR of radius R, centred on the origin, counting multiplicity

within F R.

of degree m for which both g(z) analytic

in PR"

Suppose

converges

and has m poles,

Let d(z) be the monic polynomial = ~(z) c(z) and h(z)

that a sequence

such that ELi/l] h converges

in a

to h(z)

= ~(z) g(z)

S = {LI,L2,...}

in F R as i+~.

is given

Then [L~-m/m+l] l

to c(z) for some infinite subsequence

are

!

S' = {LI,L2,...}c S

on any compact set ~ satisfying c {z: o(z)~O and IzI
convergence

is that we have established

of [L-m/m+l]

approximants

c(z) and the [L/I] approximants

= {q(z)}2c(z).

.

to the meromorphic

to the analytic

It is a little surprising

a link

function

that the link is with

63 h(z) rather than g(z) = o(z) c(z). theorem establishes

At any rate, with m=l, this

our previous conjecture about rows as far as the

third row for functions meromorphic

in a region with at least one

pole. For the proof of the previous theorem, we refer to our preprint [ii].

In outline, we note that, for m=l,

=

Q[L-I/2] (z)

Let o(z)

=

g(z)

=

~ gjz j j=o

h(z)

=

~ h.z j j--o J

gj

=

c j _ 1 + ~lCj

hj

=

gj-1

I

z2

z

1

CL_ 2

CL_ 1

cL

CL_ 1

cL

CL+ 1

z + ~i oo

Then

and

.

+ ~lgj

Q [ L - 1 / 2 ] (z)

=

za(z)

q(z)

1

hL

hL+l

gL+l

gL

gL+l

CL+I

l =

CL+ 1 d e t

IIzc~(z)

It h~,÷lt - te2~ i [] -I hL

where

I fo11e1211

r

ell

el2

le21

e22J

--

1

gL+l

e2~jl

CL+l (g L gL+l )

Then it turns out that Q[L-I/2](z)

is dominated by the term

-CL+ 1 h L g(z), if L is chosen so that lhL/hL+iI is suitably large. This is not surprising, since CL+ I is dominated by the contribution of the pole of c(z) and h L and gL are expansion coefficients of holomorphic functions. Once again, the foregoing remarks lead to a conjecture, which we entitle the role of the poles conjecture. "Suppose that a function c(z) is given, which is meromorphic in a circle F R of radius R, centre the origin, and has m poles, counting multiplicity, within r R.

Let ~(z) be the monic polynomial of degree

m for which g(z) = ~(z) c(z) and h(z) = ~(z) g(z) are analytic in F RSuppose that an infinite sequence [Li/~3 h converges to h(z) in P R.

S = {LI,L2,...} is given such that Then we conjecture that

[L~-I/w+I] c converges to c(z) for some infinite subsequence S' = {L~,L~,...} c S on any compact set ~ satisfying

c {z: ~(z)~O,

[zI
Baker and I prove in our preprint that this result holds also in the case of ~=2 and h ( z ) b e i n g

a holomorphic function of order

less than i. de Montessus' Theorem in TWO Variables The final topic I wish to mention is the analogue, due to Chisholm and myself [9], of de Montessus' Canterbury Approximants.

theorem in convergence of

We define a C.A, in this context by

fmlm2/nln2 (x'y)

=

~I ~2 a..xiyj i=o j=,o i] ~I ~2 b..xiyj i=o j=o

iJ

=

65 In this expression, a(x,y) is a polynomial of maximum degree n I in x and maximum degree n 2 in y and the set (aij} occupies a rectangular block of dimensions

(ml+l)×(m2+l)

in the (i,j) lattice space.

Similarly, the denominator is a polynomial and {bij} occupies a (nl+l)×[n2+l) block of lattice space. are (ml+l)[m2+l ) + (nl+l)[n2+l )

We define boo=l, so that there

1 coefficients to be determined.

Suppose that we seekapproximants to f(x,y)

=

~ c. xiy j ~ i=o j --o iJ

Then we cross multiply to give

? { i.o.j=o ..

,i--o j=o

T j=oT I i=o =

~ ~ d..xiy j i=o j=o iJ

We require as many of the dij to be zero as possible, and look at the dij lattice space to see what the equations should be.

First we need

the rectangular block (O~i~ml) @ (O~jzm2) which determines the {aij} , once the bij are known. equations. into

Second, we need (nl+l)×(n2+l)

- i further

Without going into details, this second block is divided

a triangular region and a trapezoidal region, and these are

appended to the first block to give the correct number of selfconsistent equations for {aij} and {bij}.

The details are explained

in [9,12,133. The approximants so defined to have a variety of useful properties,

such as reduction to Pad6 Approximants

and factorisation,

as is explained in Prof. Chisholm's contribution to this book.

In

seeking to generalise a theorem about the convergence of rows of the Pad~ table to meromorphic functions, we seek a generalisation of the

66 notion of rows and of meromorphic functions. of several variables

A meromorphic function

is a section of a sheaf of germs [14].

Such

precision is necessary because complex functions of two or more variables

are much more complicated than in the one variable case,

and great caution is required.

With an eye on the possible, we

state a theorem about functions of the form f(x,y) = A(x,y)/B(x,y), where A(x,y) and B(x,y)

is analytic in a domain

IxlgRl,

lyI~R2 to be specified

is a polynomial: B(x,y) =

~I ~2 i=o j =o

B..xlyJ 1J

with

B

= I. O0

To ensure that it is genuinely a polynomial of degree nlxn2, we require that B(x,O) has n I zeros, counting multiplicity, (~=l,2,...,nl)

which are ordered so that O
at p~l) ~ ...

g Ip~l) l - - < R 1 and similarly that B(O,y) has n 2 zeros, counting 1

multiplicity,

at p~2)

(v=l,2,...,n2)

which are ordered so that

O < Ip~2) Ig Ip~2) l ~ ... g Ip(2) I < R 2. We further require f o r ~ a s o n s n2 ' which are by no means immediately obvious, that f(x,O) have no poles equidistant from the origin except multipoles, requirement for f(O,y). IP~I

and a similar

In equations, this means that if

= IP~i) I, then n~V-I (i) = p~i) for ~=2,3, "'''ni and i=1,2. We have two reasonably strong theorems, which are:-

Theorem 1 Let B(x,y) = gl(x) g2(y), so that B(x,y) factorises.

Le~ RI,R 2

be any numbers for which Rl>Ip(1) l and R2>Ip(2) l. Then the C.A's nI n2 [ml,m2/nl,n2] converge uniformly to f(x,y) as min(ml,m2)÷~ , on any compact subset of {x,y:

Ix]~Rl,

]y]gR2, B(x,y)#O}.

@7 Theorem

2

Let f(x,y)

= f(y,x),

which means

that f(x,y)

R=RI=R2, m=ml=m2' n=nl=n2' Pi-Pi-(1)__pi(2)

Then we may set

Let

n R > Ipnl

~ i=l

Ipi/Pl I

which means

that A(x,y)

has to be analytic

(Ixl
@ (lyI
Then the C.A's

to f(x,y)

on any compact

The proofs There

is symmetric.

subset

are also other theorems

plain.

We have analogues

several

constraints

[m,m/n,n]

o5 {x,y:

of these theorems

in a larger domain

Ixl~R,

lyl~R,

uniformly

B(x,y)~O}.

are too long even to outline.

for similar

cases.

of de Montessus'

on the nature

converge

than

But the spirit

theorem,

and location

is

but with

of the singularities

allowed.

Postscript Theorems have preferred L=O,I,2,...

about

rows of the Pad~ table have been

to term the set of [L/M]

a row because

the Taylor

along the line of writing; be that "Columns Only time will I suspect convergence

that the Baker

a few years.

of convergence

incredibly

and endurance.

reason

tell which nomenclature

(also called the "poles

clearly

expansion

is usually

Maybe

difficult,

and it may

a better

title.

is the more popular. conjecture

about

(or did I mean columns?)

the role of the poles

out" conjecture) theorems

expressed

is not too strong,

Graves-Morris

of rows

I

with M fixed and

of the Pad~ Table" would have been

of subsequences

proved within

discovery

this

approximants

discussed.

will

take

for Canterbury

conjecture

longer.

Approximants

being both a challenge

will be

The is

of ingenuity

68 REFERENCE S i.

"The Essentials of Pade Approximants", Academic Press N.Y. (1975).

G.A. Baker, Jr.,

2.

G.A. Baker, J.L. Gammel and J.G. Wills, 31 147 (1970).

3.

G.A. Baker, J. Math. Anal. AppI. 43 498 (1973).

4.

A.F. Beard.n, J. Math. Anal. Appl.

5.

O. Perron, "Die Lehre von den Kettenbr[ichen", Vol. II p.270, B.G. Teubner (1957).

6.

R. de Montessus de Ball.re, Bull. S,c. Math. France 30 28 (1902).

7.

E.B. Saff, J. Approx. Theory 6 63 (1972).

8.

W.B. Gragg in "Pad~ Approximants and Their Applications", Academic Press (London) p.l17 (1973).

9.

J.S.R. Chisholm and P.R. Graves-Morris, Proc. Roy. S.c. A 342

J. Math. Anal. Appl.

21 469 (1968).

341 (1975). I0.

G.A. Baker in "The Pad~ Approximant in Theoretical Physics" eds. G.A. Baker, Jr. and J.L. Gammel, Academic Press, N.Y.

(1970). ii.

G.A. Baker and P.R. Graves-Morris, "Convergence o£ Rows of the Pad~ Table", Brookhaven preprint AMD 675 (1974).

12.

R. Hughes Jones, "General Rational Approximants Kent preprint (1973).

13.

P.R. Graves-Morris and R. Hughes Jones, "An Analysis of Two Variable Rational Approximants", Brookhaven preprint AMD 674

in N-Variables",

(1974). 14.

L. Hbrmander, "An Introduction to Complex Analysis in Several Variables", D. van Nostrand, p.161 (1966).

THE

USE

OF

PADE

APPROXIMATION L.

IN N U M E R I C A L

INTEGRATION

(*)

Wuytack

Summary. The probiem of the numerical wiil be considered. approximation,

evaluation

of the integral

A nonlinear technique,

b I = f f[t)dt a

based on the use of Pad@

will be given and discussed.

The value of the integral

satisfies I =y(b) , where y(x) is the

solution of the initiai value problem y' =f[x) solution is approximated

with y(a) =o.

using Pad@ approximation.

one-step method is obtained to approximate properties of this method are given,

This

Consequently

a

the vaIue of I. Certain

e.g. its order of convergence.

1. Introduction

Most classical formulas for approximate integration of a definite b integral I =f f[x).dx are linear (see [3]), which means that I is a approximated

by a linear combination

of the values of the integrand

f or I ~ W l . f [ x 1] + w2.f[x 2] + ... + Wn.f[Xn]

The points xl,x 2 ....,x n usually belong to w1,w 2 .....w n are called weights. derivatives combinations

[a,~

and the numbers

In some cases the values of the

of f are also taken into consideration of the values of f and its derivatives

points are formed to approximate

'

and then linear at certain

the value I of the integral.

(*) Work supported in part by the FKFO under grant number 2.0021.75

70

Using such techniques, a sequence 11,12 .... of approximate values can be constructed having I as limit.

The convergence of the sequence

I K can be accelerated by several techniques.

Well-Known is Romberg

quadrature where the sequence {I K} is constructed by using the trapezoidal rule and the convergence of this sequence is then accelerated by using a linear extrapolation technique.

Also nonlinear

acceleration techniques can be used, e.g. rational extrapolation and the e-algorithm.

A comparison of these different techniques for

accelerating the convergence of { I K} can be found in

[~

and

[4 •

In many cases the linear methods for approximating I give good results.

There are however situations,

e.g. if f has singularities,

for which linear methods are unsatisfactory.

Sometimes it is then

possible to modify a classical method in order to adopt it to the special situation on hand (see e.g.

[4 ,[7] and

[4 ).

Another Kind

of approach, which will be followed in this paper, is to use a nonlinear technique.

This means that I will be approximated by a non-

linear combination of the values of f and its derivatives at certain points.

In this paper some methods will he described which are based on the use of Pad@ approximation.

The first method is based on the

approximation of the integrand f by a Pad@ approximation and then b performing the integration f r(xJdx. This approach is called a "direct" and considered in section 2.

The second method is given

in section Z and is based on an "indirect" approach to the problem under consideration,

by reformulatin Z it as an initial value problem.

The resulting differential equation is then solved by using Pad@

71

approximation.

In order to apply this technique,

the derivatives of

f must be Known, which might be sometimes less interesting in practice. Therefore a modification of this method is given in section 5, having the same order of convergence but without the need to compute derivatives.

In section 4 some convergence properties are proved.

2. A direct method

Let R m be the class of (ordinary) rational functions r = P where p n q resp. q is a polynomial of degree at most m resp. n and such that P is irreducible. Let r be the Pad6 approximant of order (m,n) for q b f in R m , then I = f r(x).dx can be considered as an approximate n r a value for I. I

r

In general it is however not easy to find the value of

, if it exists.

If the poles of r are Known then the partial

fraction decomposition of r can be formed.

This sum can then be

integrated term by term in order to get Z . r

This process is

numerically less interesting and several difficulties can be encountered.

An application of this technique to the computation

of Fourier Transforms can be found in

[I] .

3. An indirect method

x Put y[x] = f f[t].dt then I =y[b]. a [a,~

then y is continuous on

at a point x of

[a,~,

If f is Riemann integrable on

[a,~ .

Furthermore if f is continuous

then y is differentiable at x and y'[x] =f[x].

The computation of I can now be done by computing the value in b of the solution y of the followin@ initial value problem y'[x)

=f(x)

, y(a)

=o

.

: (1)

72

In order to find a solution of (1] numerically

a discretizatlon

technique can be used. positive

Let x, =a +i.h for i = 0 , 1 ....;M with M a l b-a integer and h = T " Let Yo =y[a) and an approximate

value of y[x i) for i > o ,

be denoted by Yi"

We now describe a method

f o r computing Yi' f o r i = 1 , 2 ..... M.

Assume that Yi is Known and that the values of the derivatives exist for k > o .

Then,consider

s i [ h ] =Yi + h ' f [ x i ) Pi Let r. = - - b e z qi

f[k)[x i]

the series s. defined as follows l

h2 h3 f " [ x _ ) +~T!'f'(xi) +~!' i + ""

the Pad@ apprcximant

of s, of order [m,n]. z

[2)

This implies

that r i is an element of R m and that n

si[h).qi[h) -pi[h] =OChm+n+ki+lj

[3]

for some integer value of K i, which is as high as possible.

If qi[xi+1) # o then the value of Yi+1 is defined as follows

Yi+l =ri[xi+l]

"

(4)

Since Yo is Known and since r i exists for every i, this technique allows us to compute yl,y 2, ....YM'

The value YM can be considered

as an approximate value for y[b) or I. section that lim YM =I h~ o

It will be seen in the next

if certain conditions

ere satisfied.

73

A variant of the above technique has been used by P.J.S. Watson in [9], for the case where h = b - a

or M = I.

In the next exampie

we illustrate what Kind of formulas we get if [4) is

used

to

compute approximate vaiues for I.

Example

Let m =n = 1 then the Pad@ approximant r i of order (1,1)

of (2) is the rational function associated with the irreducible form of ~ q

where

f'(x.) p=yi.f[xi)

÷ h. [f2[xi) - Y i . T

]

and

~'[x i) q =f[x i ] -h.-----~----.

The formula [4) gives 2.f2(x.) Yi+q = Yi + h . 2 . f ( x i

) - h.f'(x 1 i)

for

i=0,1

. . . . . M-1

It is clear that YM is a nonlinear combination of values of f and its first derivative at the points x,. z As a consequence of [5) we also get the following formula for approximate integration between x i and xi+ 1 :

Xi+l

2.f2(x. )

1

fxl f[t).dt ~ h.2.f(xl ) _h.f,(xi )

(5)

74

4. Convergence

properties

In this section some convergence

properties

of the method described

in section 3 will be given. Consider any one-step method of the following form

Yi+l =yi + h'g(xi'h)

for i=0,1

for computing a solution of (I) numerically.

. . . . . M-1

(6)

Assume that g(x,h) is

defined and continuous for every (x,h) in [a,b] x [o,h o] , where h ° is some positive real number.

Under this condition

the following

result can be proved.

Theorem 1.

Let Yi be d e f i n e d by (6), then

x in

[a,~

if

and o n l y i f

lim

h~ o x=xi

Yi =y(x) f o r every

g[x,o) =f(x).

This property is a special case of a theorem about the convergence o% one-step methods for the numerical equations

solution of ordinary differential

Csee [4 ,p.?1).

Due to the definition

of r. in section 3, it is clear that l

be written in the form [6) with g{x,h) = f { x ) + h .

g(x,h) = [ri(x) -yi ] /h or

f'(x] + 3 ~~ . f"(x) .....

g[x,o) = f i x ) , consequently

[4) can

This implies that

theorem I can be applied and

~yM

=y{b)"

About the order of convergence we can prove the following result, the case where K. =o in (3) for i =0,1,2 .....M-1. l the case of normal Pad~ approximants.

for

This case is called

75

Let r. be the normal Pad@ approximant l

Theorem 2.

of s. of order &

[m,n) and Y i + l be d e f i n e d by [ 4 ) , then Y ( X i + l ) - Y i + l ~O(hm+n÷1) as h ~ o ,

f o r i = 0 , 1 , 2 . . . . . M-1.

A proof of this theorem is given in [10].

5. One-step methods w i t h o u t usin~ d e r i v a t i v e s

Consider any one-step method of the form (6) for computing of (I).

In order to find the value of g[xi,h)

that derivatives

of f must be computed.

it might be possible

This is e.g. the case if (6)

is derived by using the method in section 3. with m + n > l . compute derivatives less interesting. another expression,

a solution

of the integrend can be compllcated

The need to and numerically

Therefore one could try to replace g(xl,h) without derivatives,

the same order of convergence.

in (6) by

hoping to Keep a method with

This technique

has successfully

been

applied in some cases.

If e.g. the derivative quotient,

in (5) is replaced

by its forward difference

we get

2.f2(x.) 1 Yi+l = Yi + h • 3.f(xi ) _ f(Xi+l 3

It

can be proved (see [10]) t h a t t h i s

(7)

o n e - s t e p method has the same

o r d e r o f convergence as the method d e f i n e d by using ( 5 ) .

The f o r m u l a

(7) can a l s o be c o n s i d e r e d as a n o n l i n e a r method f o r a p p r o x i m a t e integration

of f between x i end xi+ 1 , namely as

Xi÷l I xi

f(t).dt

2.f2(Xl) ~ h.3.f(xi ) _f(xi+l

)

76

6. RemarKs

In [10] some numerical examples are given, illustrating the usefulness of the nonlinear techniques derived in this paper.

For "smooth"

integrands the classical linear methods give in general better results than the nonlinear techniques.

If the integrand has a pole near the

interval of integration than the nonlinear techniques can give better results.

Due to possible singularities in formulas of type C5) and

(7), care must be taken in applying these formulas.

Difficulties can

sometimes be avoided by a careful choice of the stepsize h.

Other nonlinear techniques for numerical integration are considered in [10 I.

Author's address Wuytack L. Department of Mathematics University of Antwerp Universiteitsplein I B-26]0 WILRJK (Belgium)

77

References

[I] CHISHOLM, J.S.R.: Applications of Pad6 approximation to numerical integration.

The Rocky Mountain Journal of Mathematics 4

C1972),159-167.

[2] CHISHOLM, J.S.R,; GENZ, A. and ROWLANOS, G.E.:

Accelerated

convergence of sequences of quadrature approximations. Journal of Computational Physics 10(1972),264-307.

[3] DAVIS, P.J.; RABINOWITZ, P.: Numerical integration. Publ. Co., London,

[41 FOX, L.:

Blaisdell

1967.

Romberg integration for a class of singular integrands.

The Computer Journal 10C1967),87-93.

[5] HENRICZ, P.:

Discrete variable methods in ordinary differential

equations.

[61 KAHANER,

O.K.:

John Wiley & Sons, New York, 1962.

Numerical quadrature by the s-algorithm.

Mathematics of Computation 26(1972),689-693.

[71LYNESS,J,N.;

NINHAM, B.W.:

expansions.

Numerical quadrature and asymptotic

Mathematics of Computation 21(1967),162-176.

[61 PIESSENS, R.; MERTENS,

I.; BRANDERS, M.:

Automatic integration

of functions having algebraic end point singularities. Angewandte InformatiK (1974), 65-66.

[91 WATSON, P.J.S.:

Algorithms for differentiation and Integration.

In "Pad~ approximants and their applications" MORRIS, P.R., editor),

[I01WUYTACK,

93-98.

(GRAVES-

Academic Press, London,1971.

L.: Numerical integration by using nonlinear techniques.

Journal of Computational and Applied Mathematics. To appear.

DETERMINATION

OF SHOCK WAVES

BY CONVERGENCE ACCELERATION by Pr.

Max BAUSSET

-

TOULON

(France)

-MAY 1 9 7 5 -

The difficulties

of the determination

of stationary detached shocks

arise from the global character of the problem to be solved as it is impossible to determine shock waves in the vicinity of a point. On the contrary for attached shocks determination These difficulties non-s~at~anary

is possible step by step from the vertex of the body. can be avoided by considering

flow. If a body situated in a motionless

so that the field of initial velocities causes a shock wave the determination

is not null,

shock problems

in a

fluid is set into motion

this motion immediately

of which in the vicinity of the initial

time is a local problem. The evolution of the shocks corresponding

to this motion in the vicinity

of the starting point of the body is presented here. On analytical of the stationary detached shocks waves related to an analytical be obtained by a process of convergence

accelaration,

representation

convex body can

then by passing on to the

limit when the permanent motion is reached. The data will be supposed to be such that operations

can be considered as possible within the fields where it is being

operated.

-

E~E~_~_~_~e_!~_~_~e~i~!~e_S!~i~

:

The space is related to orthonormated

fixed axes.

The equation of the body is : .%~ s h o c k

[Y (1)

x = f(yz) +

~(t)

oI

[.i o''x body

82

in which f is supposed to be uniform and the body occupies

the region

:

x ~

f (y z) +

~ (t).

The initial data are such that : (2)

$(O) = O

~' (0) >

0

This body is plunged into a non viscous compressible, supposedly perfect fluid. This motionless representing

respectively

the pressure,

and and

the density and ratio of specific heats.

The latter will be supposed to be constant throughout ristic quantities

non heat-conducting

is defined by quantities p, p

of this fluid (velocity,

the motion.

The characte-

pressure and density) will be desi-

gnated as V, p and p at the point of the spatio-temporal

coordinates

x, y, z,

and t. The fluid and the body are motionless before t = O. Since at the initial time

~' (0) > 0 the motion of the body immediately

propagates

(3)

itself through the fluid.

x =

F (y z t )

For reasons of calculations write (4)

causes a schock wave which

Its equation will be :

symmetry which will appear below,

let us

: ~ (x y z t) (x y

z t)

~

f (y z) +

=

F

so that when considering

(y z t)

$(t) - x - x

covector r = ] x, y, z ] the vectors normal at each

instant to any point of the body and the shock are defined by :

(5)

N,

= ~r *

=

N~

= ~r ~

=

[-1,

Fy,

F'zl

f'z In these conditions,

the normal number of Mach M at time t at any point

of the shock wave and the upstream number of Mach at infinite by :

32~ are defined

83

6t (6)

= _ _ ~'(0)

M (~) =

l -

C(~r~

c designating Let us write

x ~r~) 2

2 the speed of sound so that c p = yp at any point x, y, z, t. 2 (y + I) ~ = y - 1 and relate pressure and density to their va-

lours at infinite

introducing

P (x y z t) =

will be written

under

dr+ t

(7)

:

R (x y z t) =

the continuous

equations the form

motions

of dynamics,

of the considered

fluid are

mass and energy conservations

which

:

2 ! - ~ 2" R. ~ r P = O I +~

-2 c

dtR - R.. T r (~r v) = O

(I - 2 )

dt

dimensions

P

In these conditions, defined by the classic

without

desigmating

R. dtP

(I + 2 )

the corpuscular

The solution and the boundary

p. dt R = O

derivative

of shock problems

of limits

In the absence which

+

and Tr

the trace of matrix

~r V.

is a result of the study of this system

on the body and on the shock wave.

of viscosity,

leads to the equations

the body is necessarily

a stream surface,

:

(x y z t) = 0 (8)

6r~ x V~ = ~t ~

in which

V~

is the value of V on the body

The classic motion,

conditions

of shock phenomena

mass and energy are preserved

Theae usual conditions

~ = O.

while

can be expressed

mean that the quantities

crossing

the surface

by the equations

:

of

of the wave.

84

~ (xy

z t)

=0

(9) 6t#

-- 2

V~

~r ~ ~r ~ x ~r ~

P

+2 ~2

=

~r ~

--

'c

R~

= 21(

~2~r~-I

6t~

--

~/

2

I 7t° _ 2,)

+

2 (6 t *)

in which

V~ ) P#

and R~

are the values of V, P and R on shock ~

At any point of the spatio~temporal the shock and for t to equations

~ 0

equations

= 0.

region included between the body and

(7) are identities in x, y, z and t. Added

(8) and (9), they permit the calculation of the partial n-order

derivatives of quantities V, P, R and

~ at time t = 0.

In effect, if one place oneself at the initial time when the fluid is motionless everythere except on the body which is set into motion, the position of the shock wave coincides with the position of the body. Thus one has for t = 0 the relations

(lo)

:

~ (x y z o) =

which will be designated

~ (xy

~

=

z o)

F(y z 0) = f (y z)

~o"

O

They entail the following equations V~

= V o

: R~

$o

= o

R $o

(11) P$

o

= P

~o

~

r

~

o

= ~

r

~o

85

From the given equations (8) and (9) considered at time t = O one deduces the following relations :

6r~ °

x

= (6t~)t = O =

V~o

~'(O)

(12) 6r~ ° x V~

= - (| - 2 )

l~--]--, t - 0

o

c

0r~ ° x ~ J o

(6r¢ ° x

6r#o)

(6t~)t=O

If one notices that -.(6t$)t_O = F' (y z O) one deduces by elimination of V the t initial value of the shock velocity at any point :

(13)

F' (y z O)

t

~'(O)

+ (~

2(1_ 2 )

(O)

(1_~2)

)2

+

6r~ °

x

--

6r~ °

- Second order derivatives : . . . . . . . . . . . . . . . . . . . . . . . .

The total number of the partial n-order derivatives of a function A(x y z t) being C n n+3' one will be brought to consider the table : Jl

(14)

6n rn

6nA II = II 6xn

A ( x y z t )

~n A ;

6Y n

6nA ;

6zn

which will he a line or a matrix 3 x 3 according as A has a scalary or vectorial value. Relations (|l) then permit to calculate the spatial derivatives of (x y z t) for t = 0 in the form :

n

(15) ~r n (6r~ o) =

In

-~x - n (6r~o) ; ~ y

n

n

(6r# o) ; ~--~ (6r~ o)

1

from which the initial values of all the spatial partial derivatives of the function representing the shock wave :

(16)

6 p+q F (y z O) 6y p

6z q

=

dP+q f (~ z) 6y p

~z q

can be deduced. But concerning the temporal or spatio temporal derivatives of F it is necessary

86

to use the derivatives of the boundary of limits on the body and the shock. If one designates by ~ one differential which according to the case can be

~r'

~t' or dr, one deduces from equations (8) and (9) the following linear

equations in relation to

~(~t~), ~(~t~) and

~(~r~) :

+ ( x y z t) = 0

(17)

~r ~ x

6V#

=

6(~t ~) -

6(~r~) x

v~

(x y z t) = 0

V~ = ~. ~r ~

x

~(~t ¢)

+

~. 6r ¢

x

~r ¢ "

~(~r ¢)

+ Y "

~(~r ~)

(18)

P~ = el. ~(6t ~) +

81 •

~(~r ~)

R~ = e2" 6(6t~) +

82"

~(~r ~)

in which to following quantities which are dependent only on the first order derivatives of ~ that are known at the initial time are :

= - (I - 2),

~ 2 ~r ¢ x 6r¢ + (6t~)2 ~ #

x

8 = 2(I- 2),

~r ¢ "(6t#)2

~r ¢

y = (I- 2 ) 6r~

x

~r ~ .(6t~)2

6 ¢ t

~1 =

x 6r ~

.... -2 (6 r ~ x 6 ~¢)2 c

(19) 8L = - 2(|+ 2 ) c --2

Y2 =

2c2"

(dt~) 2 (~r ~ x ~r ~ )2

(| - 2 )

r (6t0) 2

. 6r ~

2 6r ~ x dr~ ~2 = -2 ~2(I- ~ ) (~t~)4

87

According to the choice of differential 6, equations lines and matrices

(|8) include the scaleries,

: t!

~t(~t ~) =

#t 2

~t(~r ¢)

(20) ~r<6r ~) =

=

Cx2

,,

¢,, xy

¢,, xz

¢"yx

#'y2

# 'yz

zx

zy

Cz 2

J ¢"tx #"ty

¢"tz 1

~r(~t¢) = ~t(~r¢)

which contain all the partial second order derivatives of function ¢. As to the first members,

they contain the derivatives of V, P and R on the body or the

shock, i.e. the quantities

;

~v~

~v~

dt(V~) = ~

~'t + ~t

for the scalery derivative and the tables :

I 6rCV ~) =

~v~ --~ 8x

~

~v~

; 87

6z

for the vectorial derivative and the similar quantities for P and R and for function ~. Considering the equations

(7) of the motion valid on the body ~ = 0

with the preceding differential relations, equations. Namely and

one has eleven sixteen indeterminate

~V, ~P, ~R, on the shock or on the body as well as ~(~t ¢)

~(~r¢). Consequently the'process used does not permit, as in the case of

first derivatives,

to calculate all the second order derivatives of V, P, R

and F placing oneself at any time.

If one places oneself at time t = 0 the preceding relations entail :

dt(V~o) =

(21)

~V~o

~V¢o

~x

$'(0) + - - ~ t

~V~

dt(V ~ o

)

6V¢

o F~(O) + ~x

o ~t

88

for corpuscular derivatives as well as equations :

_

,,

go

;

6x (22)

;

=

6y

6V0o

6V~o

~t

~t

o

-

6z

O

6x

~y

=

~ r (~r f)

and the following value :

(23)

6r(~r~o)

=

0

0

0

0

f,,2 y

f,, yz

O

f" zy

f,,2 Z

One sees then that at that time the ensemble of second order partial derivatives is determined. In particular, one obtains : ~t(~t~o ) = Ft2(Y z O)

~rt(~o) =

i0

)

Fty(Y z 0)~ Ftz(y z O)

....

I

If one introduces the vector normal to the body for t = 0 : N~o= 6r f and the normal initial number of Mach defined from (6) by : (24)

M (x y z O) = iF~(.~rf, x

i ~-flL 2

the results are as follows :

(25)

iv t! O; Fty(Y z O) ) Ftz(y z O)

2 M c <6rf x 6-~-} M2 + |

2

0 ;g "

go

6y 6N~

go

6z

for all initial spatio-temporal derivatives. The initial value of shock acceleration is expressed as follows : (26)

,, Ft2 (y z O)

in which A

n

and C

n

M2

A6 A8 (M2-1) . ~"(0) - -2 c ~., (f) + ~2 .~.~2(f) (l- ~2)C 4 ~4 " I ~66

are n-degree polynomial expressions and ~ l(f) and ~2(f)

quantities related to the geometry of the body :

80

A6

F 7 I( l + ~2) M 2 _ U21

(~2 M 2 + l - ~2). (M2_l)

(27) A8 = (p4_ p2) M8 + (_2~4 + 3~2 + l ) M6 + (3~4 + 2~2 - I ) + ( - 2 ~ 4 + ~2+1)M2

-(1-

~2)2

C~ = (3 p2 + l) M4 -

(2~ 2 - 3) M2 - ~2

c6 °

+ 1)

+ 1) [ ( 3

M4 +

-

~ l (f) = 2 6r f x ~r(~r f) x 8rf. ~ r f

-3)

x ~r--~ - I

- T r <~r(~rf)>

(28) "~2 (f) = ~r f x 6r(6rf) x 6rf. <0rf x ~r--~-| If one places oneself in the particular case of plane or axisymmetrical flows one finds again in (13) and (26) formulae established by CABANNES.

-_6~£~_~!2~_2~_~!~!~

:

The process used for the calculation of initial second order derivatives can be applied to superior order derivatives. The total number of the n-order derivatives of wave function F(y z t) n is Cn+ 2. Now if one differentiates identities (17) and (18) in yaz, and t (n-l) times and then if one places oneself at time t = O one obtai~e~ 6 Cn+in-I identities in relation to y and z. Similarly if one derives (n-2) times the equations of the motion which are identities in relation to x, y, z and t one obtains 5 C n-2 n+l

relations.

Placing oneself again at time t = 0 and on the shock or the body one finally obtains : (29)

f (n)

=

6

n-I

Cn+ I

+ 5

n-2

Cn+ I

equations. These are linear in relation to the initial partial derivatives of V n-I p and @ numbering 5 Cn+ 2 and to the initial values of the (n-l) order derivatives of F(y z t) numbering Cn+l°n-; Indeed, the coefficients of these equations are dependent only on derivatives the order of which is inferior to (n-l). Consequently, giving to n the integer values successive from n = 2 one can calculate all the preceding indeterminates as one can see that the number of equations equales the number of indeterminates.

90 •"ne total number of the equations to be resolved is therefore : (30)

n ~ f(n) = -6 + n(n+l)(n+2)(Dn+ 19) 2 4 !

N =

In particular, one can thus calculate derivatives F~ n) (y z 0). This calculation including all the initial scalcry derivatives of the three characteristic functions V, p and p can be simplified as it has been shown in the case of second order derivatives considering the lines : (31)

~(n)(6r~ )n t

= l~(n+l)tnx ; ~ tny(n+l) ; ~ tnz(n+l)l

as well as tables (14) for A = V, p, p in the form :

r ptq

6xP6tq

6yP6tq

6zp 6t q

One can see that one has thus to resolve according to the values of n successively :

n

2

~

(n)

3

4

5

6

....

23

56

II0

150

301

....

23

79

189

379

680

....

scal~ry equations the number of which can be reduced thanks to (31) and (32). In particular A, B ..., H, ... designating polynominal expressions in relation to M the degree of which is equal to the index, one obtains : F'" t3

(33)

(y z 0) =

+ I BIO,$ C12

'' ( O ) . ~ ( f )

+--3 F]4 ~ 4 ( f ) + C.E16 . in which ~ i ( f )

M2 -~"' (0) + ~ 6(l-p 2)(M 2 +I)

+ 73 DI2 . ~ ( f ) El2

%

~"

~8

(o) "[! ( f)

+ (~2(f)) 2

+ 73, FI2 . ~ 3 ( f ) G|2

+

~3. D]4 ~ 5 ( f ) + - - 3 HI6 . ~ 6 ( f ) C16 . c ,LI 6

are still quantities dependent on the body. One obtains :

~3(f)

= 16r f x 6(2)(6 f)x ~ r ~ 2

r

I

+

(6rf x

(34) + Tr <6 r 2(2) (~~ f)> - d~t <~r(6rf)>. <~rf x ~r(6rf ) x ~--~>

91

given the considerable complexity of calculations _(IV) impossible to develop ~ 4 t the shock wave. -

; it has been

except for y = z = 0 corresponding to the apex of

A~pr_o_xi_m_a_t_e_equa_t!_ons__of_non_-s__ta_ti_on_a_rz_~he~

:

The proceding calculations permit to write the equation of non-sta stationary shocks in the form of a 4-degree polynominal expression : F (y z t)

t2 " f (y z) + t F t' (y z O) + - 7 Ft2

=

(y

z O)

+

(35) t3 F"'"

+-~

t3~Y

z O) + t 4 FIV

~

t4

(y z O)

This represents with tolerable approximation these shock waves for three spatial variable flows and its evolution in the vicinity of the starting point of the body. Now the considered shock waves exist whatever value t has as long as velocity remains supersonic. It is therefore interesting to approach the equation of these waves by an analytical expression valid within the longest possible lapse of time, If function F (y z t) is uniform, one can accelerate the convergence of the associated Taylor series. WYNN'S E-algorithm corresponding to PADE'S diagonal approximation will be chosen. Thus the fractionary approximation of the equation of non-stationary shocks will be determined in the form : t!

If(yz) Ft2(Y Z O) - 2 F't (y z 0)] t - 2f(yz).F'(Yt Z O) (36)

[|,1] F (yzt) = t!

t. Ft2 (y z O) - 2 F' (y z O) t whose field of definition is that of PADE'S first order diagonal approximations. It is interesting to notice that in the case of very high velocities of the body

~ M

-> co

(y Z O)

one can write : ~ ~

Ft2(Y z O) '~

f~r ~

3!a2+ 1

x ~J>

2|

(1 + p 2 ) , £ 1 ( f )

F't (y z O) ~

+ (1- !a2).~2(f )

One can then see that, for an analytical body, approximation (36) remains valid for t

> O. It is clear that such is not the case if the body's velocity is

~ndeterminate.

92

-

D~_t_ermination_of_d_et_a~_hed_s_ta_tio_~_a_~z_s_h~h~

:

It is possible to infer from the foregoing results approximate formular determining by the movements

the positions of the detached stationary shocks created

of blunt analytical bodies in translation motion.

In effect, when the second derivative

~"(t) vanishes

for t ->

the motion of the body tends to become unifo©m. It is then logical to admit that the motion of the fluid in relation to a mark linked to the body tends to become stationary. stationary

shock results from the knowledge

lary logical to admit that the stationary

Consequently

the position of a

of F(y z t) for t--~oo. It is simi-

flow is reached all the more rapidly

as the velocity limit of the body is reached more rapidly. One will place oneself in the case when is chosen so that : t

~< 0 :

$ (t) = O

t > : ~ (t) = t. ~' (0)

In any space of time when F(y z t) is an analytical variable dependent on time, one can write

: oo

(37)

F (y z t) ~

~

aj(yz).t j

j=O and one has indicated a calculation process of a. coefficients. J co ral methods allowing to replace the analytical function Ya.t j

There are seveby a sequence

L

of functions

converging towards F(yz)

in the field of

O j

convergence of the entire sequence a.t j and which are definite whatever 3 value of t is and which have a limit when time increases indefinitely. Within the framework of the

(38)

A

= n

n ~ 0

a.t j J

n ~

e-algorithm

the

theory supposes

0

is a converging sequence whose terms are dependent as parameters. Among others this theory proposes

to replace this sequence by another converging

more rapidly towards the same limit or converging

to a more extensive field

than the initial sequence. PADE'S defined by :

n-order diagonal approximation

fused in (36) for n = 17 is

93

t n 1A l ) ..............

t ° An

a 2 ) .................

an+ l

an+l

a2n

Pn

(t)

t °

Qn

(t)

tnAo

al

(39)

i

t

n

a1

an which

leads

in p a r t i c u l a r

Ql(t)

a2

................

an+ !

an+l

~ ................

a2n

~

lim

P (t) A n--5-----= (-I) n

t =o

Qn(t )

A

=

the c o f a c t o r

ao

,

ald ........

an

a2

~

a 3 ~ .......

an+ |

an

~

an+|) ......

a2n

of a

o

in

(41)

A

n

•

A.

As to SttANKS'S t r a n s f o r m a t i o n s , sequence

.....

2 ao(ala 4 - a~) - a 1 ( a l a 4 - a 2 a 3) + a 2 ( a l a 3 - a 2) 2 a2a 4 - a 3

Q2(t)

and 6 b e i n g

~

~..•......•

to

lim t =

:

t

a2

P2(t) (4O)

n-1

~

: 2 aoa 2 - a i

P1(t) lim t ~

with

~ ...............

In, n| A n =

the sequence

B

n

=

they

are

defined

:

An+l"

An-l-

An+l + A n - I

A2 n

- 2 An

n

>

1

by associating

to

94

The process

can be iterated by considering _

C

(42)

= Bn+| Bn-I

B2 n

n

the sequence

:

~ 2

n

Bn+ 1 + Bn_ 1 - 2B n and so forth. In each of these iterations t

->

the first term B! C 2 ... has a limit for

oo

lim

2 aoa 2 - a I

B I (t)

a2 (43)

2 aoa 2 - a I

lim C2(t )

22 (ala 3 - a2) 2 a4a 2 - a 3

a2 One can then see that PADE'S

a4 2 a2

approximation

If,I] and SHANKS'S

first

order B 1 have the same limit which we shall call first order approximation. On the contrary,

second order approximation

differ.

- ! l ~ ! ~ _ ~ l ~ 2 ~ _ ~ ! _ ~ ! ~ ! ~ l _ ~ h ~

When the studied the choice between

Concerning the stationary

method

numerical

which is the case here,

and another

results

the approximation

detached wave, point

Placing

is determinate,

an approximation

a comparison with accurate

an indefinite

function

:

relative

of function

the preceding

can only be made through to know particular

cases.

x = F(y z t) representing

calculations

have been conducted

to

(x, y z) only up to n = 2.

oneself

at a mark linked

to the body,

it follows

from (35)

and : _

(44)

F't (y z O) -

~2

M 2

+

1 -

~2

1

~'(0) = c.

~Sr f x 8 rf ~'2 M

that in the vicinity

(45)

of the initial

time one can write

F (y z t) = f (y z) + ~t -~2

M2 + 1 - -~2

:

<6rf x ~--~>~l~ + tSF, ' t2 (y z O)

M in which

the expression

of the second derivative

$"(O) = O. In these conditions gives a position

of the detached

PADE'S

follows

(or SHANKS'S)

stationary

shock

:

from (26) with

first order approximation

95 (46)

x = f(y z) -

2( ~2 M2+ I - ~2)2.(M2 + 1).d 4 M 2. A 8 . ~

2 (f) -

(~r f x 6r f)

I

M 2. (M 2 + l)~A6.~l(f)

In this expression the velocity of the body and the conditions of the motionless fluid interven through the normal number of Mach which plays the role of an auxiliary parameter.

There results a certain complication even

for very simple analytical bodies. As shown above the permanent motion will be reached all the more rapidly as the body's velocity is high. In this hypothesis

the preceding rela-

tion is consideratly simplified and leads to the approximation

(47)

x = f (y z) + (I +

2 ~2(3 ~2 + I) ~ 2 ) . £ I ( f ) + (I- ~2).£2(f)

:

<~r f X ~--~> + ~ ~22>

The field of validity of (46) remains to be specified in relation to ~4~as well as to r. As in the case of (47) it is linked to that of PADE'S approximations

for the accelerated sequence.

It can easily be verified that if analytical function f(y z) is chosen as uniform and convex everywhere,

the field of definition of (47) remains limi-

ted. The application of (46) or (47) to the two dimensional flows (or revolutions)

leads for the body's curve

(48) ~(Y) =

~(Y) -

x =

~ (y) to

2 ~2(3 ~2+ I)(I + ~,2)2y + ~ (I + ~2)y . ~,, _ 2y- ~"-0 '2 + ~ (I+ ~2).(|+ ~,2)~,

I ~

and : 2 (~2 M 2 + I -~2)2(M 2 + I),C4(I + ~,2) . y (49)

(y)

=

~(y)

-

I[M2 A8 - 2(M2 + I) A6]-y.~'2~"+ M2(M2 + l)A6(l+~'2)y~"l + ~ M 2 (M 2 + I) A 6 (I + ~ , 2 ) ~

with

M = ~<;

+ +'~ 1

coefficient ~

being equal to 0 or 1 wether the problem is 2-dimensional

axisymmetrical

in relation to Ox.

or

96

- Distance

of vertices

:

In order further

to specify

relation

been chosen so that for y = z = O one has The corresponding R 2 the principal notations

(46), suppose

that the body has

: 6rf = l-l, O, O[

point will be called vertex of the body. Designating radii of curvature

at this point,

R l and

one has with the utilized

:

(50)

l

Net I~r(~r~)l

H = Rl R2

The vertex of the shock surface will be the point of the wave situated

on axis Ox. The calculations

have been conducted

One can therefore work out not only PADE'S SHANKS'S

second order approximation

vertices

one obtains

Designating

(51)

hI =

[l,l] but also

(43~.

h I the first order approximation

of the distance

:

-

In the second order approximation

h 2 = h I.

In which one has

approximation

Iformulae

M 2o

[52)

at the vertex up to n = 4.

I +

3 ]-~ .

[(I + ~2) M2o - ~2](M~ - I)

considered

P4" $8 Q4" T8

here one has

:

J

:

P4 =

M4 - (4v2 - I) M 2 - 4 (l _~2) o o

Q4 =

MY + (2~2 + I) M 2 o o

S8 =

M 8 + (-6~ 4 + 8B 2 + I) M 6 + (18B 4 o o

(53) + 2 (I _~2)

- 12B 2 + 9) M 4 + o

+ ( - 1 8 ~4 + 16 ~2 + 2) M2 + 6 (I _~2) o

2

of

97

T 8 = M 8 + (18 p4 _

8 p2

+ 2) M G

o

- (38~ 4

- 36 ~2 + 7)

M4 +

O

o

(53)

2

+(22 p4 _

17p 2 + 10) M2

-

2 (1 - ~2)

O

and in which M

is the normal initial number of Mach at the vertex

:

O

D~ 2

(54)

M

=

~

+

2(I -!a 2)

o

I

It may seem paradoxical only dependent

on ~r(6rf)

]

+ I

4(l-Ij2) 2

that the second order approximation

and not on third order derivatives.

is

This follows

from the fact that one is placed at point y = z = 0

Again, expressions

placing oneself in the field of high velocities

had to the approximate

1 hl = ~

2p2(3B 2 + 1) 1 + B2

formulae

the preceding

:

+ ~I~_.,Zt

(54)

- Curvatures the vertex . . . . . . . . . . . . . . . . .of . . . . .shocks . . . . . . . . . . .at .

In order to calculate

:

the principal

radii of curvature p! and O 2

at the point of the shock wave situated on Ox, it is necessary the spatial second order derivatives

of F(y z t). Now,

known onl~ up to n = 2 w h e n one derives fore only possible

to apply PADE'S

1 +

Pl

- -

P2

(to 2 =

~

-

2n)

PlP2

R6

-

m3. $8 _ 2 ~ . T 8

(55) 1

.

m2 ,~

R6

~2 . Sg- ~. R6

also in relation

[I, I] transformation.

2 1 - -

to calculate

these derivatives to time.

are

It is there-

98

with

: R6 = 4 M2

[(3~ 2 +

I) M 4 -

o

S8 = _ ( ~ 4

(2~ 2 - 3) M 2 - F2]

o

_ 6~2

O

_ 3) M s +

(4~+

3~ 2 + 7) M 6 +

o

(56)

+(4B4

_ 3H2

(-6~ 4 - 7~ 2 +7)

M4 +

o

_ 1) M 2

O

_ B4 + ~2

O

T8

= _(~4

- 9~ 2 -4)

M8 +

(4~ 4

+ 4~ 2

+

|I) M 6 +

O

+(-6~ 4

-

I0~ 2

+

o

|0) M 4 +

(4~ 4 - 4~ 2 -l)

M2

o

which

lead

_ ~4 + ~2

o

to :

(57)

CONCLUSION

:

The presentation

of an application of the

determined physical problem demonstrates

E-algorithm

the difficulty

to a well-

of the choice between

such and such approximation.

The advantage of the acceleration processes that they permit to place oneself immediately and the initial value of numerical

In the preceding vertex)

formulae

results from the fact

closest to the accurate solution

calculations.

(distance of vertices,

curvature at the

are applied to the case of a spheric or paraboloidal

leading angle, equal to O, eomparaison with the numerical

body with a

results obtained

from computers

through longer methods demonstrates

approximations

seems to be correct and that even with reference

without being excellent

that the convergence of

the first order approximation

to curvature

gives reasonable

results.

The method used here is a direct one, the shock being obtained from the given body, we have limited ourselves

to first and second order

99

approximations,

the parameters of the problem remaining indeterminate.

It must be noted that PADE'S methods applied up to the twelfth order by computer lead to remarkable results for the inverse problem. They are comparable to the results obtained here in the considered cases, i.e for a perfect diatomic gas, an infinite upstream number of Mach and on axisymmetrical flow, the chosen shock surface being a paraboloid.

K.I. BABENKO, G.P. VOSKRESENSKII~ A.N. LYUBIMOV and V.V. RUSINOV -

Spatial Flow of an Ideal Cas around smooth Bodies Moscow 1964

G.A. BAKER

The theory and application of the PADE method Adv. Theo. Phys.

H. CABANNES

! 1965

Tables pour la d~termination des ondes de choc d~tach~es La recherche a~ronautique 36 - Paris

J.P. GUIRAUD

1953

Possibilit~s et limites actuelles de la th~orie des ~coulements hypersoniques - 0 N E R A

Publ. 99 - Paris

1961.

M.M. HAFEZ and H.K. CHENG On acceleration of convergence and shock-fitting in transonic flow computation - Univ. of Southern - California - 1973.

H. PADE

Sur la representation approch~e d'une fonction par des fractions rationnelles. Ann. Sci. Ecole. Norm. Sup. 9. Paris

L.W. SCHWARTZ

Hypersonic flows generated by parabolic and paraboloidal shock waves. N.A.S.A°

D. SHANKS

1892

1973

Non-linear transformations of divergent and slowly convergent sequences - J. Math. and Phys. 34 - 1955

100

M. VAN DYKE and H. GORDON Supersonic flow past a family of blunt axisymmetric bodies N.A.S.A.

A. VAN TUYL

Tech. Report 1959.

Use of rational approximations in the calculation of flows past bl~nt bodies - A.I.A.A.

P. WYNN

5 - 5

1967.

The rational approximation of functions which are formally defined by a power series expansion - Math. of Comp.

14 -1960

CYCLIC ITERATIVE METHOD APPLIED TO TRANSONIC FLOW ANALYSES~

H. K. Cheng, University of Southern California Los Angeles, California and M. M. Hafez, Flow Research, Los Angeles, California

Inc.

SUMMARY This paper reviews recent works on acceleration techniques for iterative solutions of elliptic and mixed-type problems, using algorithms related to Pad~'s fractions.

The study focuses on the question of how to speed up convergence of

relaxation methods currently available for transonic and related flow computations, with minimal alterations

in computer programming and storage requirements.

The theoretical basis of the work is similar to the power method, but allowance is made that moduli of some of the eigen-values can be very close to one another and to unity.

The study contributes to a clarification of the error

analyses for the sequence transformations of Aitken, Shanks,and Wilkinson, and to developing a cyclic iterative procedure applying the transformations to accelerating large linear and nonlinear systems.

Use of the first and second

order transforms similar to Shanks' (corresponding to the second and third rows in the upper half of Pad~'s Table)

is shown to be effective, but their subtle

differences from the latter prove to be crucial. Examples illustrating the accelerating technique include transonic flow as well as model Dirichlet problems.

Reduction by a factor of three to five in

computing time is possible, depending on the accuracy requirement and the order of the transformation.

The possibility for reducing the computer storage require-

ment via Wynn's recursive identities is examined for a linear system in Appendix A.

+This research was supported by the O f f i c e of Naval Research under Contract Number N000I 4-67-A-0269-0021.

102

I.

INTRODUCTION

Many current computation methods in fluid dynamics make procedures, wherein iterations.

use of relaxation

solutions are obtained after a sufficiently large number of

One recent advance in this respect is, perhaps, the calculation of

plane transonic flow by Murman & Cole l

and the subsequent extensions by many

workers.

These methods~using type-dependent schemes

(For example, see Refs. 2-4.)

in the discretization

and following line-relaxing procedures~succeed

shock waves in supercritical flows.

in capturing

The computer storage and the number of oper-

ations of the programs are low enough to make the computation possible even for a modest institution.

However, the computer time of 400 -lO00 iterations required

for the more complicated problems may still demand I/2 to 2 hours on an IBM 360 (or 370), and 10 - 40 minutes on a CDC 6600.

Use of acceleration technique with

a savings in computer time by a factor of 3 or 4 is certainly worthwhile, especially if one has a great number of problems to solve. The convergence rate of these relaxation procedures will depend on the largest eigen-values of the iterative matrix, ~X.~ ~ radius.

referred to subsequently as the spectml

The error (norm) at the k th iteration is, in most cases, gauged by I~,,Ik

The need of acceleration follows from a rather well-known fact that the spectral radius tends to unity, as the mesh size vanishes.

(See for example, Refs. 5-8.)

The central question in the following is, therefore~ how to speed-up convergence of relaxation methods currently available for the transonic flow and similar computations with minimal alteration in computer programming and storage requirements. Our acceleration technique is basically a cyclic iterative procedure; transformations related, but, not identical, to the nonlinear transforms of Shanks and Aitken, and to the Pad~ fraction are applied at the conclusion of each iterative cycle. Although the genesis of our study may be traced back to the Pad~ fraction, 8 its rational basis is derived from the power method of Fadeev & Fadeeva , but, special allowance is made in the error analysis that the magnitudes of some of the eigen-values can be very close to unity, and to one another, and may also repeat themselves. An algorithm using ~d~'sfraction has been employed recently by Martin & Lomax to accelerate their relaxation method for transonic flow. 9,10

Their basic

iterati~e procedure takes advantage of a fast, elliptic solver; but, it still makes use of the type-dependent schemes similar to other line-relaxation programs (this is, itself, quite novel). expansion

In their acceleration procedure, however, a three-term

in an artificial parameter

~

is used; solutions to a 3rd-order

103

perturbation problem are then used to generate the Pad~ fraction (at

E = l).

A

savin~ in computer time by a factor of nearly two has been reported for certain (but not all) cases studied. Much of the material used below is taken from our work on "Convergence Acceleration and Shock Fitting for Transonic Aerodynamics Computations", AIAA paper 75-51o

An

updated version of this work has been distributed as an

University of Southern California Report. II

This talk will concern primarily the

acceleration technique, and we shall take this opportunity to examine more closely the underlying ideas, and their subtle differences from those of Shanks 12 & Pad~ 13. What we hope to convey in the following is that only the most elementary of the transforms and their equalities have been used; they, nonetheless, have been very helpful. II.

REMARKS ON PADE FRACTION AND SHANKS SEQUENCE TRANSFORMATIONS

The use of transformation to improve the convergence characteristics of sequences is a recurrent theme of this proceeding. which b e a ~ a

One class of transforms,

close relationship to the key equations in our method,

Shanks and the corresponding Pad~ fractions.

12,13,&14

is that of

The simplest among these

is the "el" transform (singled-out in block on the left side of the diagram inserted below), which predicts the convergence limit ~ iterates

(:~k-,' ~k~ and

in some quarter

~k~l •

as the

~z

from three successive

This formula has a long history and is referred to

_ process of Aitken.

14

Considering

(~k as the

k-term partial sum of a series for an analytic function, Shanks identifies one of his transformed sequences ~,

of (~:)k (to which the "el" belongs) with that of

the rth row of the Pad~ table above the diagonal. ~ SHANK'S HIERARCHY

e,

PADE FUNCTION

e.

e~ % 2

Z

_

%o4).$..,-

_

to, "r. r2,to2 r2, r~-

t',.a -

rm~

-

-

_

"rnrl -

e, (~)

- <- 1

=

\

~ ( ~ ) = Do +~t + b~t~+

. . . .

rm. = P m i=1

i=s

41n Shanks' original paper the e n is written as ek, the transform.

with k denoting the order of

104

Much work has been done in uncovering the many important and interesting properties of, and identities among, the elements in the Pad~ Table, hence, the en-sequences.

(See for example, Refs. 15-17.)

But much has yet to be learned

about the error estimates in general applications. could give in this regard is thi~:

If the

~'5

A rather superficial

remark one

are the iterative solutions to

a nonlinear scalar equation

then the e I transform predicts the limit

(I) with an error comparable to the

square of the error in the original sequence.

The transform in this case is

simply a derivative-free variant of Newton's method.

But, this superlinear

accuracy does not hold for a system of equations involving more than one unknown; the accuracy of the transforms in this case must be established on a different basis. There is a second observation related to the accuracy of the e transform, n which is also quite well known after Shanks original workl~ namely, the transform e n of

~

represents the exact limit

~

, if the sequence

the transient behavior for successive k in the exponential

~

has precisely

involving k th powers

of qi~

It is apparent that convergence will require the magnitude of each

~[

than one; it also follows that the prediction would be exactly correct, sequence

~

happens to be the partial sum of n

to be less if the

geometric series.

The stipulated exponential transient cannot be one of general validity~because there is no a priori reason that the iterates of a general scalar equation should not approach its limit algebraically!

However, for iterative solutions to a system

of algebraic equations of interest, a similar exponential transient does apply to each component of the solution near the convergence limit.

This basis is provided

by the power method to be discussed below in Section 3. In passing, we may observe that, owing to the storage limitation, application of the e n or equivalent transform beyond e I and e 2 may not be easily accommodated in a computer program.

Therefore, our acceleration scheme has been limited to

these corresponding to elements far removed from the diagonal

in the Pad~ Table.

This is in contrast to most applications of P~de fraction in fluid mechanics today. 17-20

Ill. THE POWER METHOD In our application velocity potential

~

involving a large algebraic system, the unknown is the , and its k th iterate is

~k

They may be considered as

105

"vectors" with components as many as the number of total grid points used in the relaxation method, say N. the "vector"

The new iterate at the (k+])

iteration,

is a function of

~

determined by the difference schemes and the iterative procedure used. ~/~

as a perturbed solution

We regard

from the convergence limit

%=O+E~,

E~--,-o.

Near the limit, the nonlinear iterative equation yields a linear, recursive relation for the error vector

~kel"- Q ~k where Q is the Jacobian matrix of the function g of

~

, independent of k and ~k.

If the eigen-values of this matrix~[sdredistinct, we may represent the initial error vector eo by a linear combination of the eigen-vectors N

Eo = 2 ~ ( ~ This leads to a form for the error vecto~

~ = ~(

~,~

It shows that the error vector decays (~?~plifies) convergence,

exponentially in k.

For

the magnitudes of the eigen-values must be less than one,

just like the requirement on the ~

in Shank's

exponential transient.

This

is the main base for the power method of Fadeev & Fadeeva~as well as the two of our transformations to be discussed below. The linear recursive equation for the error vector has an exact analog in the discretized version of the time-dependent system

C ~ = A ~0 , From t h i s ,

with

Q = exp. ( ~ t C - ' A ) .

we may see the prospect f o r a c c e l e r a t i n g a pseudo-unsteady f l u i d

mech-

anics problem. Let us adopt Fadeev & Fadeeva's r e s u l t and o r d e r the e i g e n - v a l u e s a c c o r d i n g t o their

a b s o l u t e magnitudes

14,1> l~J >I~,I > .... After

long enough i t e r a t i o n s ,

a s s o c i a t e d w i t h the f i r s t

> I~,I > I~,,,I> .....

> 14,1.

i . e . ~ l a r g e enough k, one may omit a l l

eigen f u n c t i o n I/',.

our f i ~ t - o r d e r t r a n s f o r m , which p r e d i c t s the l i m i t

This~ a f t e r ~

but one term

ellminating~,3~,~leads to

from two successive i t e r a t e s

108

with an unknown eigen-value

i~.i

If one chooses to eliminate

i~.i by three

successive iteraties, he will recover the el-transform corresponding to the first row of the Pade Table

c~= ~ + I

However, more useful estimates of

~i

-

"~" I

are obtained by averaging over all compon-

ents through proper sum and inner products illustrated as, with

N

I/ i

-

~K=E K

-

Js

A justification for ~'i ' is given in Appendix B. To generate formula corresponding to the e 2 and higher-order transforms, we simply include more and more higher-order eigen functions into the "transient" representation, and obtain

where Pn = 1 and pj's are constants for the entire field.

One of our modest con-

tributions, delineated in Ref. II, is to analyse the remainder and confirm the classical IV.

results for repeated and closely spaced eigen values.

APPLICATION TO A CYCLIC ITERATIVE PROCEDURE In the application,

algorithm:

the transforms are used as a part of an iterative

the procedure consists of several cycles, each makes k' iterations

(say lO - 30).

The transform is applied at the end of each cycle to yield an

estimate of the limit, to be used as initial data for the next cycle.

The

sketch in Fig. 1 illustrates the method when the first-order transform is used, which needs data from three stages of iterations.

Note that these three values

can be taken from values at k-m> k, and k+m, for some integral m.

Additional

storage for whole sets of field data is required, and it varies from I to 4 sets, depending on the order of the transform and ways the eigen-value estimates are handled. In passing, we note that if the component,

~process

is strictly applied for each

i.e. at each grid point, not only more storage is required but the

redundant eigen-value estimates

implicit in such process

sistency and delay the approach to the limit.

We find the

would lead to incon~process

coverges

much more slowly in most cases. V.

EXAMPLE:

A DIRECHLET PROBLEM

In a study described in our report 11, we have tested this cyclic-transform technique on line-relaxation methods applied to a model Dirichlet problem, using

107

various relaxation parameters and sweep directions.

These numerical experiments

show that a reduction in iteration number by a factor of three to five is generally possible.

In the example of Fig. 2, a line Gauss-Seidel procedure is applied to

the system based on a 9-point central difference scheme, for which neither the optimum

relaxation parameter, nor the spectral

to the best of our knowledge.

radius,

A typical convergence history of the unaccelerated

results, using a 1/30th mesh, is shown as a solid curve. is the iteration number k.

is theoretically known

The abscissa of the graph

The accelerated result based on a 2nd-order transform

is shown in short dash with circles, which approaches the limit within 1% in 30 iterations~as compared to VI.

EXAMPLES:

three to four hundred for the unaccelerated one.

TRANSONIC THIN AIRFOIL PROBLEMS

We shall study below the results of application in transonic small-disturbance theory governed by the von K~rm~n equation.21 The discussion

is confined to the

flow over a symmetric circular arc airfoil, which has an embedded supersonic region.

The basic program to be accelerated is one similar to that of Murman and

Cole I, using an x-mesh of 2½% chord, and a y-mesh near the wing 2% chord.

For the

result shown in Figure 3a, the relaxation parameter is taken to be 1.4 in the subsonic region and

0.9 in the supersonic region.

This slide gives the conver-

gence history for the velocity perturbation near the mid chord.

The unaccelerated

result shown in solid curve takes 140 iterations to approach the limit within I%; cyclic acceleration using the first-order transform presented in thin solid curve takes 60 iterations for the same accuracy.

For the results using 2nd-order trans-

form, only data at the end of each cycle are shown in circles; this takes only 40 iterations to reach the limit within I%. The results in Figure 4b differ from the preceeding one in that, here, a uniform relaxation parameter~uJ= regions.

0.95, is used in the supersonic and subsonic

The convergence rate for the unaccelerated program in solid curve is

low, as expected, taking 400 iterations or more to reach the limit within I%. This is to be compared with the 65 and 30 iterations for the two accelerated solutions. We have also studied the acceleration of transonic solutions involving circulation,

i.e. airfoil at incidence.

The basic line-relaxation program is

the same as before, except for a doubling in the number of grid points to account for the asymmetry and the use of a somewhat different pair of relaxation parameters. One sees from Fig. 5 that the use of the first-order transform in solid curve achieves a convergence within I% at 150 iterations for the circulation, whereas the unaccelerated one may take more than 400. We would like to emphasize that the above examples

involve shock waves which

are "captured", so to speak, by the numerical procedure - thanks to the "numerical

108

viscosity" inherent in the computer program. near the shock is lost.

Because of this, the flow detail

A shock-fitting method, which modifies the computer

program to fit the shock as a surface of discontinuity, has been developed. 11 The natural question to be asked is whether acceleration and shock-fitting techniques can work together.

The answer is an affirmative one.

In Figure 5,

we present the shock and sonic boundaries from our iterative, shock-fitting solution, computed for a slightly supersonic Mach number.

The unaccelerated

result obtained after 240 iterations compares well with that obtained by Magnus & Yoshihara, who used a shock capturing method based on an unsteady approach.

With

acceleration based on a 2nd-order transform, the very same shock-fitting solution is recovered in 64 iterations. VII.

CONCLUDING REMARKS In summary, our study with the transonic flow and other examples show that

the cyclic acceleration techniques based on sequence transforms may effectively increase the convergence rate and the efficiency of the relaxation methods, with minimal programming and storage changes.

A reduction by a factor of three to

five in computer time is possible, with and without shock-fitting.

In fact,

where accurate description for the shock is important, the time saved by acceleration with shock fitting can be 6 to 36 fold.

One observes that the above

demonstration involves only the use of some of the most rudimentary forms of sequence transforms.

With an increase in data storage capacity (or facility),

it should be possible to employ the more sophisticated higher order transforms and their recurrence relations which are discussed in other parts of this Proceeding.

In the meantime, possibilities for reducing the data storage require-

ment for the higher-order transforms do exist.

This is supported by a study

described in Appendix A below for an iterative procedure applied to a linear system, making use of Wynn's recursive relations for the APPENDIX A.

~-algorithm.

IMPLEMENTATION OF WYNN'S ~-ALGORITHM FOR APPLICATIONS TO ITERATIVE MATRIX EQUATIONS

In Ref. 22, Wynn uses the ~-algorithm as an acceleration technique for iterative vector and matrix problems. +

The effective use of the transforms in

the cyclic iterative method discussed in the text, as well as the corresponding elements in the

~-algorithm,

are limited, in practice, by the increased storage

requirement for the higher-order transforms •

However, the possibility for using

higher-order transforms without the increas ing storage remains, and is confirmed below for a linear system.

This is accompl ished through application of Wynn's

+The symbol " ~ " employed in this Appendix, is not to be confused with the error vector " E~" used in the text.

109 rhombus rule, and other identities for a linear iteratlve equation system. The 22 result provides an alteration from Wynn's origlnal procedure with a substantial savings in data storage. Wynn's Recursive Relation Applied to Vectors and Matrices The power of the

~-algorithm lies in the fact, established through many

examples, that if the sequence

~o,

~ , ~z .....

~x ....... i.e.,

is slowly convergent, then the numerical convergence of the sequence 4

, "'" ,

to the l i m i t rapid.

~25

,

...

,

(or a n t i l i m i t ) ,

with which sequence { O x t

In the s c a l a r c a s e , the q u a n t i t i e s

$,l

~_f

is a s s o c i a t e d ,

~(k? s a t i s f y

E~÷,~ =

-z

,

_(o)

i.e.,

E(k}

E~ ,

is f a r more

the rhombus r u l e 15

I @

(A.l) which is closely related to the Shanks' transform.

As is well known, the elements

generated in this manner in the E-algorithm may be identified with those on the upper half of the Pade Table (cf. Sec.

II in text), hence, those in ShankS' e n

transform , 2n

,n (A.2)

J with

~(k)

= ~

~

E~ )

and setting

In cases in which

=

0

~:)x is a vector or a matrix, the algorithm is still mean-

ingful, provided the inverse of the entity is consistently defined.

Wynn has

considered the following alternative definitions. (i)

Primitive Inverse: independently;

In this case, each component is considered

it amounts to a simultaneous application of the scalar

~-algorithm to components of the array. (ii)

The Samelson Inverse of a Vector:

In this case, the inverse of the

~ : (X, XI,....,XN) is taken (after K. Samelson) to be

vector

N

X-'~(~ where (iii)

~

Xj X J ' ~ ' c a ' ~ ' ' ' ' ' ' ~ " ) '

is the complex conjugate of

~

(A.3) .

The Normally Defined Inverse of a Square Matrix:

This was not

recommended for large systems. Wynn discusses in Ref. 22 applications of the algorithm to numerical analyses, including boundary-value problems,

initial-value problems, Fredholm and Voltera

integral equations, and differential equations.

110

~/ynn's_Procedure for Acceleratincj Relaxation Solutions Of particular

interest are Wynn's application to the acceleration of the

Jacobi and Gauss-Seidel

relaxation methods for iterative solution of large systems 22

of linear algebraic equations.

For subsequent discussion,

it is convenient

cIk) to arrange the array of ~s into

the familiar pattern suggested by the rhombus rule, illustrated at the middle of the page, where the original column near the left.

sequence

{~kl

is given on the first non-zero

With the identification

given by Eq. (A.2) elements on each

column corresponds to those belonging to Shanks'

e - transform of the same order n The diagonal elements in the Pad~

(with the order increasing towards the right).

Table rn, n are identified with elements on the "roof top" with even subscript, i.e.,

_(o) Eis .

with

_--'-~o~c

0

o

-r~

~J

o

~2

E~

~ ~

,

~

~s

E?

In Wynn's applications,

a sequence of vectors or matrices

is obtained from

an iterative procedure for a linear system, say,

and stored as

E(~) before the acceleration

if we have three iterates be determined as

E~)

Do

J

~

procedure

, and

~2

For example,

according to the rhombus rule (which in this case is ident-

ifiable with Pad~'s rll or Shanks ~ e I { ~ l } i.e.,

is applied.

, a better estimate will then

if one wishes to obtain

E~) ~

).

with n ~

If more resolution

is needed,

4, more iterates (with k ~ 4)

have to be generated from Eq. (A.4) and stored. Underlying this procedure

is the assumption that at the end point (towards

the right) of the application of the rhombus rule, one shall arrive at (or near) the limit ~

satisfying the equation

(~) ~ (~ (~ + ~

This assumption can indeed be justified. components of ~ ,

(A.4b)

In fact,

inasmuch as the number of

say N, is finite, the exact solution ~

Do and 2N (and only 2N) successive

iterates,

i.e.,

can be predicted from

(~o ' ~ ' ~2~-'.',~,-..,

~,

111 ~A/+I ) . . . . > ~)2N-I, ~)aN, using rhombus rule.

This follows from Eq. (A.4a),

f o r whicha corollary of the Cayley-Hamilton theorem (cf. of Ref. l l )

Eq. (3.9) on p. 5,

gives ~/' N

where p j ' s

j:o /=0 (A. 5a~ in the c h a r a c t e r i s t i c polynormal of the i t e r a t i v e

are the c o e f f i c i e n t s

matrix Q N ] I ('"X'-~j) )=1

--

~ + ~,/~.'e Pz .A-='F''"

+~,V/~.N .

(A.Sb)

Now, the right hand member of Eq. (A.5a) is precisely eN(~),__ identifiable with E ZN <~-N)'

Hence, the N-component vector (~

~Nel consecutive iterates

is p r e c i s e l y recovered from any

of Eq. (A.4a) upon reaching the end of a p p l i c a t i o n

of the rhombus rule (at the 2Nth column of the

E(/~)

array, f o r any I<). Note that

the v a l i d i t y of Eqs. (A.5a) and (A.5b), hence, the conclusion,requires the eigenvalues,v~.j15 ~ to be n e i t h e r d i s t i n c t nor completely real.

is

also p o s s i b l e to recover the e n t i r e

---~(~")JE(5A)

On the same basis,

set o f the e i g e n - v a l u e s

it

from the r a t i o

in the limit k - ~ o o

The above shows that the a p p l i c a t i o n of the ~ - a l g o r i t h m amounts to providing a 2N f i n i t e steps process f o r solving a set of require, however, the storage of ?-N 2

N

l i n e a r equations.

This would

pieces of data, which may not be desirable

f o r a large system.

Generating

~2s "(/() By I t e r a t i o n s

Our procedure f o r applying the (higher-order) IE-algorithm without the penalty of an increased computer storage r e l i e s on a theorem of the E - a r r a y . Namely, i f the successive (vector)

I£ -elements on the f i r s t

(non-zero) column

are generated by a l i n e a r matrix i t e r a t i v e law (equation), say,

~K*, ---- Q ~)~ ÷ b ,

(A.6)

successive vector elements on any other even column obey, and can be generated from, the same i t e r a t i v e law, i . e . ,

~s

= Q ~,s

+ b

(A.7)

This special aspect concerning l i n e a r i t e r a t i v e systemswas not considered in Wynn's work. 22 The f o l l o w i n g proof through induction makes use of a recursive r e l a t i o n o f Wynn 15

T

corresponding to the "missing i d e n t i t y of Frobenius".

In Eq. (3.8b~ of Ref. 11, ~'i = I" should be w r i t t e n as " i = n+1".

112

Let E, W, N, and S denote the four elements around an element C in the E-array,

in the order of East, West, North,and South.

Wynn's recursive identity

for elements of the even column is]5 , I_.~ S-C

N-C

=

~ +. s._.2__ E-C ~/-C ,

(A.81

or) ! E

=

÷

L~ + . _ L _ ._/_t_ N-C $-C W-C

C

(A.9)

To render (A.8) or (A.9) applicable at the second even column (from the left), one may introduce an additional with the auxillary

column

column with

E(.k~_=

O.

E(~)--~oo, which is consistent

Let the superscript * refer to a

i

successive

iterate, e.g.

~ ) ; = ~k*l "

It will be established first that if the

iterative law Eq. (A.7) holds in two neighboring even columns, in the next even column.

it will also hold

In particular, we want to show that, if Eq. (A.7) is

applicable to W, N, S, and C, it will also hold at E

which is related to the

others through rhombus rule, or better through Eq. (8) or (9). both sides of Eq. (A.9) by Q, and add

~ ;

Now, multiply

we have

Q E "I" ~2 = Q (NI c +

l I)~-Qc+D" S-c

(A.I01

w-C

But the first term on the right is

Q (CN-c)-'~C5-c)-'- (w-c)-')-' (All~

=

If we assume that Eq. (A.7) hold at N, S. W, and C, then the R,H.S. of Eq. (A.IO) becomes

+ -

-

N'~-C* which i s ,

C*

÷

5"-C*

according to the recursive

V¢'-C* relation

Eq. ( A . 9 ) ,

E*.

Hence, Eq. (A.IO)

yields

b,

E* = q E *

(A. 12'~

confirming that Eq. (A.7) holds along the next even column (to the right of the two even columns containing N, S, C, and W).

It remains to show that the same

iterative law applies along the first two even columns fk) E_ --p-oO

,

~(k) -o "- ~

involving elements •

113 This

is t r u e under Eq. ( A . 6 ) .

E~s

Hence,

:

Q

~s

+ D ,

(A. 13)

p r o v i d e d i t holds f o r 5 = O. Departure From Wynn's O r i g i n a l

Procedur..e:__ S i g n i f i c a n c e

The s i g n i f i c a n c e o f Eq. (A.13) w i t h t h e accompanying p r o v i s i o n l i e s f a c t t h a t a p a r t o f the elements on even columns in the generated a l t e r n a t i v e l y

by i t e r a t i o n .

In o t h e r words, use o f i t e r a t i o n s

successive even columns may be exchanged w i t h the s t o r a g e r e q u i r e d f o r which is l a r g e f o r the h i g h e r o r d e r t r a n s f o r m s . m a t r i x o p e r a t i o n s in Eq. ( A . l l ) , the (N-C)-'~

etc.,

in the

E - a r r a y may now be along @o = ~)K,

We note in passing t h a t the

hence, t h e t h e o r e m ~ h o l d s a l s o in cases where

are d e f i n e d by t h e " p r i m i t i v e

i n v e r s e " and o t h e r a l t e r n a t i v e s

mentioned e a r l i e r . With the theorem Eq. (A.13), any element in the £ - a r r a y from three initial iteratles

* = @ o , ~'o = ~, , and Eo =@z,

can be recovered

applying the same

iterative law to generate elements in the intermediate even columns.

This proce-

dure is illustrated in the diagram above Eq. (A.4a) for a case in which the end ¢ (o)

point is ~6

; the downward arrows indicate generation of elements by iterations,

the net-work otherwise signifies application of the rhombus rule, Eq. (A.I), in the usual manner.

The most crucial feature of such a procedure is, perhaps, the

fact that, in the process of generating intermediate elements in the even and odd columns, the data storage never exceeds the requirement for storing three pieces _(k) of l:S , for any s and any k. We note that, a similar procedure based on Eq. (A.8) alone may also be used to recover elements in the even column; this however requires storage for four E ~

instead of three as in the one described

above. This modified

8-algorithm may be used in the cyclic iterative procedure

for relaxation methods described in the text, in which the algorithm will be applied to predict the limit from successive iterates, handicapped by the excessive storage requirement. is to use the procedure continuously

but will no longer be

One potential application

(in one long cycle), corresponding to an

unbroken ZIG-ZAG path along the "roof top" of the

~ -array.

This reprents a

new relaxation procedure,of which the problems of stability and rounding errors deserve attention in future study.

114

APPENDIX 8.

Consider a n o n l i n e a r

ALT~I~ATIVE CRITERION FOR DETERMINING THE RELAXATION PARAMETERS

relaxation

equation

=

(B.1)

Let us assume

and let ;7(~) be a functional associated with Eq. (B.I).

Then t h e

stationary condition ~c$)

=

o

should p r o v i d e a system o f e q u a t i o n s f o r t h e unknown p a r a m e t e r s ~ l , ~ J 2 J e t c . , may r e p r e s e n t ,

f o r example, a minimum e r r o r p r i n c i p l e .

iterative

m a t r i x has a dominant e i g e n - v a l u e ,

criterion

is i d e n t i c a l

For a l i n e a r

t h e ~1 o b t a i n e d from l e a s t squares

w i t h t h e i n n e r p r o d u c t form o f t h e /~'l ' i . e . ,

~f

in

o u r work. 11

ACKNOWLEDGEMENT This paper is based on a study at the University of Southern California supported by the Office of Naval Research, contract number N00014-75-C-0520.

Eq.(B.I)

system, whose

Fluid Dynamics Program, under

115

REFERENCES

1)

Murman, E. M. and Cole, J. D., i'Calcu]ation of Plane Steady Transonic Flo~ i

AIAA Jour., Vol. 9, no. l, 1971, pp. 114-121. 2)

Krupp, J. A. and Murman, E. M., "Computation of Transonic Flows Past

Lifting Airfoils and Slender Bodies," AIAA Jour., Vol. 10, No. 7, 1972, pp. 880886.

3)

Garabedian, P. R. and Korn, D. G., "Numerical Design of Transonic

Airfoils '~, in Numerical Solution of Partial Differential Equations - II, Academic Press, 1971. 4)

Jameson, A., "Numerical Calculation of the Three Dimensional Transonic

Flow over a Yawed Wing", Proceedings AIA A Computational Fluid Dynamics Conference, pp. 18-26. 5)

Young, D., Iterative Solutions for Large System of Linear Equations,

Academic Press, New York, 1971 6)

Varga, R. S., Iterative Matrix Analysis, Prentice-Hall, Englewood Cliffs,

New Jersey, 1962. 7)

Wilkinson, J. H., The Algebraic Eigen-value Problem, Clarendon Press,

Oxford, 1965. 8)

Fadeev, D. K. and Fadeeva, V, N., Computational Methods of Linear Algebra

(translated by R. C. Williams), W. H. Freeman & Co., San Francisco, 1963. 9)

Martin, E. D. and Lomax, H., "Rapid Finite Difference Computation of

Subsonic and Transonic Aerodynamic Flows", AIAA paper No. 74-II, 1974 lO)

Martin, E. D., "Progress in Application of Direct Elliptic Solver to

Transonic Flow Computations", to appear in Aerodynamic Analyses Requiring Advanced Computers~ NASA SP-347, 1975. ll)

Hafez, M. M. and Chen9, H. K., "Convergence Acceleration and Shock

Fitting for Transonic Aerodynamics Computations", Univ. So. Calif., School of Eng'r., Rept. USCAE 132, April 1975. ]2)

Shanks, D., "Nonlinear Transformations of Divergent and Slowly Convergent

Sequences", Studies of Applied Math., (J. Math. Phys.), No, 34, pp. 1-42, 1955. 13)

Pad~, H., "Sur ]a representation approchee d'une fonction par des

rationelles", Ann, Ecole Nor (3), Supplement, ]892, pp. ]-93. 14)

Aitken, A. C., "Studies in Practical Mathematics If, Proc. Royal Soc.

Edinburgh, Vol. 57, ]937, pp. 269-304.

116

15)

Wynn, P., "Upon System of Recursions Which Obtain Among the Quotients of

the Pad~ Table", Numer. Math., Vol. 8, 1966, pp. 246-269. 16)

Gragg, W. B., "The Pad~ Table and Its Relation to Certain Algorithms of

Numerical Analysis", SIAM Review, Vol. 14, No. l, 1972. 17)

Baker, G. A., Gammel, J. L., and Willis, J. G., '~An Investigation of the

Applicability of the Pad~ Approximation Method", Jour. Math. Anal., Vol. 2, 1961, pp. 405-418. 18)

Van Dyke, M. D., "Analysis and Improvement of Perturbation Series",

Quart. Jour. Mech. Appl. Math., Nov. 1974. 19)

Van Tuyl, A. H., "Calculation of Nozzle Using Pad~ Fractions", AIAA Jour.,

Vol. II, No. 4, 1973, pp. 537-541. 20)

Cabannes, H. and Bausset, M., "Application of the Method of Pad~ to the

Determination of Shock Waves", in Problems of Hydrodynamics and Continuum Mechanics, in Honor of L. I. Sedov, English ed. published by SIAM, 1968, pp. 95-I14. 21)

von K~rm~n, T., "The Similarity Law of Transonic Flow", Jour.

Math. and

Physics~ Vol. 26, 1947, p. 3. 22)

Wynn, P., "Acceleration Techniques for Iterated Vectors and Matrix

Problems", Math. Comput., Vol. 16, 1962, pp. 301-322.

LIMIT .... °

-

-

.....

dp -'" -"~'" ~-2m

J

~. ~" ~

~bk- rn 0 (1)k (STORED)

.°

~

®"

°

,i

i

,

|,

----

i

i

I

I,,

I

@k+rn , |

,,,

k ~

Figure I.

,

,

illustrated

f o r the f i r s t - o r d e r

,

,

ITERATIONS

C y c l i c a c c e l e r a t i o n technique a p p l i e d t o an i t e r a t i v e every k' i t e r a t i o n ) .

I

solution,

transform (the c y c l e is repeated

117

ALGS-2 LIMIT

.12

4

.lO

t

, -',2oj !,, .08

, ,/3o ~.06 LINE

.04

i ~, / / / / ~ '

G-S

J J/~,

I/

/ I

/

;Z,t;JJ-

/

.02

I

z o

/ 20

40

60

80

lO0

• ITERATIONS

Figure 2.

Comparison

of unaccelerated

model Dirichlet

problem.

and accelerated

line SOR solutions

to a

118

--'r---

, | .2262

O

(J)X 1.22:

C IRCULAR-ARC AIRFOIL AT Kc = 1 . 8 ,

x :

-0.025,

y ~ 0

I

'i, 20

LINE S0R ,(~)= 1.4 & 0.9 ALSOR (IST-ORDERTRANS.) ®

0

~

ALSOR (2NO-ORDERTRANS.)

I'

K

i/

~ALSOR-I

LSOR

140

350

lql i

~ ALSOR-2

60

40

I0-2

120

100

10-3

! I

J I .16

J

....

J ,

, 160

L

80

l

•

i

,

240 k ~ ITERATIONS

Ca) Figure 3.

Demonstration solution at zero

of cyclic

to a supercritical

transforms:

technique

transonic

incidence with a similarity

and second order un iform.

acceleration

on line relaxation

flow over a circular-arc

parameter

airfoil

K c = 1.8, using first

(a) cO = 1.4 and 0.9, and

co = 0.95,

119

1.2262

g~x

0

0

I . 22

I

CIRCULAR-ARCAIRFOIL AT

Kc

x

1.8 ,

= -0.025 ,

y

1.ZO L. LINE S0R

,

C,I,~= 0.95 ,

UNIFORM

ALSOR-I

Q

1.18

Q

O

ALSOR-2

k LSOR

K ALSOR-I

65

30

10"2

120

60

io-3

230

140

10-4

400 ?

800

?

?

1.16

l

i

m!

I

120

le~l

K ALSOR-2

_ _ L ~ _ _ L

180

(b)

Figure 3 (continued)

I

240

.

_d

....

L__

I k

......

I

~ II~R~T~GNS

I,

120

,98

f

f

J

.90

J J J J ALSOR-|

J

LINE

SOR

J J

,80

,70

i. . . . .

i

i

I

4o

Figure 4.

Test of cyclic acceleration cu

i

~o

arc airfoil

i

i

~zo

_

i

~

. . . . .

k

t~o

technique On line SOR solution with

= 1.8 and 0.8 to a supercritical at incidence with K

order transform:

_J

Convergence

transonic flow over a circular-

= 1.8, ~ / ~ = 0.1454, using firstc history of the circulation.

i

121

3.o

2.38Y MAGNUS & YOSHIHARA

CI RCULAR-ARCAIRFOIL Kc= -- 1.829 2.0

SHOCK-FITTING AX = ,05

k

= 240

(k = GO w i t h a c c e l e r a t i o n )

~ Y = .1

= 8O

k

= 40

1.O

/.'S /

~.,,"/ ,Y -1 oo

0

X

(L.E.)

Figure

5.

Application solution

of with

cyclic

1.0

(T.E.)

acceleration

shock fitting.

technique

to

a line

relaxation

A Technique for Accelerating Iterative Convergence in Numerical Integration, with Application in Transonic Aerodynamics

E. DALE MARTIN NASA Ames Research Center Moffett Field, California, USA

Summary A technique is described for the efficient numerical solution of nonlinear partial differential equations by rapid iteration. In particular, a special approach is described for applying the Aitken acceleration formula (a simple Pad~ approximant) for accelerating the iterative convergence. The method finds the most appropriate successive approximations, which are in a most nearly geometric sequence, for use in the Aitken formula. Simple examples are given to illustrate the use of the method. The method is then applied to the mixed elliptic-hyperbolic problem of steady, inviscid, transonic flow over an airfoil in a subsonic free stream.

1. Introduction The numerical solutions of nonlinear partial differential equations such as those governing fluid flows frequently are obtained most efficiently by iterative methods. The rate of iterative convergence of the method chosen is an important consideration, and various means of accelerating the iterative convergence have been useful. One popular device for accelerating convergence of a sequence of numbers such as provided by iteration is Aitken's extrapolation formula (or A 2 process) [ 1], whose use is described in most books on numerical methods [2] and which is identified [ 3 - 5 ] as a simple Pad~ approximant if the successive iterates are partial sums of a power series. Shanks [ 3 ] provided generalizations of Aitken's transformation and studied their use. In [6] Wynn gave a simple algorithm for rapid computation of one of the nonlinear transforms studied by Shanks, and later Wynn [7] discussed application of this acceleration technique to vector and matrix problems, including application to boundary-value and initial-value problems. The present paper describes a special technique for applying the Aitken extrapolation formula for accelerating iterative convergence in the numerical solution of partial differential equations. The method was first introduced and used in [8] and then used in a modified form in [9] with additional results given in [9,10]. Although the application to be discussed is in a numerical finite-difference solution, the general method applies equally well, for example, to analytical solutions or to numerical solutions by finite-element methods. The use of the simple Aitken formula with three successive iterates is emphasized

124

(even though the elegant e-algorithm of Wynn with longer sequences could be used), because the eventual applications are expected to be those numerical problems requiring significant computer storage. The Aitken formula, using only three iterates, requires less storage than other forms of the e-algorithm. Often the use of the Aitken formula with iterates obtained arbitrarily by successive approximations does not lead to a significantly improved approximation. However, because Shanks [3] showed that the formula works best if the sequence is "nearly geometric," the present approach seeks to obtain successive iterates that are in a nearly geometric sequence. (Because of the work of Shanks in popularizing the Aitken formula and his valuable demonstration of the special applicability to "nearly geometric sequences," our past work has referred to the simple extrapolation formula as the "Aitken/Shanks formula.") The sequence of approximations can be most nearly geometric if obtained from a power-series construction. Therefore, the basis of the present approach is the construction of successive approximations derived from formal power-series expansions to obtain as closely as possible a nearly geometric sequence. The technique is based on the concepts of perturbation-series expansions (in the sense of Poincar~; see Bellman [ 11 ] ). An artificial parameter is introduced in such a way as to obtain three problems to solve for terms of a nearly geometric series, for use in the Aitken/Shanks formula. Expansion in powers of an artificial parameter has also been considered by Genz [5] to develop a mathematical proof(unknown by the authors of [8] at that writing), but the central idea in the present approach is that the artificialparameter expansions are used, in combination with an "artificially extended form" of the equations to be solved, as a device to determine most appropriate successive approximations. This technique produces the nearly geometric sequence of solutions, even in nonlinear problems. The previous application of the Aitken/Shanks transformations to acceleration of iterations in numerical integration by Wynn [7] used simple straightforward iterations. The results of such a procedure with use of only the simplest acceleration formula are described below for an example problem and are compared with the present method. The present approach based on perturbation series requires that complete perturbation solutions be available on the entire computation field (or entire domain of the equations) at each iteration. This concept therefore adapts well to a finite-difference method using "direct elliptic solvers" [ 12-15 ] in the iterative procedure to determine the solution simultaneously at all points on the entire computation field (rather than in successive traverses over the field as in a point- or line-relaxation method). Such methods have been referred to as "semidirect" [8 10]. After several simple examples to illustrate the method, it is applied to the problem of inviscid flow over an airfoil in a subsonic free stream, including conditions for which the flow equations are of mixed type (elliptic in an outer region, with an embedded hyperbolic region and a shock wave). This transonicaerodynamic-flow problem has also been treated by Hafez and Cheng [16] using the Aitken/Shanks acceleration formula, but in a quite different way, in combination with a line-relaxation method.

125

2. General F o r m u l a t i o n of Method Consider the general partial-differential or difference equation system and the accompanying boundary conditions represented by LU-F(x)

= NU

BU = G(x)

in R,

(2.1)

on B,

(2.2)

where U = U(x) is a vector function of the position vector x, L is a separable, linear, elliptic differential or difference operator, F(x) is a given vector function and N is a possibly nonlinear operator such that the operation NU is a vector of the same dimension as U and has c o m p o n e n t s that m a y involve U, x, and derivatives o f the c o m p o n e n t s o f U with respect to the c o m p o n e n t s o f x. Assume for simplicity that B is a linear operator. The b o u n d a r y condition (2.2) is applied on B, which includes all appropriate b o u n d a r y segments o f the domain R. For illustration o f this notation and o f the m e t h o d , simple one-dimensional examples are given in the n e x t section. Examples treated in the earlier version o f [8] included (i) the scalar Laplacian as L with a scalar, if, as U, and (ii) a Cauchy-Riemann operator matrix as L with two c o m p o n e n t s of U, denoted as u and v. The right side of (2.1) can be complicated and can make the equation system hyperbolic or parabolic in some regions [ 8 - 1 0 ] . In the formulation o f a problem to be solved, L and F(x) are chosen judiciously and m a y be the result o f "scaling and shifting" transformations [ 17,9] for increasing the rate o f iterative convergence or of addition of terms [9,10] for stabilizing iterations. For treatment with additional terms, an extended Cauchy-Riemann solver for use in present calculations has been described in [ 18 ]. In the m e t h o d s to be discussed for the iterative solution of eqs. (2.1) and (2.2), suppose U l ( x ) , U2(x), and U3(x) are successive approximations to U(x) in R. Let u(x) and Un(X) be respectively each a single scalar c o m p o n e n t o f the vectors U(x) and Un(x) (n = 1,2,3). T h e n one form o f the Aitken/Shanks extrapolation formula [1,3] for an improved approximation u*(x) to u(x) is

u*(x) -

UlU 3 - u2 2

(2.3)

u 1 - 2u 2 + u 3

Application o f the formula in this way to individual c o m p o n e n t s of U at each x separately is referred to by W y n n [7] as use o f a "primitive inverse" o f the e-algorithm. W y n n concludes that use o f the primitive inverse is competitive with use of other more complicated inverses. The work o f Hafez and Cheng [ 16] considers coupling o f the matrix elements in the numerical solution, which is related to the more complex inverses o f the e-algorithm. 2.1 Artificially extended equation. For obtaining power-series solutions to (2.1) and (2.2) that are most appropriate for use in the Aitken/Shanks extrapolation formula, it has been found convenient to artificially extend eq. (2.1) by inserting b o t h an artificial parameter e and an "initial a p p r o x i m a t i o n , " Uo(x), to U(x) as follows. Let LU-F(x)

= (1-e)NU o+eNU

in R

(2.4)

126

along with condition (2.2). Note that the solution U to (2.4) with (2.2) depends on e (as well as on the specified function Uo(x) ).: U = U(x,e). However, at e = 1, the solution to (2.4) with (2.2) is the same as the solution to the original equations (2,1) with (2.2). Furthermore, if Uo(x) is close to the solution U(x), then (2.4) is nearly the same as (2.1) and the solutions then are nearly the same. Thus, either of the conditions e = 1 or U o = U makes (2.4) the same as (2.1). Both o f these facts can be used to advantage in the m e t h o d s to be discussed. 2.2 Method 1. The simplest iteration scheme is a straightforward m e t h o d o f successive approximations. Although this m e t h o d can be combined with use of a relaxation parameter (see [8,9] ), for simplicity here we omit that useful device. If we let e = 0 in (2.4) and define U o ( x ) as a previous iteration, we obtain the following equations for the iterative solution denoted as Method l(a): LU n - F

= NUn_ 1

BU n = G(x)

in R,

on B,

(2.5a) (2.5b)

where subscript n denotes iteration number. If, as is frequently done, the Aitken/Shanks formula is used to a t t e m p t to accelerate the convergence o f the iteration, we denote as Method 1(b) the solution of (2.5) for three successive iterates and substitution of the results for one c o m p o n e n t o f each U n into (2.3). (This designation of Method l(b) is useful for a comparison in an example problem below.) 2.3 Method 2. The new approach for applying the Aitken/Shanks formula, first introduced and used in [8] and in a modified form in [9], is referred to as Method 2. The two versions are called, respectively, Methods 2(a) and 2(b) for later convenience. Consider the solution to (2.4) with condition (2.2). The solution evaluated at e = 1 is a solution to (2.1) with (2.2). The specified Uo(x) can be used as an initial approximation to U. For obtaining a m o s t nearly geometric sequence of approximations, assume that U(x,e) ~ UI'(X ) + eU2'(x ) + e2U3'(x) + . . . .

(2.6)

Successive approximations to U(x) are then defined by n-term truncations of the series (2.6): n

Un = E

ei-1 Ui'(x)

(2.7)

i=l A l t h o u g h (2.6) is equivalent to a Taylor series or asymptotic series expansion about e = 0, its convergence or lack of convergence at e = 1 is n o t of particular significance for applicability of eq. (2.3) (see [3] ). If the series (2.6) is substituted into the problem o f eq. (2.4) and condition (2.2) and coefficients o f powers o f e are collected, one obtains equations to solve for the Un':

127

LU 1 - F

= NU o

in R ;

BU 1 = G(x) on B ;

(2.8a)

LU 2' = NU I ' - N U o

in R ;

BU 2' = 0

on B ;

(2.8b)

LU 3' = N 2 ' { U 2 ' , U l " I

in R ;

BU 3' = 0

on B;

(2.8c)

in which N 2' is defined by the perturbation expansion (2.9)

NU = N U I ' + e N 2' { U 2 ' , U I ' ! + 0 @ 2 ) . With the definitions (2.7) and N2 {U2, U1} -= N U I ' + e N 2' { U 2 ' , U I ' ~

(2.10)

one can also solve the following equations for the successive approximations, Un: LU 1 - F

= NU o

in R ;

BU 1 = G(x) on B

(2.11a)

LU2-F

= NU 1

in R ;

BU 2 = G(x) on B

(2.11b)

LU3-F

= N2 {U2,Ult

in R;

BU 3 = G(x) on B

(2.11c)

in which it has been assumed that

e=

1. Note that if the right side o f eq. (2.1) is linear in U(x), then

the problems for the successive U n in eqs. (2.11) are the same as (2.5) for Method 1. We denote as Method 2(a) the solution o f eqs. (2.1 1) for three successive iterates and substitution o f the results for one component o f each U n into (2.3) to obtain an improved approximation. (If NU is linear in U, this is the same as Method l(b)). Note that when the solution is near to convergence at any x, significant errors will be introduced by the loss o f significant figures in applying eq. (2.3). An alternative procedure (denoted as Method 2(b)) that eliminates the difficulty near convergence is to replace eq. (2.3) by the equivalent expression (at e = 1): (u2') 2 u*(x) = u 1'

(2.12) u3' _ u2' '

where each Un'(X) is a single component o f the vector Un'(X). That is, eqs. (2.8) are solved for Un'(X), and (2.12) is used for extrapolation. In a numerical solution, u*(x) can be used as the next Uo(X) in a repetition of the sequence.

3. Example Problems and Comparison o f Methods This section gives simple analytical one-dimensional examples for illustration and comparison o f the methods. 3.1 Example 1. Consider the nonlinear problem (d/dx+l)u

= (1/2)u 2

u(O) = 1.

in 0 ~ < x < oo,

(3.1a) (3.1b)

128

The iterative solution by Method 1 is f o u n d f r o m ( d / d x + 1)u n = (1/2)u~_ 1 , Un(0 ) = 1 .

(3.2)

The analyticalsolfitionsfor n = 1,2,3 (assuming u o = 0) are: Ul(X) = e-x ,

(3.3a)

u2(x ) = e-x [1 + p ( x ) ] ,

(3.3b)

1 u3(x ) = e-x [1 + p(x) + p2(x) + ~ p 3 ( x ) ] ,

(3.3c)

where p(x) = ( 1 / 2 ) ( 1 - e - X ) .

(3.4)

For Method 2, the artificially extended equation is: (d/dx + 1)u = ( l - e ) ( 1 / 2 ) u o 2 + e(1/2)u 2 , u(O) = 1.

(3.5a) (3.5b)

Substitution o f u = Ul'(X) + eu2'(x) + e2u3'(x) + . . .

(3.6)

into (3.5) leads to ( d / d x + 1)u 1' --- (1/2)Uo 2 ,

Ul'(0) = 1 ,

(3.7a)

( d / d x + 1)u 2' = (1/2) [(Ul') 2 - Uo21 ,

u2'(0 ) = 0,

(3.7b)

( d / d x + 1)u 3' --- Ul'U2' ,

u3'(0) = 0 ,

(3.7c)

(d/dx + 1)u 1 = (1/2)Uo 2 ,

Ul(0) = 1 ,

(3.8a)

(d/dx + 1)u 2 = (1/2)Ul 2 ,

u2(0) = 1 ,

(3.8b)

(d/dx + 1)u 3 = (1/2)u22 - (1/2)(Ul-U2)2,u3(0) -- 1 .

(3.8c)

or equivalently, with e = 1 and eq. (2.7),

The analyticai solutions to (3.7) with u o = 0 are: Un'(X) = e-X[p(x)]n-1 ,

(3.9)

where p(x) is given by (3.4) and where the solutions u n to (3.8) are given by (2.7). Evaluations o f these solutions at x = 1 and applications o f the appropriate forms of the Aitken/Shanks formula are given in Table 1. The results for the extrapolated solution u* m a y be compared with the exact solution to (3.1), u(x) = 2 ( l + e X ) -1 ,

(3.10)

129

Table 1. Results o f E x a m p l e 1 at x = 1 (u o = O)

l(b)

2(a)

2(b)

EQUATIONS:

(3.2) & (2.3)

(3.8) & (2.3)

(3.7) & (2.12)

n

Un(1)

Un(1 )

Un'(1 )

METHOD:

1

0.3678794412

0.3678794412

0.3678794412

2

.4841515202

.4841515202

.2325441579

3

.5247721376

.5209005060

.1469959430

u*(1) =

.546583145

.537882842

.5378828426

Exact u(1) =

.5378828428

.5378828428

.5378828428

evaluated at x = 1: u(1) = 0.5378828428 to ten significant figures. We note first that the extrapolated solution u* by Method l(b) is somewhat closer to the exact value than u3, but not significantly closer. We note further that the third approximation, u3, by Method 2(a) is not as good an approximation as u 3 in Method l, but that the extrapolated solutions by Methods 2(a) and 2(b) are exact except for loss o f 1 or 2 significant figures. (Method 2(a) is less exact because o f loss o f significant figures in (2.3).) The striking accuracy o f Method 2 in this example occurs because the sequence o f solutions produced by t

i

Method 2 is precisely geometric, i.e. Un+l/U n = constant for all n at a given x. The difference from Method 1 is seen by comparing eqs. (3.2) with (3.8), in which (3.8c) has an additional term that produces the geometric sequence. 3.2 Example 2. Consider next an example which is linear (so that Method 2(a) would give the same results as Method l(b)), but for which the iterative sequence is "nearly geometric." Let us use Method 2(b) for this example (eqs. (2.8) with (2.6), (2.7), and (2.12)). The problem is du 2 = - x ~du d--x--~-2u

in 0 < x < ~

, u(0) = 0 ,

(3.11)

which is written in this way in analogy to more complex problems in which one may put a very simple operator on the left and the rest o f the terms on the right for iteration. (One can also shift the term 2u to the left side, with very similar results.) The artificially extended equation is du du - 2u) d-x _ 2 = ( l - e ) ( - x ~duo - 2u°) + e(-x~-~-

in 0 ~< x < oo I

u(0) = 0

(3.12/

130

Substitution of (3.6) leads to (with u o = 0): dul'/dx-2

Ul'(0) = 0 ,

(3.13a)

du2'/dx = -x d u l ' / d x - 2u 1' ,

= 0,

u2'(0) = 0 ,

(3.13b)

du3'/dx = -x d u 2 ' / d x - 2u 2' ,

u3'(0 ) = 0 .

(3.13c)

The analytical solutions are Un'(X) = (-1) n+l (n+l)x n

(3.14)

and the successive approximations are given by (2.7). The sequence (3.14) is not geometric, but since £imn--~ [Un+l(X)/Un'(X)] exists at given x, the sequence is "nearly geometric" [3]. Evaluation o f the solutions (3.14) at x = 0.5 gives (Ul' , u2', u3' ) = (1.00, -.75, .50) so that the successive approximations are (u 1, u2, u 3) = (1.00, .25, .75). Substitution o f the u n' into (2.12) gives u*(.5) = 0.55, which compares well with the exact solution to (3.11), u(x) = (2x + x 2) (1 +x) "2 ,

(3.15)

from which u(0.5) = 5/9 = 0.555555 . . . .

4. Transonic F l o w Over an Airfoil

For application of the methods described above, consider two-dimensional, steady, inviscid flow over a thin symmetrical parabolic-arc airfoil in a subsonic free stream. At high subsonic Mach numbers, part o f the flow can be supersonic, so we consider the transonic small-disturbance equations, which are nonlinear elliptic partial differential equations in subsonic regions and hyperbolic equations in supersonic regions. Transition o f the velocity field from a subsonic region to the embedded supersonic zone is smooth, but transition from the supersonic to subsonic region is usually discontinuous, through a shock wave. The improved finite-difference method of Murman and Cole [ 1 9 - 2 2 ] captures the shock waves (in a fully conservative way) but spreads the rapid transition over several mesh points. In [8] a semidirect finite-difference method, based on the use of a fast direct Cauchy-Riemann solver [ 15], was applied to solving the equivalent o f Murman's transonic finite-difference equations [ 21 ] iteratively for the perturbation velocities, u and v. (The iteration procedure has been formulated in such a way that at nonelliptic points terms on the right side of the difference equations cancel out the elliptic character o f the left side when the iterated solution converges.) Both Methods l(a) and 2(a) described above worked well for subcritical and for slightly supercritical (local Mach number > 1) flows, except that Method 2(a) could be used only before any part of the solution was nearly converged. In [9] the method was extended to strongly supercritical flow by the addition o f stabilizing terms to the difference equations and to the Cauchy-Riemann solver [18]. Also introduced in [9] was the method version denoted here as Method 2(b), which can be used when the solution is nearly converged. In smooth subsonic flows the acceleration technique is effectively used repeatedly. However, in transonic

131

flows with strong shock waves, the acceleration technique is n o t helpful at the beginning o f the iteration when the shock wave and its location are n o t well defined. Therefore in [9] it was considered desirable to use the straightforward iteration Method l(a) until the m a x i m u m residual is reasonably small, so that the supersonic region is nearly defined, and then use Method 2(b) to extrapolate three iterates to a final solution. A fully conservative second-order-accurate formulation has been introduced in [ 10], and so a fomulation that includes either Murman's fully-conservative first-order-accurate formulation or the second-order formulation will be used here. 4.1 Governing equations and b o u n d a r y conditions. Let the dimensionless X and Y axes be respectively along and normal to the airfoil chord, the free-stream Mach n u m b e r be Moo < 1, and the dimensionless velocity c o m p o n e n t s in the X and Y directions be U,V. One m a y then define perturbation velocity c o m p o n e n t s u,v through a Prandtl-Glauert transformation with 13~ (1 - M2 ) 1/2: U = l+(r//3)u,

V = rv,

Y = y//3,

X = x,

(4.1)

which a m o u n t s to shifting and scaling o f certain terms (cf. [ 17, 8 - 1 0 ] , so that the transonic small disturbance equations take the form fx + gy = 0 ,

Uy - v x = 0

(4.2a,b)

where f = f(u) = u - a u 2 ,

g = g(v) = v ,

a = r ( 7 + 1)M2/2/3 3 ,

(4.3a,b) (4.4)

in which a is a transonic similarity parameter and r is an airfoil thickness ratio. Eqs. (4.2) are often written in terms o f a perturbation velocity potential q~ defined by u = ~bx, v = ~by, and all the developm e n t s to be described apply as well to that potential equation. The equation system (4.2) is elliptic, parabolic, or hyperbolic depending on w h e t h e r u - UCR is negative, zero, or positive, where the transformed critical velocity is u C R = 1/2a. The corresponding pressure coefficient is Cp = -2(r//3)u. The linearized surface b o u n d a r y condition for the symmetrical parabolic-arc airfoil, whose upper surface is given by Yb(x) = r F ( x ) = r(0.5 - 2x 2) in - . 5 ~< x ~< .5 (with F(x) = 0 in [xl > .5), and the conditions at infinity are v(x,0 +) = F ' ( x ) , u,v+0

as

x2+y2~oo.

(4.5a) (4.5b)

Eq. (4.2a) is written in a "conservation-law" (or divergence) form, in terms o f flux c o m p o n e n t s f and g. Therefore discretized forms o f (4.2a), for numerical solution, can represent in a fully conservative way either that differential equation or the corresponding integral form. These discretized forms can thus be formulated correctly to represent transitions between elliptic and hyperbolic regions [21,10 ].

132

Since only the term fx in the system (4.2) determines the type o f point (depending on the local value o f u), one can write the general type-dependent difference equations in the form: (fx)T + (gy)C -- 0 ,

(Uy)C - (Vx) C = 0 ,

(4.6a,b)

where subscript C indicates a central-differenced representation of a derivative and subscript T, which indicates type-dependent differencing, may be replaced by E, H, P, or S at points defined respectively as elliptic, hyperbolic, parabolic, or shock points [21,10]. At all points where the difference equations are clearly elliptic or hyperbolic, subscripts E or H are used. Transition points from elliptic to hyperbolic (progressing downstream from left to right) are P points, and transitions from hyperbolic to elliptic are S points. For defining the finite-difference operators, Fig. 1 shows a staggered u,v mesh, with the shaded area indicating a mesh cell for eq. (4.2a). The center o f a mesh cell is the point at which ~b would be defined on a conventional mesh and is the point that is designated E, H, P, or S. The indices j and k indicate respectively the x and y directions. Second-order-accurate central differences are (Ux) C = (uj, k - Uj.l,k)/Ax,

(Vy)C = (vj,k - Vj,k_l)/Ay ,

(Uy)C = (Uj,k+l - Uj,k)/AY,

(Vx)C

(4.7)

(Vj+l,k - Vj,k)/Ax •

In general, (fx)T is represented by AX(fx)T = Afj,k = ( f G ) j , k - ( f G ) j - l , k

(4.8a)

(fG)j,k = f((uG)j,k) = ( U G ) j , k - a ( u 2 ) j , k

(4.8b)

where

and where u G is either a "hyperbolic form" u H or an "elliptic form" u E. With (i) the definition (4.8) for the difference operator (fx)T, (ii) a condition to determine whether each (UG)j, k is represented by u E or UH, and (iii) specifications o f u E and u H to obtain the finite differences (4.8a) to the order o f

uj_2, k+l

°

Uj_l, k+l

+

Vj - I , k

uj_2, k

•

+

Vj-I, k-I

•

~ ' k

uj

Uj, k+ I " I I I ~j,k------Vj+l,k

,k

Fig. 1 - Differencing mesh and mesh cell.

~-

~j+l, k

vj+l,

Z~y

k-i ~-

133

accuracy desired, all four type-dependent operators are obtained. As derived in [ 10], (i) the secondorder-accurate elliptic operator, (ii) either the first-order or second-order-accurate hyperbolic operator, and (iii) the corresponding parabolic-point and shock-point operators are all produced in (4.6) with (4.7) and (4.8) by the following relationship: (4.9)

(UG)j,k = (1 -Oj,k)Uj, k + Oj,k[XUj_ 1,k + (1 - X)uj.2,kl , where oj, k = 0 (and UG=UE)

if

~j,k
1

= 1 (and u G = u H )

if

Uj,k > u C R ,

)

(4.10)

~ , k = (Uj,k + 6 Uj_l,k)/(1 + 6 ) ,

(4.11)

and where X = 1 for the first-order-accurate hyperbolic operator, X = 2 for the second-order-accurate hyperbolic operator, and 6 is a parameter that may be varied from 0 to oo but is derived as unity for Murman's first-order-accurate operators [21]. As an example to illustrate, suppose X = 1, 6 = 1, and Uj,k < uCR and Uj-l,k > uCR" Then for the shaded mesh cell in Fig. 1, eqs. (4.8) - (4.10) give A~,k = Uj,k - uj-2,k - a(u2k - u22,k ) which is equivalent to Murman's [21 ] first-order shock-point operator. In a similar way the KruppMurman first-order parabolic operator [20] is also obtained. Both the first- and second-order-accurate hyperbolic operators given by (4.8) - (4.10) with X = 1 and X = 2 are equivalent to upwind difference operators originally proposed by Murman and Cole [19]; the fully conservative second-order P and S operators were introduced in [ 10]. Analysis of all these E, H, P, and S operators [ 101 has verified their consistency, accuracy, and stability in the examples computed. Because of the slow iterative convergence of the second-order-accurate iterative method to be described, two methods of adding artificial viscosity have been proposed and used [10]. Both leave the scheme fully conservative and formally second-order-accurate. The boundary conditions for the finite-difference equations (4.6) are the same as (4.5) but with (4.5b) replaced by a far-field condition on an outer rectangular boundary B: uj, k = UB(X,y)

or

vj,k = VB(X,y)

on B

(4.12)

where, for example, u B and v B are given by a Prandtl-Glauert solution (see [ 15,8,9] ). For solution o f eqs. (4.6) with (4.7) through (4.11 ) and with conditions (4.12) by the semidirect methods, one must rearrange the equations so that the left side is an appropriate elliptic operator and provides a stable iteration scheme. One first adds (Ux) C - (fx)T to both sides o f (4.6a) to obtain (Ux)C + (Vy)C = (Ux)C - ( f x ) T ,

(4.13a)

(Uy)C - (Vx) C = 0 .

(4.13b)

134

This set contains a central-differenced elliptic operator on the left side regardless of the local type o f the equations. The nonlinear type-dependent term has been shifted to the right side where, in an iterative procedure, it can be computed from a previous iteration. Although the iteration of these equations [8] converged well for subsonic and slightly supercritical flow, it was found [9,10] that terms with parameters multiplying uj, k and Uj_l,k needed to be added to both sides o f (4.13a) to produce iterative convergence at higher Mach numbers. A more specific form of the diffe1:ence equations, in which the second-order-accurate relations (4.7) have been substituted, is Dj,k(U,V) = Rj,k(U),

Ej,k(U,V) = 0 ,

(4.14a,b)

in which Dj,k(U,V) - (1 - a l ) U j , k - (1 +a2)uj-l,k +/2"l(vj,k - Vj,k_l),

(4.15a)

Ej,k(U,V) = (uj ,k+ 1 - Uj,k) - #(vj+ 1,k - Vj,k),

(4.15b)

Rj,k(U) -= (1 - a l ) U j , k - (1 +a2)Uj_l, k - A ~ , k ,

(4.15c)

and where A~,k is defined by eqs. (4.8) - (4.11) and /~ -- Ay/Ax. The formal order of accuracy o f eqs. (4.14) depends on the value o f h used in (4.9). 4.2 Equations for Method l(a). As described in section 2.2 above, the straightforward iteration Method l(a) for eqs. (4.14) is simply Dj,k(Un,Vn) = Rj,k(Un_l),

Ej,k(Un,Vn) = 0 .

(4.16a,b)

For determining each oj, k in (4.10), eq. (4.11) uses Un. 1. The presence o f alUj, k and a2Uj_l, k on both sides o f eq. (4.16a) allows the interpretation and treatment of these terms as an off-centered time derivative, Ou/Ot, multiplied by a constant. When the solution converges, these terms cancel out. The semidirect Method 1(a) proceeds by solving the left side of (4.16) in terms of the known right side by an "extended Cauchy-Riemann" solver [18] for u n and vn at all points simultaneously. The iteration with a 1 or a 2 ¢ 0 needs a reasonable (but very roughly approximate) initial approximation (Uo) , such as a Prandtl-Glauert solution. Ref. [ 10] gives variable specifications o f a 2 for best convergence. The boundary conditions on (4.16) are Vn(X,0 +) = F ' ( x ) , un = uB

or

vn = vB

(4.17a) on B .

(4.17b)

4.3 Equations for Method 2(b). The artificially extended form, (2.4), of eqs. (4.14) is Dj,k(U,V) = (1 - e)Rj,k(U o) + eRj,k(U),

(4.18a)

Ej,k(U,V) = 0 .

(4.18b)

135

For Method 2(b) assume that (4.19a) (4.19b)

u(x,y,e) = Ul'(x,y) + eu2'(x,y) + e2u3'(x,Y) + -.. , v(x,y,e) = Vl'(X,y) + ev2'(x,Y) + e2 v3'(x,Y) + .... The successive approximations are then (for n = 1 , 2 , 3 . . . ) n

Un= ~

n

ei-lui'(x,y),

Vn= Z

i=l

(4.20)

ei'lvit(x,y). i=l

Substitution of (4.19) into (4.18) leads to i

Dj,k(Un', v n ) = R.n_1 ,

Ej,k(Un', Vn') = 0 ,

(4.21)

where: R o = Rj,k(Uo)

(4.22a)

R 1 = Rj,k(Ul' ) - Rj,k(Uo)

(4.22b)

(with u o being used in (4.11) in determining oj, k for use in Rj,k(Ul')) and R 2 = (1 -C~l)(U2')j, k - (1 +~2)(u2')j, k - (Af2)j,k,

(4.22c)

Af2 = (f2)j,k - ( f 2 ) j - l , k '

(4.23a)

(f2)j,k = (1 - Oj,k) [(u2')j, k - 2a(ul'u2')j, k ] + aj, k { [X(u2')j.l,k + (1 -X)(u2')j_2,k] - 2a[X(Ul')j.l,k + (1-X)(Ul')j_2,k] [X(u2')j_l, k + (1 -X)(u2')j.2,k] } .

(4.235)

The boundary conditions are: Vl'(X,0 +) = F'(x) ;

Vn'(X,0 +) = 0 (n = 2,3) ;

(4.24a) (4.24b)

u 1' = u B

or

v 1' = v B

on B ;

u n' = 0

or

v n' = 0

on B (n = 2 , 3 )

.

(4.24c)

With some reasonable approximation for (Uo)j, k, such as a nearly converged solution b y Method l(a), eqs. (4.21), with n = 1,2,3, give three successive approximations Ul', u2', u 3' at each j,k to use in (2.12) to obtain an extrapolated solution. 4.4 Results and discussion. A research computer program written to solve the transonic small disturbance equations by the methods described above for a biconvex airfoil at zero incidence, includes the option of switching after some iterations by Method l(a) to the extrapolation technique, Method 2(a). A conversational version of the program, for interacting with the program, was run on an IBM 360/67 computer, and computing times were measured on a Control Data 7600 computer.

136

Pressure distributions have been computed for a range o f subsonic and transonic Mach n u m b e r s from b o t h first- and second-order-accurate formulations. Examples by Method l(a) are shown on Fig. 2 for a thickness ratio o f 10 percent and Moo = 0.825. For this calculation the boundaries were at one-half chord upstream and downstream o f the airfoil edges and at 3.5 chords above the airfoil. The results computed on a 39 ×32 uniform m e s h compare well with a line-relaxation program [22], which uses a variable and finer mesh. On a very coarse ( 1 9 × 3 2 ) mesh, with only 10 mesh intervals on the airfoil chord, the first-order-accurate results, of course, are n o t good. The shock is badly smeared, and an anomalous j u m p behind the sonic point that is characteristic o f the first-order P operator is exaggerated on the coarse mesh. However, the second-order-accurate results are very s m o o t h through the sonic point and are surprisingly accurate. .8

(a) .6 . . . . . . . . .

A

C~

.2 -Cp 0

I M(z)= 0.825, T =0.10

-,~

-A -.6

•

-.8 1

-.75

I

J

-.50

-.25

i 0

I

.25

l

.50

o

Isi" ORDER, 3 9 x :52 MESH

•

2nd ORDER, 3 9 x 5 2 MESH

--

MURMAN et al, t VARIABLE MESH, .75 REE 22

X

.8

(b) .6

A .2

-Cp

MQO= 0 . 8 2 5 , T =0.10

0

o

ISt ORDER, 19x 32 MESH

•

2nd ORDER, 19x 32 MESH

-.4 MURMAN et el,

-oB

-.75

A

I

-.50

-.25

0

I

I

I

.25

.50

.75

X

Fig. 2 - Pressure on a thin biconvex airfoil.

VARIABLE MESH, REE 22

137

Figure 3 shows an interesting effect o f switching to Method 2 before the iteration has converged enough, when the types of all points are n o t yet quite the same as the final types. Method l(a) was used for nine iterations; then Method 2(b) was used to obtain the three successive terms at each point and the extrapolated solution shown in Fig. 3. A property o f the A i t k e n / S h a n k s extrapolation as used in Method 2 is that all the significant figures of the three successive approximations at any point contain information about the exact solution, even t h o u g h those successive approximations themselves are n o t very close to the enact solution (see example problems above in section 3). It thus appears possible in Fig. 3 that this procedure m a y be picking up the fact that the exact solution to the equations (or the solution on a very fine mesh) has the well-known logarithmic singularity just behind the shock, even t h o u g h the converged solution on the coarse m e s h smears,over this singularity. Even the finer m e s h used by the program in [22] was not fine enough to pick up the singularity, partly because that point apparently occurs between the m e s h points for this case. This p h e n o m e n o n illustrated in Fig. 3 is n o t an isolated case but is a typical occurrence in Method 2. It m a y be that the numerical solution in Fig. 3 is as good as representation o f the exact solution to the equations as is the fully converged solution in Fig. 2(a) (circles). The most significant property o f the semidirect m e t h o d is the relatively short c o m p u t i n g time required. On the 3 9 × 3 2 mesh, the time per iteration was measured as 40 milliseconds in a very inefficiently coded program, b u t for various reasons discussed in [ 10] it is expected to be reduced to 20 ms. (The direct solver requires only 14 ms) The subcritical cases were sufficiently converged in 3 iterations or less, and a slightly supercritical case (first-order-accurate, using Method 2) required 6 iterations. The

.8 .6 .4

.2 .Cp .0

I M¢O=0;825, z" =0.10

-.2

o

-.4

--.6

-.8 -.75

® ,

- . 50

® R

~.25

0 X

I

I

.25

.50

ISt ORDER, 59 x 32 MESH MURMAN et ol, VARIABLE MESH, REF. 2 2 i

.75

Fig. 3 - Pressure distribution resulting from Aitken/Shanks extrapolation (Method 2(b)) before iterative convergence.

138

first-order-accurate case shown in Fig. 2(a) required 20 iterations by Method 1 and, as described above, the results of Fig. 3 required only 9 iterations by Method l(a) followed by 3 more by Method 2(b). At this writing, the program has not yet been written for the above formulation that includes the second-order-accurate formulation in Method 2. It is expected that when this is done, the program can be run rapidly with the first-order (X = 1) operators on the very coarse mesh using Method 1, then switched to second-order (X = 2) and Method 2 for final extrapolation.

5. Concluding Remarks It has been shown that a special procedure (Method 2) is effective for obtaining most appropriate successive approximations for use in the Aitken extrapolation formula for accelerating the iterative convergence of numerical solutions to nonlinear partial differential equations. The procedure is based on the combined use of artificial perturbation-series expansions and an artificially extended equation. It was shown in a previous paper [8] that one version of the technique was very effective for accelerating iterative convergence when the solutions are smooth. The method, in a modified version, has now been applied with some success to a strongly supercritical transonic flow problem, in which the flow equations are of mixed type and whose solutions have shock-wave discontinuities. The method is expected to be extended to more general flows, including lifting airfoils and three-dimensional flows.

References

1

Aitken, A.C.: On Bernoulli's numerical solution of algebraic equations. Proc. Royal Soc. Edinburgh 46 (1926) 289-305.

2

Henrici, P.: Elements of Numerical Analysis. John Wiley & Sons, Inc., New York, 1964.

3

Shanks, D.: Nonlinear transformations of divergent and slowly convergent sequences. J. Math. and Phys. 34 (1955) 1-42.

4

Johnson, R.C.: Alternative approach to Pad6 approximants. Pad~ Approximants and Their Applications (ed. by P. R. Graves-Morris). Academic Press, London, 1973; 53-67.

5

Genz, A.C.: Applications of the e-algorithm to quadrature problems. Pad6 Approximants and Their Applications (ed. P. R. Graves-Morris). Academic Press, London, 1973; 105-115.

6

Wynn, P.: On a device for computing the em(Sn) transformation. Mathematical Tables and Other Aids to Computation 10 (1956) 91-96.

7

Wynn, P.: Acceleration techniques for iterated vector and matrix problems. Mathematics of Computation 16 (1962) 301-322.

8

Martin, E. D.; Lomax, H.: Rapid finite-difference computation of subsonic and slightly supercritical aerodynamic flows. Presented as a portion of AIAA Paper No. 74-11, 1974. Also AIAA Jour. 13 (1975) 579-586.

9

Martin, E. D.: Progress in application of direct elliptic solvers to transonic flow computations. Aerodynamic Analyses Requiring Advanced Computers, Part II, NASA SP-347, 1975; 839-870.

l0

Martin, E. D.: A fast semidirect method for computing transonic aerodynamic flows. AIAA 2nd Computational Fluid Dynamics Conference Proceedings, 1975; 162-174.

11

Bellman, R.: Perturbation Techniques in Mathematics, Physics, and Engineering. Holt, Rinehart and Winston, New York, 1964.

139

12

Buneman, O.: A compact non-iterative Poisson solver. SUIPR Rept. 294, Inst. for Plasma Research, Stanford Univ., Stanford, Calif., 1969.

13

Hockney, R.W.: The potential calculation and some applications. Methods in Computational Physics, Vol. 9 (ed. by B. Alder, S. Fernbach, and M. Rotenburg). Academic Press, New York, 1970; 135-211.

14

Buzbee, B. L.; Golub, G. H.; Nielson, C.W.: On direct methods for solving Poisson's Equations. SIAM J. Numer. Anal. 7 (1970) 627-656.

15

Lomax, H.; Martin, E.D.: Fast direct numerical solution of the nonhomogeneous Cauchy-Riemann equations. J. Comp. Phys. 15 (1974) 5 5 - 8 0 .

16

Hafez, M. M., Cheng, H. K.: Convergence acceleration and shock fitting for transonic aerodynamics computations. AIAA Paper No. 75-51, 1975.

17

Concus, P.; Golub, G.H.: Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations. SIAM J. Numer. Anal. 10 (1973) 1103-1120.

18

Lomax, H., Martin, E.D.: Variants and extensions of a fast direct numerical Cauchy-Riemann solver, with illustrative applications. NASA TN D-7934, 1975.

19

Murman, E. M.; Cole, J. D.: Calculation of plane transonic flows. AIAA Jour. 9 (1971) 114-121.

20

Murman, E. M.; Krupp, J.A.: Solution of the transonic potential equation using a mixed finite difference system. Lecture Notes in Physics 8 (ed. by M. Holt). Springer-Verlag, Berlin, 1971; 199-205.

21

Murman, E.M.: Analysis of embedded shock waves calculated by relaxation methods. AIAA Jour. 12 (1974) 626-633.

22

Murman, E. M.; Bailey, F. R.; Johnson, M. H.: TSFOIL-A computer code for 2-D transonic calculations, including wind-tunnel wall effects and wave-drag evaluation. Aerodynamic Analyses Requiring Advanced Computers, Part II, NASA SP-347, 1975; 769-788.

THE RISE OF A BUBBLE IN A FLUID John L. Gammel Department of Physics, Saint Louis University St. Louis, Missouri 63103 I.

Introduction The standard numerical approach to problems in hydrodynamics is to replace

the appropriate partial differential equations by a set of finite difference equations.

This approach has several difficulties inherent in it.

For example, ficti-

tious viscosities are introduced and the cumulative effect of these viscosities may result in large errors for late times. As a new approach, we advocate the exact solution of the partial differential equations by means of power series expansions. I

This approach has the disadvantage

that the power series may not converge for late times or in regions of space where the flow pattern varies rapidly.

I{owever, by use of Pad~ approximants or powerful

generalizations of these approximations which we have developed for application to hydrodynamics,

this difficulty has been overcome.

The problem which we consider as an illustration of our method is that of the rise of an incompressible volume of gas (that is, a bubble) which is initially spherical.

The problem is to calculate the shape of the boundary and the height of

the center of gravity at subsequent times. vertical or z-axis.

The problem has axial symmetry about the

To label a point on the boundary, we use the polar angle 8

which is the angle between the vertical axis and a line joining the origin (the center of the initial sphere) and the point.

The sphere has initially unit radius.

At subsequent times, the shape of the boundary is specified by x = x(t, cos e) (i)

y = y(t, COS 8) By known methods of hydrodynamics x and y are obtained as power series, 1 y ffi cos 8 + t2( - ~ +

+ t

6

3 9 3 ~ COS 28) + t4(- ~ COS 8 + ~ COS 38)

9 (=~.~o~ +~1

cos 28) +

142 and x is best calculated from

~x ~2x + ~7(~2y

i =0

De ~t2

,

~fl ~t 2 + ~)

which expresses the fact that all forces other than gravity are perpendicular

to

the boundary. Section II deals with methods of su~m~ing these series and the information which may be gained about the rise of the bubble and its shape at various times.

II.

Methods of Series Summation A.

The ordinary Pade approxlmant. Consider the function f(x) = - / l + 2 x

-v l+x

= 1 + 1

~x-~

5 2

+ ....

(3)

Suppose one has only the first three terms in this expansion and wants the value of f(x) for some large positive value of x, say x = ~.

Substituting

series will no__~tresult in a rapidly converging expression, w f(x) = J(l+2x)/(l+x)

x = ~ into the

[Were it known that

, the source of the trouble would be obvious because x = -i/2

and x = -i are branch points of f(x) so that the series cannot converge for IxI > 1/2.]

The Pade approximate method consists

in writing f(x) as the ratio of

two polynomials,

f (x) =

N O + NIX + .. + N x m 1 + DlX + .. + DnX

where D O ffi 1 by choice

m (4) n

(were D O ~ i, the numerator and denominator

could be divided

by it with the consequence D 0 = 1), and fixing the values of the N's and D's so that were the right hand side of Eq.

(4) expanded in a power series in x, the

result would agree with the right hand side of Eq. series, f(x).

through order m + n. For example,

The result is called the [m/n] Pade approximant

for the function of Eq. NO + NlX

l+DlX: and cross multiplying

(3), that is, the original power

(3), the [I/i] Pade approximant

1

l+~x--~x

to

is

5 2

+ ....

and equating powers of x yields

(5)

143

coefficient

of x0:

N0 = 1

coefficient of xl:

1 N1 = ~ + D1

coefficient of x2:

0

=

_

5

1

(6)

~ + ~ D1 ,

from which D 1 = 5/4 , N 1 = 7/4 , NO=

(7)

1

Thus 7 ~ii_+2

x

+x

l+~x

=

5 l+~x

(8)

=

1.4.

(9)

For x = =, /2

The source of this surprising accurate result is that #(l+2x)/(l+x)

has a cut, that

is, a dense sequence of zeros and poles, running from x = -1/2 to x = -i. approxlmant

has only one zero, at x = -4/7, and one pole, at x = -4/5, both of which

lle between x = -1/2 and x = -i. charges:

The Fade

The zeros and poles are rather llke electric

when viewed from a great distance a complicated

distribution

of zeros and

poles looks rather like a single zero and pole.

The point x = = is so far away from k the cut that a single zero and pole suffice to represent the actual analytic structure very well. Of course, one never relies only on the [i/I] Fade approxlmant: es the convergence of a sequence of Pade approximants [2/2],

[3/3],

approximants

... sequence).

From Eq.

are finite at x = ~.

is finite at x = =.

(8) and also Eq.

such information

that (i/~2~xl~l+x)

is usually available.

form a square array, and the approxlmants

from this array is dictated by physical information.

the [l/l].

(4), one notes that these

We have used the information

In physical problems,

The [m/n] Pade approxlmants

(in our example,

one discuss-

to be selected

144

B.

The location of the zeros and poles. In the example of Eq. (3), the zeros and poles of the [i/i],

... approximants approximation analysis:

lie between -1/2 and -i, and they become more dense as the order of

increases;

in principle

branch points.

[2/2], [3/3],

that is, they form a cut.

Cuts have no real meaning in

they can be located as one pleases so long as they Join

The Fade approximents

have a definite opinion of their own about

where the cuts go. In many applications, awkward.

the location chosen by the Pade approxlmant

is at best

Consider f(x) = J ( l

- x) 2 + 1

This function has branch points at x = 1 ± i.

(i0)

The Pade approxlments

cut the func-

tion along an arc of circle running from 1 + i to 2 to 1 - i (the circle is entered at x = 1 and also passes through the origin). figure i.

This situation is illustrated

in

Beyond

// //

/

/

/

/

? /

/

kkk\ \% kk%\

Figure I x = 2, the Pade approximants

will not converge to f(x), and even if they did, it

could not be to that value of f(x) on the principal Riemann sheet. A way of moving these cuts is the following. applications,

This is very important for

because such a phenomenon is encountered

in the s,,mmation of the

expression for the height of the center of gravity of the bubble

(see Eq. (2)).

145

This fact is reported at length in a Mission Research Corporation report.

(An

Accurate Early Time Solution for a Rising Fireball Model, by C. Longmlre, G. McCartor, N. Carron, and F. Fajen, Report Number DNA 2967T, October, 1972 (MRC-R- 20). ) The scheme for moving the cut is very simple. odd powers separately,

Consider the even powers and

that is, consider F+(x 2) = y1 (f(x) + f ( - x ) )

,

(11) xF_(x 2)

= i

~ (f(x) - f(-x))

The location of the branch points is not changed, so that the singularities of F+ and F

are at x 2 = (l±i) 2 = ± 2i.

to F+ and F

Now the zeros and poles of the Pade approximants

(whose numerators and denominators are now polynomials

in x 2) are

located on the imaginary axes, or, in the x-plane, along the dashed rays shown in Figure I.

Now the Pade approximants

converge for all real positive x.

Actual numerical data will be presented in Section III.

C.

Pade approximants

for functions of two variables.

J. S. R. Chisholm of the University of Kent, Canterbury,

has proposed the

following extension of the defining equation (Eq. (4)) for functions of two variables: f(x,y) =

N ~ ~=0

As before, we may choose bOO = i.

N ~ a V=0

N x~y~/ ~ 0=0

N ~ b TxayT . r=0

(12)

The a's and b's are defined by requiring that

when the right hand side of Eq. (12) is expanded in a power series in x and y the result agrees with the power series expansion of f(x,y) through order 2N, and that in order 2N + i, the average of the coefficients of xYy 2N+I-Y, y = 1,2,...,N, also agree.

146

This verbally complex algorithm is best explained with an example: f(x,y) = COO + Cl0X + c01Y + c20x2 + CllXY + c02Y 2 + c30 x3 + c21x2y + Cl2XY 2 + c03 y3 + ...

,

a00 + al0x + a01Y + allxY

(13)

I + bl0X + b01Y + bllXY

Cross multiplying and equating the coefficients of various powers of x and y yields 00 coefficient of x y :

COO = a00

,

i 0 x y :

c00bl0 + Cl0 ffi al0

,

01 x y :

coobo1 + CO1 ffi ao1

,

11 x y :

c00bll + Cl0b01 +c01bl0 + ell = all

20 x y :

Cl0bl0 + c20 = 0

(xly 2 + x2y 1):

,

,

c20bo1 + Cllblo + Cllbo1 + c02blo = 0 .

(14)

These linear equations for the a's and b's generally have one and only one solution. The following properties of these approximants make them the most desirable of the possible approximants i)

The NIN Pade approximant to f(x)g(y) is the product of the ordinary N/N

Pade approximants 2)

to f(x) and g(y) separately.

The N/N Pade approximant to f(x) + g(y) is the sum of the ordinary N/N

Pade approximants

D.

to functions of two variables:

to f(x) and g(y) separately.

Movement of cuts in functions of two variables. Exactly the same tricks for moving cuts as that used in Section B for func-

tions of one variable may be used for functions of two variables. Eq. (ii), one defines

In analogy with

147

F++(x2,y 2) = 1 (f(x,y) + f(-x,y) + f(x,-y) + f(-x,-y))

yF+_(x2,y2)

1

,

(f(x,y) + f(-x,y) - f(x,-y) - f(-x,-y))

,

x.F_+(x2,y 2) = ~1 (f(x,y) - f(-x,y) + f(x,-y) - f(-x,-y),

,

(15)

xyF__(x2, t 2) = ~1 (f(x,y) - f(-x,y) - f(x,-y) + f(-x,-y)) It is important to note that the structure of the expansion of x and y (see Eq. (2)) suggests such a decomposition of these functions, because a general two

variable expansion has 1 term of order 0, 2 terms of order i, 3 terms of order 2, and so on, whereas these expansions of Eq. (2) have missing terms. other term, two expansions with the correct structure is found.

Taking every

When we come to

the calculation of torus time in Section II, we will find that taking every other term results in Pade approximants whose poles and zeros are not awkwardly located.

E.

Other generalizations

of Pade Approximants.

One sees that Eq. (4) is nothing more or less than Df - N = 0(x re+n) This is a linear equation in f.

(16)

One might also consider

pf2 + Qf + R = 0(x m+n+£+l)

,

where m, n, and £ are the degrees of polynomials P, Q, R, respectively. approximants are called quadratic Pade approximants.

(17) Such

But, more important, we might

consider - + Q {df~4 P -d2f kdxl + R = 0(x m+n+£+l) dx 2

(18)

because, if the power series expansion of f is known, so is the power series expansion of d2f/dx 2, (df/dx) 4, etc., so that Eq. (18) results in linear equations for the coefficients

III.

in the polynomials P, Q, R.

The Bubble Problem. A.

Approximations

based on partial differential equations.

From the series of Eq. (2), we obviously know x(t, cos 8) and

148

y(t, cos 0), where x and y are the Cartesian coordinates

of the point 0 at time t

(remember that 0 refers to the initial configuration). Had we great faith in Newtonian mechanics,

we might very well start out a

system of partial differential equations

~2x+ ~t 2

= 0 "'" (19)

a2

+

~t 2

= 0 "'"

But what forces act on these particles?

First, gravity

~2 x ~t2 + ... = 0

, (20)

~2y + Y1 = 0 ~t 2 "'" where y is the vertical axis. boundary,

What else?

and since the magnitude

,

The pressure acts perpendicularly

of the pressure ought to be proportional

and the direction cosines of a vector perpendicular

to the

to y,

to the boundary are +3y/~0,

-ax/aO, a2x -

-

at 2

~X7

+

+ ~Y ~ 0 + """

=

0 (21)

32y + Yi ~t 2 There are properties

- ~y ~~x +

that such a set of equations ought to have.

i)

Galilean invariant,

2)

invariant against transformations

The pressure term is not Galilean invariant, of 0 by writing

It ought to be

that is, invariant against y = y' + vt, and

0 = 0(0')

transformations

= 0 "'"

instead

of any sort on the variable

0, (22)

but it can be made invariant against

149

B2x --+ @t 2

By/De

+ ay

+ ... = 0 Bx 2 (23)

1

Bx / I e

~-

ay

To make these Galilean invariant, straint.

According

+

... = 0

Bx 2

we need to impose the incompressibility

to d'Alembert's

as a con-

principle,

(24) ~t 2 + ...

Bt2 + ...

,

and a constraint V(x,y) = constant has to be handled by writing

~Bv ~ x dx where I is a Lagrange multiplier,

+

~v By

dy

and subtracting

32x +

Bt2

,

(25)

Figs.

(24) and (25), so that

~BV ~x +

... = 0 (26)

'B'2~ + I_~-~V+ ... = 0 Bt 2 A consideration constant)

By

too long to be included here shows that the constraint

results

(volume =

in

B2x

+

De

A

+ ay

Bx 2

De

+

... = 0

,

Bx 2

(27) @2y + ~ _

1

It is very important

~

~8

- ay

~8

that ~ is a function of t only.

+

The Galilean invariance

now obvious because when y = y' + vt, the vt term is absorbed plier,

... = 0

is

in the Lagrange multi-

so that I' = I + art. What other types of terms might one consider adding?

tional to v I boundary,

2

, where v± is the component of the velocity perpendicular 2

or Vll , and perpendicular

are G a l i l e a n

"Drag" terms propor-

to the boundary,

might be added.

invariant because we interpret v I to be m e a s u r e d relative

to the Such terms to the

150

fluid at rest at infinity. Our final equations are

_ _

.38 + A-~

32x ~t 2

+

ay -28 ~+

8vz 2 -38 ~+

3x 3x 3x i 20 30 30 ~2y + ~ - ~-~ ~Y --D Bv 2 ~ @t2

YVll 2 -3e ~

=0

@x 2 30 Y V l l --D

= 0

,

(28)

We determine e, B, and y so that the expansion of the solutions of Eq. (28) in a double power series in t and cos e agrees with the correct expansions to as high an order as possible.

agreement,

3 2

'

33 = - I-~

'

= - - - 15 ~

,

(29)

is attained through order t 6 (all powers of cos e relevant

The choice of the exponents

in vA

(2))

With

y agreement

(see Eq.

2

and Vll

2

to order t6).

are also dictated by requiring

this

and the power series expansion of A(t) is also fixed.

Some figures for the rise of the bubble as calculated shown in Appendix I (figures 1-6).

in this manner are

Torus time comes quite early, at t = 1.095 or

SO,

It is extremely difficult we have not done that.

to generalize

this idea to higher powers of t, and

But by abandoning

the invariance requirements

+ ~3y - ~2e+ ~ - ~ 38

33 2 -~ 38 By~v~

_ c - 38 ~

3x 2 @e

3x 38

and going back

to

.... A 32x

_

=

0

,

Bt 2

(30) i

3

@x 38 -

@x 38

33

=

0

,

3t 2 we can fix the coefficients

in the polynomials

A, B, and C so that agreement

151

through any order in t is attained.

The danger is that the polynomial A will

develop zeros, but in some orders and at some places on the boundary this danger is not realized.

Torus time tends to come later as the degree of approximation

is, as the order of the polynomials

A, B, C) is increased,

(that

perhaps at t = 1.2 or so,

as figure 7 exhibited in Appendix I shows.

B.

Estimate of torus time by means of ordinary Pade approxlmants.

The thickness

2£ of the bubble is, of course,

of the bubble and y at the bottom,

the difference

in y at the top

that is

2£ = y(t 2, O) - y(t2,~)

(31)

Explicitly, £ = 1 - 1.875 x i0 -I t 4 - 5.438 x 10 -2 t 8 - 1.362 x 10 -2 t 12 + ... These coefficients [2/2],

(32)

are tabulated to a large number of figures in Table I.

[3/3],... Pade a p p r o x l m n t s

The [i/i],

yield £ = 0.767 for t = i in good agreement with

the results obtained from the partial differential

equation approach.

In general,

they all give £ = 0 (Torus time) at t = 1.2 or so, also in good agreement with the partial differential

equation approach

(see Table II).

Now, it could be that the zero in £ is a zero located on a cut crossing the real t axis (see the location of poles and zeros in Table II).

To study this possi-

bility, we move the cut as discussed in section II.B, that is we write £ = F+ (t 8) + t 4 F_ (t 8) Various Pade approxlmants general, 1.5.

(33)

to F+ and F_ at various times are shown in Table III.

they suggest £ = 0 at some time later than t -- 1.2, perhaps

They also appear to have no undesirable

In

t = 1.4 or even

zeros or poles, and the approximations

seem to change quite smoothly as the order of approximation

increases.

Really, although it may appear in section II that an enormous number of terms have been calculated

in the expansions

for y(t2,e

= O) or y(t2,8

are reduced in Eq. (33) to functions whose expansion coefficients term in the original expansion, approximant may be studied.

= ~), we

are every fourth

and 36/4 = 9 so that only a [4/5] or [5/4] Pade

More work is needed to generate accurate series

152

expansions

to higher order;

such work will require much computer time on fast com-

puters. The partial differential

equation method is applicable

to the rise of a

cylindrical bubble, which develops a cap after torus time; that is, it looks like

at torus time. cylindrical

According

to Gary McCartor,

the series expansions needed for the

bubble are

4 i 7 28 73 r = i + t (~ - ~ cos 2~) + t6(- ~ cos ~ + ~ - ~ cos 3~)

8 = ~ + t 2 (sin ~) + t4(- ~

1

i01 35 sin 2~) + t6(- i--6~ sin ~ + ~ sin 3~)

h = ~1 t 2 + ~ 7 t 6 + ... , x = r sin ~ , y = r cos ~ + h .

(34)

Acknowledgments A great deal of the work reported here was done by Gary MeCartor and William Wortman of the Mission Research Corporation,

Santa Barbara,

California,

and Mary Menzel of the Los Alamos Scientific Laboratory.

References i.

For previous Davidson,

theoretical and experimental work see J. K. Waiters and J. F.

J. Fluid Mech. 12, 408 (1962) and 17, 321 (1963).

Figure Captions Figures i-6.

Shape of the bubble at various times.

The time and distance scales

are set by the initial rise during which the bubble behaves as though it were rigid rising llke ~i t 2 and by the initial radius r = i, respectively,

h(t) is the height

l 1 of the point x = 0, y = ~ y(0, t) + ~ y(~,t).

Figure 7.

Thickness of the bubble on axis of symmetry vs time.

estimates from calculations

based on Eq. (30).

These are best

153

APPENDIX I

\ _i

Immw,.~"--~ ~ R N~ / -2

J

1

-1

2 x

Figure 1 t=l h(t)

= 1.372

154

\ -I

t

i

-2

-I

I

2 X

Figure

2

t = 1.05 h(t)

= 1.406

155

\ -I

c)

J

J

-2

2 X

Figure

3

t = 1.08 h(t)

= 1.450

156

\ \ J

f

/

-2 2 x

Figure

4

t = 1.09 h(t)

= 1.493

157

\

,\

-I

,,i

I

(

# I

ir

-2 -I

2 x

Figure

5

t = 1.094 h(t)

= 1.527

158

II! \ \i i

\

/ J

f

f"

/

j

-2 X

Figure 6 t = 1.095

h(t)

- 1.530

159

2.0

I.O.

I. O

O Figure

TORUS TIME

7

T h i c k n e s s o f t h e b u b b l e on a x i s o f s y m m e t r y vs time. These are best estim a te s from c a l c u l a t i o n s b a s e d on Eq. ( 3 0 ) .

160

Table I

Accurate Coefficients for Eq. (30). The order in which these are to be read is apparent from Eq. (30).

1.0

+00

-1.875

-01

-5.438456632653055-02

-1.362446243532089-02

5.11511872461845 -03

1.151623084926745-02

1.012249732304565-02

4.804358858144425-03

-9.952368481864343-04

-4.791099632885595-03

-5.506523714786576-03

-3.513584047019944-03

-2.120294683714934-04

2.692173228391798-03

3.926479535644956-03

3.128540287264697-03

8.953856051941572-04

-1.579142125257331-03

-3.09449704894324 -03

161

Table II Pade Approximants

formed from Series of Eq. (30).

Half the thickness of the bubble on the axis of symmetry is tabulated.

t

3/4

4/5

7/8

8/9

0.8

0.91347018

0.91347002

0.91347014

0.91347014

i. 0

0. 76469807

0. 76420450

0. 76424654

0. 76685838

i.i

0.61689611

0.61116700

0.59803286

0.66098736

1.2

0.22093630

0.18768951

1.3

<0

Location

< 0

0.37334144

< 0

< 0

(in t 4 plane) of the zeros and poles of the 8/9 Pade Approxlmant

Roots of Denominator

Roots of Numerator

0.75606093 ± i 0.64871509

0.75563195

± i 0.64727296

0.92782164 ± i 0.71306521

0.92546957

± i 0.70349188

1.3703819

1.3489190

± i 0.73761275

2.9883484 14.699468 -139.05115

± i 0.78104964

2.3641068 9,7374222

162

Table III

Pade Approximants to F+ (see Eq. (33))

213

3/4

415

t

011

i12

0.8

0.99095828

0.99128592

0.99106605

0.99106605

0.99106605

1.0

0.94842056

0.94792005

0.95637556

0.95636353

0.95665642

i.i

0.89559338

0.89475280

0.92423556

0.92412823

0.92994608

1.2

0.81047544

0.80916506

0.86835050

0.86804717

0.89101745

1.3

0.69269749

0.69092524

0.77897013

0.77845245

0.82240395

1.4

0.55475379

0.55272305

0.6602585

0.65956997

0.72114121

1.5

0.41774289

0.41575710

0.52715917

0.52639470

0.59747092

t

110

2/1

3/2

413

514

0.8

0.99087578

0.99109132

0.99106605

0.99106609

0.99106605

1.0

0.94561543

0.94039026

0.95619283

0.95616968

0.95664655

1.1

0.88342185

0.87617212

0.92087900

0.92069538

0.92905483

1.2

0.76615632

0.75356214

0.84900828

0.84847787

0.88127337

1.3

0.55636838

0.53389113

0.71094835

0.70988429

0.78092139

1.4

0.19739852

0.15790015

0.46976099

0.46784701

0.59723664

1.5

-0.39381695

-0.46139448

0.07036295

0.06707928

0.28885363

Roots of Numerator (4/5)

Roots of Denominator (4/5)

0.1077 ± i 1.053

0.1053 ± i 1.054

0.1410 ± i 1.975

0.0982 ± i 1.991 -36.932

Roots of Numerator (5/4)

Roots of Denominator (5/4)

0.1205 ± i 1.048

0.1183 ± i 1.050

0.2617 ± i 1.938

0.2240 ± i 1.962

35.761

163

Table III (continued)

Pad~ Approximants to F

1/1

t

212

(see Eq. (30)) 313

414

0.8

-0.18950191

-0.18944294

-0.18944316

-0.18944316

1.0

-0.19488349

-0.18943211

-0.19004561

-0.18978128

i.i

-0.19788633

-0.18046367

-0.18498208

-0.18047145

1.2

-0.20014065

-0.17087111

-0.18157475

-0.16572802

1.3

-0.20157703

-0.16549046

-0.18068947

-0.15482310

1.4

-0.20242239

-0.16291690

-0.18070385

-0.14917041

1.5

-0.20290741

-0.16165605

-0.18088619

-0.14641120

Roots of Numerator

(4/4)

0.09447 ± i 1.125 -0.3624

± i 2.801

Roots of Denominator (4/4) -0.09945 ± i 2.511 0.09920 ~ i 1.104

RATIONAL APPROXIMATIONS OF THE BLUNT-BODY

TO THE SOLUTION

& RELATED PROBLEMS

L. W. SCHWARTZ DEPARTMENT OF APPLIED MATHEMATICS UNIVERSITY OF ADELAIDE~ ADELAIDE~ I

S.A.

Introduction A body travelling at a speed greater than the speed of sound in a compressible

fluid will, in general, be preceded by a shock-wave.

This "bow-shock" wave sepa-

rates the fluid region into two parts: the fluid ahead of the oncoming body is unaware of the body's presence, while the fluid between the shock and the body is affected quite strongly.

The problem of finding the shape of the bow-shock,

details of the fluid motion and the resulting pressures

the

and forces on the body is

usually referred to as the supersonic blunt-body problem.

Interest in the problem

has been strong for the past thirty years; currently the major application

is to the

design of spacecraft. While the most general problem involves viscous, unsteady, equilibrium, heat and mass transfer and other considerations, that the solution of the "classical" blunt-body problem, of a thermally and calorically perfect compressible neglected,

chemical non-

experience has shown

involving the steady flow

fluid with viscous effects

is adequate for many purposes.

The results presented in this paper employ an inverse method of solution to this problem.

In an inverse method, the shape of the detached bow-wave

values of the flow variables

is prescribed;

immediately downstream of the bow shock can be specified

in terms of the free-stream parameters by means of the Rankine-Hugoniot relations.

These values serve as the initial conditions

partial differential

the

equations that characterize

which produces the given shock-wave

(shock jump)

for the system of nonlinear

the flow.

The shape of the body

is determined as part of the solution.

The

solution of the so-called direct problem, where the body shape and free-stream conditions

are specified initially,

Because exact initial conditions

requires,

in general,

an iterative

computation.

can be specified only at the shock-wave,

rather than

at the body surface, all solutions to the blunt-body problem are, of necessity, indirect. Numerical solutions to the inverse blunt-body problem have met with a good measure of success,

especially for axisymmetric

usually involve a finite-difference, many instabilities

flow problems.

marching-from-the-shock

plague these numerical

solutions,

However,

caused both by the particular

numerical procedure employed as well as by the mathematical problem.

These solutions

technique.

nature of the inverse

An important example in the latter category arises because the flow behind

any detached shock must be at least partially governing equations are elliptic

in character.

subsonic.

In the subsonic region the

An elliptic system of equations with

initial (Cauchy) data is known to be mathematically

ill-posed. 1

Slight errors in the

166

initial data can lead to unacceptably An alternate,

semi-analytic,

via a series expansion method. to be a (multiple) efficients.

large deviations

in the solution.

solution to the blunt-body problem can be obtained

Here the solution for any flow variable is assumed

Taylor series in the space coordinates with undetermined

co-

If the shape of the bow shock is given by an analytic relation, values

of the flow variables

and arbitrarily many derivatives

of these variables running

along the shock may be found at once from the Rankine-Hugoniot system of differential

yield arbitrarily many derivatives the derivatives

relations.

The

equations governing the motion may then be manipulated to of the variables

normal to the shock in terms of

along the shock.

It is interesting to note that the first attempts to solve the blunt-body problem,

in the late forties and early fifties, employed this series approach 2'3'4

Their results were inconclusive,

however, partly because they computed their series

by hand and thus were able to extract only five or six terms. discovered,

Moreover, Van Dyke 5

in 1958, that a limit line appears in the upstream analytic continuation

of the flow, that is, "ahead" of the shock.

This limit line, while only a "mathema-

tical fiction", will very frequently lie closer to the shock than does the body and will therefore determine the "radius" of convergence successful

of the series expansion.

solution based on the series expansion method must therefore

an analytic continuation procedure. for this purpose.

A

incorporate

Van Tuyl 6 suggested the use of Pad~ fractions

This idea was used by Moran 7 along with automated computation

the series coefficients

to produce successful blunt-body

of

solutions for problems with

axial symmetry. In this paper we report on an extension of Moran's work to non-symmetric configurations.

We also discuss two related problems which, because of their

relative simplicity,

allow one to elucidate certain features of the analytic struc-

ture of their solutions.

We make use of a battery of techniques,

Sykes plots 8, Euler transformations

and "series completion";

including Domb-

collectively these

procedures have come to be known as "the method of Van Dyke" in acknowledgement

of

his pioneering work in this field. 9'I0. II

The Taylor-Maccoll

Conical Flow Solution

The problem of axially symmetric supersonic

flow past a right circular cone was

solved numerically by Taylor and Maccoll in 193311. have been produced by Kopa112,

More refined numerical solutions

Simms 13'14, and others.

Provided the free-stream Mach

number is high enough to permit an attached shock which is a coaxial circular cone, the flow quantities will depend only on one space coordinate, Though the analogy is by no means complete, the Taylor-Maccoll ered to be a model "one-dimensional

the azimuthal angle e. problem may be consid-

blunt-body problem" and, as such, merits study

before proceeding to more general cases. The problem is formulated in spherical polar coordinates with the axis of the cone corresponding to the origin of the azimuthal angle e.

Let w,u denote the

167

velocity components

in the r and @ directions,

They are determined

respectively.

as

the solution of the 2nd order nonlinear equation: (aZ-uZ) ~d--~ + w +a z w + ~-~ cot @

(z)

= 0

where u

dw --

=

de and a2 + ~ 2

(u 2 + w 2) = const = 7-i V 2 2 'm

follows from the conservation

of total enthalpy.

sound, y the ratio of specific heats, and V

Here a=a(@)

is the local speed of

the velocity the fluid would attain if

m into a vacuum.

allowed to expand adiabatically

(2)

The solid cone surface is character-

ized by zero normal velocity: u ( e c) = o . Thus if w e = W(Oc) is specified,

the equations may be integrated numerically

the solid cone until a value of 8 is attained where the velocity components compatible with the Rankine-Hugoniot general blunt-body problem,

relations.

Thus this problem,

requires the satisfaction

The solution of the direct problem,

from are

like the more

of mixed boundary conditions.

i.e. free-stream Mach number and solid cone

angle initially specified, will require an iterative computation• In a later paper Maccol112 assumed,

sought a power series solution for the flow.

He

in effect, a solution of the form

w = Z w.s a j=o 8 where 8=6-8

(3)

and substituted this expansion

in (i).

Choosing as his dimensionless

c

dependent variable the quantity W/Vm, he obtained the series coefficients O(E5).

These coefficients

and Wc/V m.

In Maccoll's

ec,Y,

formulation the parameter y first appears in the 0(84 ) term,

as a factor i/(y-l), thus indicating a very strong dependence specific heats, particularly

for polyatomic

on the ratio of

gases.

In a recent paper, this author 16 has shown, via a small modification Maccoll's procedure,

that the apparent

strong dependence on y is spurious.

the cone velocity w

as the reference,

one obtains

c

w w

through

are in general functions of the three parameters

= 1-82 +

cot @ 7--

-

+

c

c

+ cot Oc ( ~ + 40 M2 + 4 cot2@c) g5 20 ~ c

84

to Choosing

168

_~6 [ 2 30 [ 3

+ 19 -4-

cotZ6

+ M2~ ( -14 ~c + 839 c°t2

through order ~6.

+ 5 cot48 c

c c ) + M4c

+''"

(4)

Here it is seen that y appears only in the E6 term and moreover

that the effect on this coefficient will be quite minor.

of large changes in y (from a physical viewpoint)

This suggests that the ratio w/w

of 7 and hence the nature of the gas. conjecture.

(~--+ 4(7-1I]

may be sensibly independent c Exact solutions of (I) appear to bear out this

Indeed the greatest change in the normalized velocity ratio at any point

in the flow when 7 is increased from 1.4 to 5/3 (i.e. diatomic versus monatomic is about one part in 3000. has been demonstrated

gas)

It seems fair to say that a type of "quasi-similarity"

and that the only sensible dependence on 7 derives from the

shock boundary condition. The exact solutions used for these comparisons were obtained from high-order series solutions recast as rational fractions.

Machine computation using the series

approach is at least as efficient as standard finite difference methods for this problem.

These series solutions can be made to yield information concerning the

nature and location of important

singularities

through the use of Domb-Sykes plots.

This graphical extension of the familiar D'Alembert following observation:

ratio test is based on the

If

~ k(So ± s) ~

a # 0,i ....

f(s) = Z anen = [ k(e0 + E) ~ log n= 0 -

(eQ + S) _

~ = 0,i,..

(5a)

.

then a-~-n = $ an_ I

~ [ I _ i+___~ ] g0 n

and hence that the ratios of series coefficients, on a straight line.

(5b)

when plotted versus i/n, will lie

Thus if we make such a plot using the coefficients

and if the points ultimately tend towards a straight-line

in any series

asymptote, we can infer

that the closest singularity is of type (5a).

The plot yields estimates of the

nature ~ and location g0 of this singularity.

We show such a plot formed from the

coefficients

in (3)in Fig.l. A singularity

at g0 = -.156 is indicated.

corresponding

to a 3/2-power branch point

Here the singularity closest to the surface of the solid

cone lies "buried within" the cone to a "depth" of about 8.9 degrees.

In this case

the shock lies closer to the cone surface than does the branch-point.

Hence for

@

= 40 o and M c = 2, the shock will be within the series expansion from the solid

c cone.

Very often this is not true, as for a slender cone at relatively low free-

stream Mach number.

The oscillations

in the plot of Figure i are caused primarily by

another singularity lying somewhere ahead of the shock. secondary singularity

Information about this

can be obtained after first mapping away the one at g0 by means

169

of the Euler transformation =

A Domb-Sykes

~

(6)

~-~o

plot of the series for w (~) reveals that the outer discontinuity

also a 3/2-power branch-point. singular lines correspond

This appears to be true in general;

to the envelopes

is

the two

of right-and left-running

characteristics

where these both exist. Using the information that we have obtained about the nature of the important singularities

of w and the virtual lack of dependence

examine the accuracy of a low-order Pad@ approximant coefficients.

Since we k n o w that the important

solution are 3/2-power branch-points, approximation.

The series

solution

series is then cast as a rational

on y, it is instructive

singularities

a simple modification

of the Taylor-Maccoll

can result in an improved

(4) is raised to the 2/3 power;

fraction;

finally,

to the 3/2 power to recover an approximation fraction is converted to a branch-point

to w.

to

formed from the first few series

the rational

the new power

fraction

is raised

Thus each zero of the rational

of the proper kind.

Using only four terms of

(4), we form

c)

where the mnemonic

(w2/3)} 3/2

notation [ 3/1] signifies

and one pole.

w3 and w~ are coefficients

ate normalized

profiles

(?)

a rational

from (4).

fraction with three zeroes

Figure 2 compares t h e approxim-

obtained from (7) with exact results

angle at both a high and a low surface Mach number.

for a cone of 5 o half-

The greatest

error is about 4%

for the low Mach number case.

For M = i0, on the other hand, the m a x i m u m error is c Two other case comparisons, for e = 400 , likewise show agreement to c within I percent. (See reference 16). less than 1%.

The relatively explained,

large error for the small Mach-number

and, in principle,

and cone-angle

removed b y a method of "partial

(4) we observe that each coefficient

w. J

contains

case can be

series completion".

a term

(_i) j+l cotJ-2ec j for j12,3...

(8)

From the Taylor series expansion

log(l+~ cot @c) _ cotZ@

c

~

one may infer the presence of a logarithmic Assuming that all the coefficients

= Z

cot @

c

(-l)J+ic°tJ-2@c

j=2

singularity

Ej

(9)

j

at ~ = -tan @ . c in the series contain terms of the form (8), the

In

170

series may be partially completed and the logarithmic singularity removed through the use of (9).

For @ <
It should be emphasised that (7) is a rational approximation derived from

analytic principles, rather than a curve fit and should therefore by systematically improved when carried to higher order. III

Hypersonic Flows with Parabolic Shocks Perhaps the simplest class of genuine blunt-body flows can be generated by a

parabolic or paraboloidal shock wave placed symmetrically in a uniform free-stream of infinite Mach number.

Because the free-stream Mach lines have zero slope, the para-

bolic and paraboloidal shocks have the correct asymptotic behaviour far downstream and thus they may be reasonable approximations to the shock waves produced by actual bodies during the early stages of atmospheric re-entry.

Because of its relative

simplicity, the axisymmetric paraboloidal shock-wave especially, is often used as a test case for numerical solutions. While a series solution to the blunt-body problem in two space dimensions will, in general, require double Taylor series expansions for the dependent variables, this was not found to be necessary for the present cases.

By using orthogonal coordinates

it becomes possible to obtain results for each variable as a single Taylor series in the coordinate normal to the shock, whose coefficients may be computed recursively as exact functions of the other space coordinate.

Thus, there is no drastic loss of

accuracy away from the stagnation region; if it happens that the dependent variables are well-behaved functions throughout the entire shock layer, it should be possible to obtain an effectively exact solution to the entire problem rather than simply a good solution near the nose of the body. stream in the axisymmetric use.

We will present results valid far down-

For plane flow, on the other hand, a limit line

appears within the shock layer, suggesting the presence of an imbedded shock. The problem may be formulated in orthogonal coordinates as in Fig. 3.

For

purposes of this section the angle of attack in the figure is always equal to zero. Taking the shock nose radius as unity, we introduce parabolic coordinates according to

x = ½[i

+ ~2_

(i + s)2] (io)

r = l~l ( i + ~ ) The shock is seen to c o r r e s p o n d to ~ = O. u s i n g D~,V~, and p V~2

Dimensionless v a r i a b l e s m a y be f o r m e d

as the reference quantities.

The field equations b e c o m e

171 continuity:

{ gV(l+~s) [g2+(~l+E)2] ½pu }g +{ ~V(I+vg) [.~:~+(l+g) 2]

(lla)

2PV }e = 0

- momentum: ~v 2

uu~ -

+v

( u~+ ( l + g ) u )

{2+ (il+s) 2

+Tp~ : 0

(lib)

~2+ (l+g) 2

g - momentum: (l+e)u2

~2+(1+E)2

vv

+u

(v~+

~v

)+Tpe

~2+(1+~)2

=

0

(llc)

uS+vT = 0

entropy:

(lid)

where the additional variables T,S, and T have been introduced to remove cubic products and hence reduce the number of nested D0-1oops

Here

T = i/p

(llc)

S = pp~ - ypp~

(11f)

T = PPE - YPPs"

(lZg)

V:0 and i correspond to plane and axisymmetric that the system of equations

in the computer program.

flows, respectively.

It can be shown

(ii) can be satisfied by assuming each independent

able to be an infinite series in g where coefficients The density series, for example,

are polynomials

vari-

in X=I/(I+~2).

is of the form

J Z

p(s,X) = ~ j=0

k=0

Pjk

s

i

X

k

(12)

and thus the solution will be in the form of a triangular array of coefficients. When expansions

of the form (12) are substituted

in (ii), the differential

equations reduce to algebraic relations where the unknown coefficients

at each stage

are computed reeursively as sums of quadratic products of coefficients

of lower

order. and u

A series solution for the stream function is developed from the series for p by term-by-term

integration

of

¢E : ~(l+~)[~2+(1+~)2] Additional

½pu

details of the solution procedure may be found in reference 17.

series solutions

for both the plane and axisymmetric

Power

cases with y=l.4 were found to

0(@ 24) in about i minute of computer time for each case on the IBM 360/67. efficients

(1B)

The co-

for the first few orders may be recognised as rational numbers from their

repeating decimals.

For plane flow, we obtain for the stream function

£ = i + 6~ + (45-35X)S 2 + [BB0-(lB55/B)X + 140X 2] ~ + [4965/2 - (19055/4) X + (24175/9)X 2 - 455X 3] g 4

(14)

+ [186h8 - (92153/2)X + (681137/18)X 2 - (I304347/108)X3+IBI6X 4] g 5+ . . . . Similar expressions

for the axisymmetric

case were calculated to 0(e 3 ) by Van Dyke 18

172

and a fourth term was added by Moran 7. The results which follow were computed with [12/12] Pad6 approximants formed from the 24th order solution.

The degree of convergence of the Pad@ table is excell-

ent; the body (given by the streamline @=0) can be found to I0 decimal-place accuracy throughout the subsonic region for the axisymmetric case. converges less well for the plane case. only 8 place accuracy.

The Pad@ table

The stagnation point could be located to

She convergence drops off dramatically as one enters the

supersonic region, however, for this case.

As an additional check, the pressure at

the stagnation point, computed as a [ 12/12] approximant, to ii and 7 places respectively for the two cases. to locate the body from the solution for @.

agrees with the exact value

Two separate methods were used

The first method involves factoring the

ntunerator of the Pad@ approximant for @(~;X) and identifying the body as the leading negative real zero for various constant values of X.

The body may also be found by

reverting the series

@(c;X)

= @o(X) + @~(X)C + @ 2 ( X ) C 2 + . . .

c(@;X)

=

(15)

to obtain

c~(×)(@-%) + c2(×)(@_@o)2+...

and then recasting (16) as a Pad6 fraction with @=0. given order is about the same for the two methods.

(16)

The accuracy of a solution of The series reversion method,

however, is more efficient computationally and is more convenient for drawing streamlines. Some significant features of the results are shown in Figures

(4) through (7).

Figure (k) shows the body shape and flow-field which support a parabolic shock for Y=7/5.

The body shape was found from the expansion for the stream function, as were

the typical streamlines in the figure.

Note that the streamline @=-.I lies "within"

the body, i.e. in the analytic continuation of the shock-layer flow field. The upstream limit line, corresponding to an envelope of characteristics as in the Taylor-Maccoll problem, and the continuation of the streamline @=.3 ahead of the shock are latent in the analytic solution which is, of course, "unaware" that the shock represents a physical discontinuity.

As expected, the upstream limit line lies

closer to the shock than does the body which indicates that the various series expansions will be divergent in the vicinity of the body.

Also shown is the sonic

line and its upstream and downstream analytic continuations.

On this line the local

Mach number, given as

M2 assumes the value unity.

~

2

= y~

~-i

(17)

For the plane case, the sonic line is clearly a closed

curve, touching the upstream limit line on the line of symmetry.

The surprising

feature of Figure (4) is the presence of a limit line within the shock layer. this line the density and other flow variables have infinite gradients.

On

Unlike the

173

upstream limit line, which exists only in a fictitious flow region, this limit line lies within the physically important region between the bow-shock and the body. Thus, in order to find that portion of the body lying behind the downstream limit line, it will be necessary to permit discontinuities of the field variables in the shock layer. A secondary, or embedded, shock might be inserted just upstream of the limit line and the solution could then be continued from the downstream side of this shock. doubtful, however, that such a modification is unique.

It is

It appears clear that no two-

dimensional body can be found that will possess a parabolic bow-shock and have a flow field free of other discontinuities. Figure 5 shows the body pressure distribution for the plane case.

The circles

in both figures 4 and 5 represent a numerical solution obtain with the m~rching-fromthe-shock technique of Lomax and Inouye 19.

Their solution is graphically indistin,

guishable from the high order Pad@ - fraction results. only [ 2/2] fractions are indicated as dashed lines.

The approximate results using

This reasonably good agreement

indicates that the earliest attempts to solve the blunt-body problem, using only hand-calculated series, might have been successful had the knowledge of the upstream limiting envelope existed at the time.

Similar results for the axisymmetric case 17

show even closer agreement. The two limit lines shown in Figure 4 were found through the use of Domb-Sykes plots.

Three such plots for the density series, all indicating upstream singulari-

ties, appear in Figure 6.

For ~=0, corresponding to the line of symmetry, the plot

clearly indicates a square-root singularity at the critical value E* + = .120.

For

~=0.5, the asymptote has been drawn so as to indicate a square-root though other exponents are surely possible. extrapolation,

Because of the uncertainty latent in any graphical

the results of the third plot, for ~=~, are particularly reassuring.

Here the singularity exponent, obtained as a best-fit to the plotted points, is e=-1.88.

It was later determined that the system of equations

considerably in the limit ~

(ii) simplifies

and that the reduced system possesses an exact local

solution near a singular point ~* of the form P ~ (S-S*) -3/(3-Y) For Y=7/5, the exponent is exactly equal to -15/8=-1.875.

The inner limit line was

found using Domb-Sykes plots after suitable Euler transformations were performed. The axisymmetric case, unlike the two-dimensional flow, appears to possess a completely analytic shock layer. to great distances downstream. nose radii downstream.

It is therefore possible to compute the body shape Figure 7 shows the afterbody computed to 200 shock

It is in substantial agreement with Yakura's 20 asymptotic

solution 1

rb ( ~--/ x ~ --=1.392 Rs

s

which he obtained by modifying the blast-wave analogy to account for the entropy

(lS)

174

layer.

A similar hut higher-order numerical solution of Syehev 21 is also shown. Our

computed afterbody agrees well with Sychev's although the convergence of the approximants is less good past x/R s = i00.

This loss of convergence is related to our use

of parabolic coordinates; our body degenerates to the branch cut ~=-i in the downstream limit. IV.

Plane blunt-body flow at arbitrary Mach number and angle of attack The method of Section III has been substantially generalized to treat both

finite free-stream Mach numbers as well as arbitrary angles of incidence.

While

numerical methods have been largely successful in the treatment of symmetric configurations,

it is only quite recently that, with the use of more general finite

difference techniques and immensely more powerful computers, that promising solutions to the more general asymmetric problems have been solved even to engineering 22 accuracy While the method of the present section treats only two-dimensional configurations,

it produces solutions of very great accuracy that can serve as useful

test-ease comparisons for finite difference solutions.

Moreover, no numerical

solution appears to be of sufficient accuracy to adequately resolve the question of whether the maximum-entropy streamline wets the body surface in asymmetric flows. In this section we show that the stagnation streamline is displaced from the maximum entropy streamline

by a relatively small, but by no means negligible amount.

We

also continue the discussion of limit lines and the domain of validity of the inverse method of solution. Following Van Dyke and Gordon,

23

the bow-shock is described by

y2 = 2R x - Bx 2. s

(19)

Here B, the so-called shock bluntness, is a measure of the eccentricity.

B=0 repre-

sents a parabola and B>I, an oblate ellipse, for example. An orthogonal coordinate system with the shock corresponding to E=0 is introduced by setting x

= I

~ {i - [(i-S~2)(l + 2Be + Be2)] ½}

(20a)

y = ~(I + E)

and

which is a generalization of the transformation

(2Oh) (i0) of the last section.

The initial conditions are specified at the shock via the Rankine-Hugoniot relations: (y+l)M~ sin20

(21a)

o[o,[(e)l-(y-i)M~ sin20+2 2

(21b)

u [ o~[(e) ] = cose

(21e)

p [ o,[(e)

v [o,[(e)]

] =

y--~ sin29 - (y+l)yML

= (y-1)M~ sin2e+2 (V+I)M~ s i n 2 e

sinS.

(21d)

175

Here the uniform free stream of Mach number M

is inclined at an angle ~ to the x-

axis as in Figure 3 and @ is given by cosO =

[ cos~ + (I-B[2) ½ sins

(22)

[l+(l-S)~2 ] ½ Equations

(21) are the initial conditions for the gasdynamic equations which, in

(g,~) coordinates, become

R(2)([) $[C[ 2( + (l+e) 2] (0u)[ + C~ pu lI

(23a)

+R(1)(~) {[C~2+(l+~) 2] (~v)E+(l+E)~v} =0, R(2)(~) {tuu~ + Tp~I leo + (l+~)~l - c~v~ } (23b)

+ R(1)(~) {~j cO + (l+~)~l + (l+~) uv} : o, (23c)

+R(1)(~) ~ [ ~ . + ~p~ [cC + (l+~)~l - (l+~) u~}= o, and R(2)([) [u(pp[- yp~)] + R(1)(e)[v(pp~- Yp%)I

: 0.

(23d)

Here C=I-B,

R(1)(~) = [1+B(2~+~2)] ½ and

~(2)(~) = (l_B~2)½. While in the cases treated in the preceding section, it was possible to express

the solution for each dependent variable as a single power series in g with coefficients which are exact functions of [, in the present more general case, this was not practical. as y.

Here the series coefficients involve the parameters M ,B, and ~ as well

Moreover~ the dependence on ~ is sufficiently complex that, in the interest of

computational efficiency, the single series must be abandoned in favour of a double power series solution.

Thus, for the density, we assume an expansion of the form p(e,~) =

Z

[

j =0

k=0

Pjk

sJ[ k

with similar expressions for the other dependent variables.

(24) By inserting these

expansions in the system (23), performing the series multiplications,

and equating

coefficients of terms of like order in both g and ~, recurrence relations may be derived for the elements 0jk,Pjk, etc.

Additional details may be found in reference

24. It is a feature of the recurrence p r o c e d u r e that the number of terms in ~ which may be found exactly at each stage decreases as the order of g is increased.

Thus if

176

the computation is started by expanding the Rankine-Hugoniot relations

(21) as a

series in ~ up to a given order N, there will exist only a sufficient number of relations to determine the first-order coefficients in g to order N-I in ~. arly, at the next stage, only terms up to 0(s2~ N-2) may be found. will be, therefore, a triangular array of coefficients,

Simil-

The final result

for each variable, which

includes all elements of total order, that is the sum of the exponents of E and ~, ~N. A formal extension of the Pads concept to cases involving two independent variables may be effected by grouping all terms of the same total order together. Consider f(E,~) to be a typical dependent variable and form ~

f(£,~) = Z

~

i=0 j=0

f.. si~ j =

~J

~

k

~

Z

fk_j,jgk-J~ j

k:0 j=0

(25)

= ~ sk ~ fk-j k=0 j=0 'J which is a single power series in £ whose coefficients are polynomials in (t/s), Standard methods for treating single series, e.g. Wynn's epsilon algorithm 25, may now be used.

Note that this method is fully compatible with the triangular arrays

produced by the recurrence formulas.

Various other techniques for treating multiple

power series have been devised by Chisholm 26 and his group. The double power series, as derived, are expansions about the shock apex. Clearly, any other point on the shock could serve just as well. point, where the shock is normal to ~he free-stream,

The maximum-entropy

is perhaps a more logical choice

and might result in more uniform convergence in the results which follow. The body produced by a given shock is located by first finding the stagnation point, which is a saddle-point of the stream function.

Starting with an initial

guess, the desired condition u=v=0 may be approached via a Newton iteration:

(vu~ - uv~) i

(26a)

Ei+ I = C.l +

(usv ~ - veu~) i (uv s - vus) i ~i+l = ~i +

(26b) (uev ~ - v u~) i

Note that the partial derivatives need not he evaluated numerically but may be f o u n d b y term-by-term differentiation of the power series which is then recast as a Pad@ fraction.

The body shape can then be found by finding points which lie on the

streamline passing through the stagnation point.

Other streamlines as well as the

sonic lines can be found using Newton iteration. Typical results are shown in Figures 8 and 9.

Various total orders of solution

are shown for each case.

A parabolic shock is set at i0 ° incidence to a free-stream

with M

The 30th order solution, computed with [15/15] approximants

= 2 in Figure 8.

and the procedure of (25), is sensibly exact.

Notice that the maximum entropy

177

streamline passes below the streamline that wets the body surface.

Since the flow-

field is rotational, the stagnation streamline does not intercept the body exactly at right angles.

It exhibits a small characteristic bending away from the maximum

entropy streamline.

In Figure 9, a flat-faced /~-to-i ellipse is set at i0 ° inci-

dence with infinite free-stream Mach number. passes above the one that wets the body. by 30th order. correctly.

Here the maximum entropy streamline

The solution here also is fully converged

A high order solution is required here to locate the sonic lines

It was decided to plot the location of the limit lines for this case to

see whether the failure of the low order solutions can be better understood.

As

before, Domb-Sykes plots are used, coupled with suitable Euler transformations when necessary.

The limit lines have been mapped out in Figure i0.

The limit lines

appear both within the supersonic position of the shock layer and also in the "fictitious" upstream analytic continuation of the flow.

The line opposite the

shock exhibits a square-root singularity (6=½) as do the limit lines within the shock layer.

These are augmented by singular lines with 6=-0.43 placed roughly

symmetrically in the analytic continuation. A on the shock.

Lines AB&AG can be traced back to point

Here the shock slope = i0 °, the free-stream angle of attack.

At

this point the shock becomes a Mach line and hence it is plausible that it should be a singular point.

Note how close the limit lines come to the sonic lines which

explains the need for high order solutions in these regions.

Because of the close

proximity of the limit lines, obtaining a starting line for a method of characteristic solution would appear to be a practical impossibility for this case. Various other features of this asymmetric blunt-body solution are explored in reference 24.

Cases up to 30 o angle-of-attack have been treated.

The displacement

of the maximum entropy streamline from the stagnation streamline appears to increase somewhat faster than linearly with angle of attack. number is lowered.

It is also increased as the Mach

The body pressure distribution can also be found to good accuracy.

The series expansion method would app@ar to merit further study as a possible alternative to finite-difference techniques for the asymmetric problem.

Three

dimensional solutions are also possible where the substantial algebra required to derive the recurrence relations for the series coefficients might be alleviated through the use of a symbol manipulation language such as FORMAC. Acknowledgement Most of this work was performed while the author was a post-doctoral associate at the NASA Ames Research Center.

Financial support during this period was provided

by the National Research Council. References i.

J. S. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations,

(Yale Univ. Press, New Haven, 1923).

178

2.

J. Dugundji, Jour. Aero. Sci. 1 5 6 9 9

3.

H. Cabannes, ONERA note technique no. 5 (1951).

(1948).

4.

C. C. Lin and S. F. Shen, NACA tech. Note 2506 (1951).

5.

M. D. Van Dyke, J. Fluid Mech. ~, 515 (1958).

6.

A. Van Tuyl, J. Aero. Sci. 2_~7559 (1960).

7.

J. P. Moran, Ph.D. Thesis, Cornell Univ. Ithaca, New York (1965).

8.

C. Domb and M. F. Sykes, Proc. Roy. Soc. (London) A240, 214 (1957).

9.

M. D. Van Dyke, J. Fluid Mech. 4 4 3 6 5

i0.

M. D. Van Dyke, Quart. J. Mech. Appl. Math. 27 423 (1974).

ii

G. I. Taylor and J. W. Maccoll, Proc. Roy. Soc. London A139 279 (1933).

(1970).

12

Z. Kopal, Tech. Rept. No. i. Mass. Inst. Tech. (1974).

13

J. L. Sims, NASA SP-3004 (1964).

14

J. L. Sims, NASA SP-3078 (1973).

15

J. W. Maccoll, Proc. Roy. Soc. London A159 459 (1937).

16

L. W. Schwartz, Jour. Appl. Math. Phys.

(ZA/MP) to appear 26 (1975).

17

L. W. Schwartz, Phys. of Fluids 17, 1816 (1974).

18

M. D. Van Dyke in Fundamental Phenomena in Hypersonic Flow (Cornell Univ. Press,

19.

H. Lomax and M. Inouye, NASA Tech. Report R204 (1964).

20.

J. K. Yakura, in Hypersonic Flow Research (Academic, New York, 1962) p.241.

21.

V. V. Sychev, Prik. Mat. Mekh. 2_~4, 518 (1960)[J.

Ithaca, New York, 1966) p.52.

Appl. Math. Mech. 24 756

(1960)] . 22.

A. Rizzi and M. Inouye, A.I.A.A.J.

i_~i, 1478 (1973).

23.

M. D. Van Dyke and H. D. Gordon, NASA Tech. Rept. R-I (1959).

24.

L.W.

25.

P. Wynn, SIAM J. Numer. Anal. ~,91 (1966).

26.

J. S. R. Chisholm, Math. Comp. 27, 841 (1973).

Schwartz, Physics of Fluids to appear 1975.

179

LIST OF FIGURE CAPTIONS Figure i

Domb-Sykes plot for radial velocity series coefficients (0c=40°, Mc=2, Y=7/5).

Figure 2

Radial velocity component (ec=5° , Y=7/5)

Figure 3

Blunt-body coordinate system.

Figure 4

Flow field, parabolic shock, M =~, Y=7/5.

Figure 5

Body pressure distribution, parabolic shock.

Figure 6

Domb-Sykes plot of density series, upstream limit line~ parabolic shock.

Figure 7

Afterbody coordinates, paraboloidal shock, Y=7/5.

Figure 8

Flow field, parabolic shock, M =2, ~=i0 °, 7=7/5, ooo N=20, o D O N=26, --N=30, --- Maximum entropy streamline.

Figure 9

Flow field, elliptic shock, M =~, ~=I0 °, Y=7/5.

Figure 10

Location of limit lines for the flow field of Figure 9-

180

-8-

W=~- Wrl E h n=o

-7

-6 -5 wn ' --4

Wn-i

-3 --2

- -

0

I

I

1/50 1/20

I

I

1/10

I/5

I/n

Fig. i

--

MC/NE EXACT EQNo(7)

10

//~

[]

wc-w -

,~'/

j,

--/IV

J

0

O-O c

8s -8 c

Fig. 2

I

181

X

i C~

182

._J
--

II

Z

° ---w

:;z . _J ...I

l

m

3d

(J --ILl Z z

I II

O-CO1

h9 o

.r{

II

II

I:E X

0

o

I

L__~

I

I

I

I

I

I

I

IoJ

~-

~

~

~.

~

~

--

o~"

(/)

rr

183

Q

1

e4

tO

_

O3 OJ n "

~

ar~ no

I

,, 0

_

~

I

I

I

I

t',.-

f,O

1.0

I

I rO

0

LC~

184

0u o

tl

o

0 t •

II

II

II

o

d

"

t~ ®

O,J

.

,o,-/

,:,.-/

/

7

o

.ff 9

tO

p

il

~-I-

d

•

||

•

iI

/ /

/

/

/

ii \

\

/

I

tr) e

Lr)•

•

,

\

/

\

/,

/

/

/

/

/

/

0

oo

~: i

L() E

-o~

I

I

q

~

0

0

185

0 -0

N 0

_

> W I >or)

c_) 0 -lor)

0

_

0

_

0

tD

0

0

o~ oLn i~ x

tl/ -

tP

_

i~

0~

ID

rr

or- ,

x >...

N

i

--

1

-

0

N

Q Q

-I

t

Q

-

~

Q -~1

llii

I

0o0

~0

i

I

I

I

Ill

I

I

I

I

I

I II

~!"

o,J

--cO

~0

G

186

Y/RS

.4

~/R S

M0¢=2

-.5

Fig. 8

187

.5

~s

y=1.4,B=2 O O [] []

.4

-----

N = 10 N=20 N= 30,40 MAXIMUM ENTROPY STREAMLINE

.3 ¸

.2 ~

.1

I

.2

J

-.3"

-,4 ¸

-.5 ¸

--.6 ¸

Fig. 9

.3

.4

.s ~ x%

188

A

~=.50

I I

I

I

I

I I I I

~=.5o

.3

\

\\

\

\

= x/R S

\

x/

//~

~ \*-/ -.5"

D

Fig. i0

E

WAVE FRONT EXPANSIONS AND PADE' APPROXIMANTS FOR TRANSIENT WAVES IN

LINEAR DISPERSIVE MEDIA

G. T u r c h e t t i ,

F. M a i n a r d i

Istituto di F i s i c a , U n i v e r 6 i t & di B o l o g n a , I t a l y Gruppo N s z i o n a l e p e r l a F i s i c a M a t e m a t i c a del C . N . R .

1.

INTRODUCTION The wave p r o p a g a t i o n

interesting

aspects,

a homogeneous h a l f free

surface.

ed by t h e

in the

space s u b j e c t e d

its

dispersive

media has d i f f e r e n t

The t r a n s f o r m

invePslon

Usually

t o a known i n p u t

boundaPy-value problem

Laplace transform.

cases.

linear

and

Many o f them c o n c e r n p l a n e waves p r o p a g a t i n g

The r e l a t e d

minedw however special

in

short

condition

is a very difficult

at the

is conveniently

solution

is easily

trea~

deter-

task except for

and l o n g t i m e a p p r o x i m a t i o n s

in

some

are dePived

litePature.

In this

n o t e we r e v i e w a s i m p l e method w h i c h e n a b l e s u s ,

Pad~ A p p r o x i m a n t s ,

t o compute t h e s o l u t i o n s

lems i n s p a c e - t i m e

domains o f p h y s i c a l

on r e c e n t 21, w i t h

investigations a particular

o f most t P a n s l e n t

interest.

about viscoelastic

Our a n a l y s i s

is the Following.

In Sections

we c o n s i d e P t h e wave p r o p a g a t i o n

viscoelastic

and t h e r m o e l a s t i c

media,

i s based waves I 1 ,

aspects.

The p l a n e o f t h e work 2. and 3.

wave p r o b

and t h e r m o e l a s t i c

emphasis on t h e n u m e r i c a l

using the

in linear

w h i c h l e a d s t o t h e same t r a n s f o r m

equation. In Section series

solution, In Section

and i n t r o d u c e

4, we d e r i v e

a wave F r o n t e x p a n s i o n w h i c h p P o v i d e s a

uniFoPmly convergent 5. we d i s c u s s

the diagonal

the

i n any s p a c e - t i m e

numerical

properties

Pad~ A p p r o x i m a n t s

domain. of this

solution

in order to accelerate

the

convergence. In Section o f t h e method.

6. we p r e s e n t

some r e s u l t s

which cenfirm

the efficiency

190 2. VISCOELASTIC WAVES The basic equations of the dynamic theory of l i n e a r v i s c o e l a s t i c i t y

are,

in the unidimenslonal

~__ ax o ( x , t ) : ~(x,t)

case ( s e e f o r example 13,4,51):

a2

e a-~T u ( x , t )

(2.1)

{so +--ds(t)~}at oil, t)

:

(2.2)

a ~ ( x , t ) =T~xU(X,t)

(2.3)

O

v (x,t) =~-~u(x,t) where

(2.4)

o is the stress,

particle velocity,

J o = J(0+) •

and

~

Denoting with • u• v ,

~ the s t r a i n t

O the density•

J(t)

u

the displacement,

v

the

the creep compliance•

d e n o t e s t h e Riemann c o n v o l u t i o n . R

(response v a r i a b l e ) any one of the variables a •

we consider the problem of determining

R= R ( x , t )

with the

i n i t i a l conditions:

R(x,t) = ~

R(~•t)= 0

for

t =0

for

x =

as

x-*~

(2.5)

and t h e b o u n d a r y c o n d i t i o n s : R(x,t)

= to(t)

R(x,t)-~O

0

Taking t h e L a p l a c e t r a n s f o r m o f e q u a t i o n s for the transform response

R(x,s)

(2.6) (2.1)-(2.5)

we o b t a i n

the following differential

equation:

2

{ a~__ 2 ( s ) } ~ ( x • s ) : O Ox 2

(2.7)

where 2

s2

(s) = - 5 e

1

In i2.8) c=(e creep

(2.8)

0 +~(s)] Jo) -~

1 ~ i t ) = S--~

and

~(s)

is the Laplace t r a n s f o r m of the r a t e

d--Jit ) dt

Account i ng f o r ( 2 , 6 )

the solution

~(×,s) =~o (s) e~p[-× ~(~)] with

# (s)~O

of

for

arg(s)=O

of (2.7) reads: (2,9)

131.

191 In most cases that

~(s)

transform tlon

~(t)

is an e n t i r e

is a n a l y t i c theory

161.

function

and v a n l s h i n 9 at

of exponential

infinityt

as known from t h e

This occurs f o r v i s c o e l a s t i c

spectrum is s t r i c t l y

positive.

in t h e s - p l a n e c u t along a f i n i t e

Then F (s)

models whose melaxa B

is an a n a l y t i c

domain o f t h e n e g a t i v e r e a l

A r e l e v a n t model of v i s c o e l a s t i c i t y

t y p e so

function axis

Ill.

is t h e Standard L i n e a r S o l i d

(SLS) f o r w h i c h :

~ ( t ) = ( 1 - a ) e -at where

1

and

1/a

=* ~ ( s ) = ( Z - a ) / ( s + a )

represent the relaxation

time r e s p e c t i v e l y

131.

For

a=O

(2.10) time and t h e r e t a r d a t i o n

we r e c o v e r t h e Maxwell S o l i d ,

for

which t h e wave e q u a t i o n reduces t o t h e S t e l e g r a p h e q u a t i o n w 171.

3. THERMOELASTIC WAVES A genera I i zed dynamic t h e o r y o f I i near thermoe I a s t i c i t y i n t r o d u c e d by s e v e r a l eliminate velocity

authors

(see fop example 1 8 , 9 e l O e l l l ) i n

t h e paradoxe p r e s e n t for

thermal

order to

t h e o r y e o f an i n f i n i t e

disturbances.

In t h e u n i d i m e n s i o n a l

in a c o n v e n i e n t non-dimensional

form,

a2

a~

s

(x,t)=a- ~ u(~,t)

1101

(3.1)

S (x,t) = E(x,t) - O(x,t)

(3.2)

E(x,t) =~

(3.3)

U(x,t)

~2 c~x---~ O ( x , t ) - ~ - ~ where

in t h e c l a s s i c a l

and mechanical

case t h e b a s i c e q u a t i o n s are, a

has been

$

O(x,t)-

is the stress e

# E

~

= ~ [~

E(x,t)+~

the strainw

02

E(x,t)]

~

tion

i n t r o d u c e d t o account fop t h e a c c e l e r a t i o n

c o u p l i n g c o n s t a n t and

0(x,t) =~

E:E(xet)

with the initial

O(x,t) : E ( x , t ) : ~

and t h e boundary c o n d i t i o n s :

E( x, t ) : 0

0

the

~ the relaxaof t h e h e a t f l u x .

We c o n s i d e r t h e problem o f d e t e r m i n i n 9 t h e t e m p e r a t u r e and t h e s t r a i n

(3.4)

t h e displacemente

temperaturet constant

the thermoelastic

U

~

O:O(xet)

conditions:

for

t: 0

(3.S)

192

O(x,t)=Oo(t) O(x,t)~0 ,

,

E(x,t)=Eo(t)

E(x,t)~O

fop

x=O

as

x~®

(3.6)

Taking the Laplace transforms of equations ( 3 . 1 ) - ( 3 . 5 ) we obtain for the transform responses g ( x , s ) • equations. O=

g

two coupled d i f f e r e n t i a l

However i t is possible to uncouple the problem 121 by s e t t i n g + O

E = E

where g-+ • E'a2

~(x•s)

(3.7)

+ E

s a t i s l y t h e f o l I owing e q u a t ions :

-

1 =:+(x,s>=0 (3.8)

/~(s),

In ( 3 , 8 )

4 with

-s

2

u~

/~2(s)_ are the algebraic roots of the equation:

2 + s 4 v=O

u=l+/~(l+~)

+ (l+~)/s•

is not d i f f i c u l t

It

~(s)

(3.9)

v=~+l/s,

to prove that:

= s2 ;TEl + v+(s)]

(S. 10) I

where c + = { [ i + ~ ( I + ~ ) ± 7 ] / 2 }

1

-~ with

7= { [ 1 + ~ ( 1 + ~ ) ] 2 - 4 f l }

~r-(s) are functions analytic and vanishing at i n f i n i t y . H+2~#2_ for Y~ - Y .

~

and

We remark that

Accountln9, for (3.6) the p a r t i a l solutions read:

~t(x•s) = {aTl(s) No(S) + aT2(s) to (s) t exp[- x ,+(s)] + + (3.11)

+ + ~-(x,s) = {a~1(s) No(S) + a~2(s) ~o (s)] where

alk(S)

arg(s)=O. vanishing

(i,k=l•2) The

value•

the negative

real

aik(S) and t h e

exp

I-

x ~+(s)]

ape known f u n c t i o n s

and

p r o v e t o be a n a l y t i c

at

~+(s)

a x i s between

are analytic s=-l/~

and

~+(s) ~0 infinity

for w i t h a non-

in the s-plane s= 0

121 .

cut along

193 4. WAVE FRONT EXPANSIONS In t h e p r e v i o u s S e c t i o n s we have shown t h a t t h e problem o f t r a n sient

waves can be reduced t o t h e i n v e r s i o n of t h e Laplace t r a n s f o r m ~ R(x,s)=?(s)

where

~(s)

a(s), ~(s) a(~) ~0,

a(s) exp{-X~S~+~(s~½}

is t h e t r a n s f o r m

R(x,t)

~(~) =0

denoted by

R(x,t)=f(t)

functions

(a(s)=l

regular at

is s i m p l e r when

f(t)

and with

case).

~(s)= 1.

(Green's function),

f(t),

infinity,

for the viscoelastic

G(x,t)

f o r any i n p u t

The c o r r e s p o n d i n g

enables us t o o b t a i n

by t h e Riemann c o n v o l u t i o n :

(4.2)

~G(x,t)

The wave p r o p e r t i e s G(x,s),

of a given input function

are some a n a l y t i c

The i n v e r s i o n o f ( 4 , 1 ) function,

(4.1)

C -

of

R(xwt)

which p r o v i d e s e x p l i c i t l y

appear from t h e l i m i t the discontinuity

t h e wave f r o n t

and t h e space damping,

we can expand

a(s)

and ~ (s)

s-~

and t h e v e l o c i t y

of

G(x,s)

of

according to:

ao~O

(4.3)

?(s) = ~l/s + ~2/s 2 + . . . so t h a t the l i m i t

of

From t h e p r e v i o u s c o n s i d e r a t i o n s

in Laueent s e r i e s ,

a(s)=ao+az/S+a2/s 2 + . . . ,

as

(4.4) as

s -~is:

G(X,S)~ aoexp[- xs - kxl

(4.s)

C

with

Z= ?'--J 2c Then f o r

(4.6)

t ~(x/c) +

R(xtt)=

f(t

we get:

- x/c)

a 0 expr[" kX]

from which we r e c o g n i z e t h a t space damping and

c

By a c c o u n t i n g f o e t h e f i r s t

derived. (4.1)

is t h e wave f r o n t

f(O +) a o e x p [ - A c t ]

Of course t h e decay c o n d i t i o n

can be a n a l y t i c a l l y

(4,7)

k >0

velocity,

t h e jump o f is

insured

few teems in

R

the

at t h e wave f r o n t .

in our cases. 1/s

of

R(x+s)

which

computed, s h o r t t i m e a p p r o x i m a t i o n s a e e u s u a l l y

On t h e o t h e r hand long time a p p r o x i m a t i o n s

either

k

by t h e s a d d l e p o i n t method or by t h e

are deduced from

limit

as

s ~0.

194 Here we sketch a pecursive method to generate the whole expansion

of

G(xrs)

in powers of

the Green°s f u n c t i o n ,

1/s, which provides a wave f r o n t expansion of

uniformly convergent For any

x,

For t h i s purpose

we put: r XS] ^ G(x,s)=expt-G(x,s)

(4.8)

c

where G ( x , s ) ,

which is analytic at i n f i n i t y u is to be expanded in

Laurent series according to: G(x,s) = Wo(X) + ~ Wk(X)/Sk k=l

(4.9)

To determine the f u n c t i o n s

Wk(X) ( k = O , l , . . . )

tial

G(x,s) ,

equation s a t i s f i e d

by

we c o n s i d e r

namely, after" ( 2 . 7 ) ,

the differen-

(2.8),

(4.1),

(4.8),

I aa

- ~2so -~-c

s'-2~(s) ~2 1 ~(x,s)=0

44.10)

which is subjected to the boundary condition

O(Ows)=a(s).

Inserting

the expansions (4.3), (4.4), (4.9) into equation (4.10) and c o l l e c t i n g l i k e powers of

s e we obtain the following recursive system of F i r s t

order di f f e r e n t i a I equations: d ~1 T~ Wo + ~ Wo=0 d

~1

c

d2

"~x wk + ~ c wk=~ ~ with i n i t i a l

conditions

to prove t h a t the

where

Z

(4.11)

x k-

wk-1 - ~ c ~=. 'q)J+l wk-J

Wk(O)=a k ( k = O , 1 , 2 . . . . ) .

Wk(X)

(k=O,1,2,...)

1

I t is not d i f f i c u l t

can be expressed by:

k Wk(X)=exp(- ;~x) ~ Akh h=O

i s g i v e n by ( 4 . 6 )

and t h e

Akh

h

44.12)

are defined by the recursive

relations:

i Ak 0 = ak Akh = ~ (Ak-l,h+l

1 k-h+l

(4.13)

~$+] Ak-J,h-l.

195 From t h e i n t e g r a l

transform theory

161, we know t h a t t h e i n f i n i t e

in ( 4 . 9 ) can be i n v e r t e d term by term in any F i n i t e p r o v i d i n g an e n t i r e (4.8),

(4.9),

Function of exponential type.

interval

of

series x,

Then, by i n v e r t i n g

we o b t a i n t h e f o l l o w i n 9 r e p r e s e n t a t i o n f o r t h e Greents Fun_c

tion: oo

G(x,t)=Wo(X) 6 ( t - x / c ) +

where (~(.)

E

k=l

Wk(X) ( t - x / c k )- "l -

(k- I)!

(4.14)

denotes the Dirac d i s t r i b u t i o n .

In (4.14) the f i r s t term isolates the d i s c o n t i n u i t y associated with t h e p r o p a g a t i n g wave F r o n t , powers o f

i n agreement w i t h

(4.7),

w h i l e t h e s e r i e s of

~ = t - x / c g which i s uni f o r m l y convergent For any

x , accounts

For the response f o l l o w i n 9 the wave Front. We notice that, when the input function

F(t)

is e n t i r e of expo-

nential t y p e , a simi lap wave Front expansion can be c a r r i e d out d i r e c t l y on

R(x,t) ,

expanding

a v o i d i n g the c o n v o l u t i o n procedure ( 4 . 2 ) .

~(s)

In t h i s

case,

in Laurent s e r i e s

7(s) = Fo/s+

/s2+

(4.15)

...

and p e r f o r m i n g the Cauchy product of t h e s e r i e s ( 4 . 1 5 ) ,

(4.3),

we can

write:

7(s) a(s)= Co/S+ 01/s2+...

(4.16)

This enables us by p r e v i o u s c o n s i d e r a t i o n s t o o b t a i n t h e s e r i e s represenn ration

of

R(x,t): oo

R(x,t)= where t h e

~ Wk(X) k=O

Wk(X) ( k = 0 , 1 , . . . )

Wk(O) = Ako= 0k 5.

(4.17)

'k!X/C)2 are g i v e n by ( 4 . 1 2 ) ,

(4.13) with

•

PADE' APPROXIMANTS AND NUMERICAL CONSIDERATIONS The s e r i e s s o l u t i o n s

(4.14),

( 4 . 1 7 ) are easy t o handle For numerical

computations s i n c e t h e y are o b t a i n e d in a r e c u r e s l v e way. mate o f t h e e r r a t a made when we t r u n c a t e the s e r i e s ,

A good e s t i -

is practically

i m p o s s l b l e , but the numerical convergence i s expected t o slow down by increasing of

~

( t i m e elapsed From t h e wave F r o n t ) w i t h a r a t e

depe~

196

i

n9 on

x

•

Since the result

i s an e n t i r e

function

w h i c h must be bounded a t

infinity,

we a r e

as i n t h e e v a l u a t i o n

exp(- 4)

usin9

is

o?

in

4

oP e x p o n e n t i a l

type

Paced w i t h t h e same d i f f i c u l t y its

Taylor

e x p a n s i o n w when

4

large. In h i s c l a s s i c a l

approximations, to accelerate

work

1121, Pad~ i n t r o d u c e d a sequence o f r a t i o n a l

henceforth

called

Pad~ A p p r o x i m a n t s ( PA) ,

the convergence of the Taylor

series

which proved

fop the exponential

function. No g e n e r a l c o n v e r g e n c e t h e o r e m s a r e a v a i l a b l e t h e case o f S t i e l t j e s

functions

1131),

(see fop example

v e r g e n c e i n measure has been r e c e n t l y

f o p t h e PA e x c e p t i n

proved for

h o w e v er t h e co~

meromorphic functions

114,1SI. I n o u r case we a p p l y t h e PA t o t h e s e r i e s the analytic

properties

insure their

solutions

in 4

the exponen-

Functlon. In the a c t u a l

the series

4= 4o

computations from a comparison of the partial

with the corresponding

P a t e i s much b e t t e r

4

is

large.

the numerical convergence of the series

c o n v e r g e n c e o f t h e PA i s s t i l l with

PA we have r e m a r k e d t h a t

f o p t h e PA when

t e r m s a r e computed (we work w i t h

6.

since

c o n v e r 9 e n c e i n m e a s u r e , and a s t r o n 9

i m p r o v e m e n t o f t h e n u m e r i c a l c o n v e r g e n c e i s e x p e c t e d as f o r tial

,

the

Ion 9 t i m e s o l u t i o n

a fixed

is

the convergence

Beyond a c r i t i c a l lost

for

value

no m a t t e r how many

number o f d i 9 i t s ! )

satisfactory

sums o f

~ > 4o

while the

until

a matchin9

is obtained.

RESULTS A c c o r d i n 9 t o t h e method d e v e l o p e d i n S e c t i o n s 4 . ,

?ormed a n u m e r i c a l s u r v e y o£ t h e models i n t r o d u c e d The r e s u l t s

a r e summarized i n s e v e r a l t a b l e s

v e n i e n c e we have f i x e d variables

x, 4

the time

is finite

For t h e v i s c o e l a s t i c we have computed:

t=T

(O~x~cT,

5.,

in Sections 2,,

and f i g u r e s .

so t h a t

we have p e r 3.

For c o n -

t h e r a n g e oP t h e

O~4~T).

waves i n a S t a n d a r d L i n e a r S o l i d

(see (2.10)

197 (i)

the Green's Function

and t h e s o l u t i o n s (ii)

of the following

boundary v a l u e p r o b l e m s :

~ ( )

R.(t)=l

0

S

1 = ~ S

Ro(S) =

(ill)

i

I

s[i + ~ ( s ) ] ~ (iv) (v)

i

Ro(t) = e-at

Ro(s)

R o ( t ) = e "at cos ~ t

Ro(s)-

In T a b l e s

=

s

I - Vl we compare t h e s e r i e s

a=O.

In t h i s

R(x,t)=e_t/2 and a t than

(4.2)

results.

check For t h e P.A.

case t h e e x a c t s o l u t i o n

digits

accuracy

i s a c h i e v e d For ( i i i )

is explicitly

i s Found f o r

known 171:

T ~ SO

u s i n g no more

P.A,

In F i g u r e s 1-4 t h e r e s p o n s e s t o several

For t h e

i o { ~ ( t 2 _ c~ ) : } 2

least a five (12/12]

and P.A. s o l u t i o n s

i n o r d e r t o check t h e method we do a l s o

long t i m e and c o n v o l u t i o n

An even more s t r i n g e n t when

+~

(s + a ) 2 + 2

above boundary v a l u e p r o b l e m s ; quote t h e

s+~

values of

(ii)

and ( i i i )

are plotted

for

T.

For t h e t h e r m o e l a s t i c

waves t h e F o l l o w i n g

boundary v a l u e

is

considered:

go(t) = 0

~o(S) = 0

I Eo(t)=1

go(S)=~

In T a b l e s VII, VIII we e x h i b i t comparing t h e s e r i e s ,

P.A.

the thermal (separately

g+ # g - , E+ , E-) and Ion9 t i m e relevant

results

transient

and e l a s t i c

r e s p o n s e s by

computed on t h e e x p a n s i o n o£

1161 s o l u t i o n s .

In F i g u r e s 5 - 8 some

a r e shown.

From t h e p r e v i o u s examples plays a crucial

S

role

if

wave p r o b l e m s .

we are

we can i n f e r interested

that

t h e Padd method

i n a 91obal

solution

of

198 APPENDIX The r e s p o n s e o f a S . L , S .

R (x,t) = I

t o any i n p u t

For l a r g e v a l u e s o f

t

(A.i)

~ (s)

with

g i v e n by

we a p p r o x i m a t e ( A . 1 )

point method. FOr the Greenes ~unction result

i s g i v e n by

err(s) go(S) d s

f(S):S~-c--"~ Vl+~(s~ ,

where

to(t)

No(S): I

(2.10).

using the saddle

and the standard

reads

R ( x , t ) = [ 2 = t l f " ( ~ ) l ] -½ e- t l f ' ( ~ ) l where

s

i s d e f i n e d by

For i n p u t s most r e l e v a n t close to point

(ii) (iii)

f'(~)=O

( i i ) and ( i i i )

contribution

s=O.Replacin 9

contribution

and must be computed n u m e r i c a l l y ,

Ro(s)

to

has a p o l e a t t h e o r i g i n

(A.1)

f(s)

(A.2)

by

i s o b t a i n e d when t h e ~(s)= f'(O)s+

can be a n a l y t i c a l l y

f"(O)s 2

and t h e

saddle pointis the saddle

e v a l u a t e d and we g e t :

R(x,t) -~ ½Erfc[~W] R(x,t) -~ x V~ x+ct

whereW=-(1-

x

ct

V-a

Erfc[W~

)-(2

(A.4)

1,,:a.

x

a

c t 2 ~a

Ca

For t h e r m o e l a s t i c i n 1161 by t a k i n g

(A.3)

)-½

waves long t i m e a p p r o x i m a t i o n s

the

limit

as

s~O

of the transform

have been d e r i v e d solutions

and

read: O(x,t)

= - ~1

+~

1-

1+

E(x,t) = where

Z=x

V1 +e/?-

Erf

[Z]

Err [Z] ¢-t

•

(A.5) (A.6)

199 REFERENCES Ill

F. MAINARDI & G. TURCHETTI, Mechanics Res. Comm., in press.

1 2 1 F . MAINARDI & G. TURCHETTI, t o be p u b l i s h e d . 131 S. C. HUNTER, i n Progress in S o l i d Mechanics, e d i t e d by SNEDDON & HILL, Vol.

I , p. 3, N o r t h - H o l l a n d , Amsterdam, 1960.

141 R, M. CHRISTENSEN, Theory of V i s c o e l a s t i c i t y ,

Acad. Press, N.Y.,

1971. 151 J. D. ACHENBACH, "Wave Propagation in E l a s t i c S o l i d s " ,

North-

Holland, Amsterdam, 1973. 161G. DOETSCH, "Theory and A p p l i c a t i o n of the Laplace Transform", Sprlnger-Verla9,

N.Y., 1974.

171 E. M. LEE & T. KANTER, J. Appl. Phys. 24, 1115 (1956). 181 S, KALISKY, B u l l . Acad. Pol. S c i . 13, 409 (1965). 191 E. B. POPOV, J, Appl. Math. Mech. (PMM) 31, 349 (1967). I10) M. W. LORD & g. SHULMAN, J. Mech. Phys. S o l i d s 15, 299 (1967). Illl

J. D. ACHENBACH, J. Mech. Phys. S o l i d s 16, 273 (1968).

1121 H. PADE', Thesis Ann. Ecole Nor. ~, Suppl, 1 (1892). 1131 G, A. BAKER, J. Adv. Theor, Phys. ~, 1 (1965). 1141 J. NUTTAL, J. Math. Anal. and Appl. 31, 147 (1970). 1151 J, ZINN-JUSTIN, Physics Reports (Sect. C, Phys, L e t t , ) ~, 55 ( 1 9 g l ) , I161 F. R. NORWOOD & W. E. WARREN, Quart. J. Mech. Appl. Math, 22, 283

(1969).

200

TABLE CAPTIONS The meaning o f t h e symbols used i n t h e T a b l e s X :

distance

TAU :

time elapsed from the wave Front ;

SERIES :

r e s u l t s of the p a r t i a l sums

in

~

is the Following:

= t - x/c

(NS) of the series s o l u t i o n

,

NS :

number of terms in the p a r t i a l sums

ERS :

estimated accuracy of the series s o l u t i o n defined by

PADE :

diagonal

II_,(NS-I)I (NS)

[NP/NP]

Pad6 approximants

series s o l u t i o n in ~

computed on the

,

NP :

order of P . A . .

ERP:

estimated accuracy of P.A. d e f i n e d by

CONVOLUTION: c o n v o l u t i o n piecewise

LONG TIME :

NS=2NP+I I

II"

L.NP..- 1/NP- I ] I [NP/NP] I

of the input with the Green's f u n c t i o n computed using the series when ERS~ 10-5 ,

ERS > 10-5 ~ ERP

ERP> 10-5 .

Remark t h a t

the P.A. when

and the long time approximation when

A Gauss quadrature w i t h

NO

long t i m e a p p r o x i m a t i o n o f t h e s o l u t i o n

V i s c o e l a s t i c waves in a S.L.S. w i t h

a=.S

Table

I

Input ( i ) ,

Table

II

Input

(ii)

Table

III

Input

(iii)

T a b l e IV

Input

(iv)

with

a = ,1 ,

Table V

Input

(v)

with

a : . I , ~:T0,

for

points

is used.

(see A p p e n d i x ) .

T=30.

Green's f u n c t i o n

Thermal and s t r a i n waves w i t h

NG=8

~ = .03,

NG=20

~=1.3

(NS~21, N P ~ I O ) , Table VI

T h e r m a lwaves f o r

Eo(t)=l

go(t)=O

T a b l e VII

Elastic

Eo(t )= 1

go(t) = 0

waves f o r

for

T = 5

201

TABLE I : S,LS(i ) T = 3 0

TAU

SERIES

NS

0

2.593E-03

1

2

2,123E-02

10

4

5.042E-02

6 8 I0

ERS

PADE t

NP

2.593E-03

0

1.E-06

2.124E-02

4

3.E-04

2.28E-02

12

7,E-06

5.042E-02

5

6. E-05

5.30E-02

7.222E-02

16

9.E-06

7.221E-02

7

3 • E-06

7.53 E-02

7.451 E-02

19

1,E-05

7.451E-02

9

3 • E-08

7.74E-02

5.973 E-02

23

4.E-07

5.973E-02

11

3.E-10

6.19E-02

12

3. 867E-02

25

3.E-05

3.867E-02

12

1. E-I 0

4 . OOE-02

14

2.059E-02

25

6.E-03

2,062E-02

12

I . E-O8

2.13E-02

16

9.012E-03

25

2.E-02

9.124E-03

12

6. E-07

9.45E-03

18

2.021E-02

25

3.E+O0

3.344E-03

12

I.E-06

3,47E-03

20

1.579E+00

25

2.E+O0

1.003E-03

12

1.E-05

1.04E-03

22

3.145E+02

25

2.E+O0

2.400E-04

12

3. E - 0 4

2.50E-04

24

2.965E+04

25

2.E+O0

4.911E-05

12

1,E-O1

4.57E-05

26

1,516E+06

25

2.E+O0

4.538E-06

12

2.E-01

5,75E-06

28

3.984E+07

25

2.E+O0

2.080E-07

12

7,E-01

3.85E-07

.0

ERP .0

LONG TIME --

202

TABLE fl : SLS(~ii) T = 3 0 TAU

SER IES

NS

0

5,531E-04

1

3

5,893E-02

11

6

2. 767E-01

9

£RS

NP

5,531E-04

0

9. E-06

5,893 E-02

5

2. E-05

9.27E-02

16

4.E-06

2. 767E-01

7

5,E-07

2,49 E-Ol

5. 858E-01

2o

8. E-06

5.858£-01

9

6. E-08

5,22E-01

12

8. 288E-01

25

4. E-06

8.288E-01

12

3. E-11

8,16E-01

15

9,497E-01

2.5

2. £-04

9.497E-01

12

2. E-09

9,72E-01

18

1 . 001 E+O0

25

4. E-02

9,898E-01

12

5. E-09

9• 99E-01

21

2,117 E+01

z5

2. E+O0

9.986E-01

12

7.E-08

I • OOE+O0

24

2,788E+04

25

2. £+00

9.999E-01

12

7, E-07

I , OOE+O0

27

9,103 E+06

25

2, £+00

1, O00E+O0

12

I , E-06

1 • OOE+O0

ERP

LONG TI ME

.0

ERP

LONG TI ME

PADE'

.0

TABLE Ill : SL.,S(iii) T=..~0 SER I ES

NS

o

5,531E-04

1

3

4,980E-02

12

6

2,182E-01

9

TAU

ERS ,0

PADE'

NP

5,531E-04

0

,0

9. E-07

4.980E-02

5

2,E-05

7.34E-02

15

5.E-06

2.182E-01

7

3,E-08

1.87E-01

4,402E-01

2O

8.E-06

4,402E-01

9

4,E-08

3.71E-01

12

6,031E-01

25

5,E-06

6,031E-01

12

7, E-11

5.88E-01

15

6,786E-01

25

4, E-05

6,786E-01

12

6,E-12

6.91E-01

18

7,073E-01

25

3. E-02

7,018E-01

12

8. E-09

7,07E-0l

21

-3,681E+01

25

2, E+O0

7 • 065 E-01

12

2, E-09

7,07E-01

24

-9,250E+04

25

2.E+O0

7,071E-01

12

2. £-06

7.07E-01

27

-6,870E+07

25

2.£+00

7,071E-01

12

3.£-06

7,07E-01

203

TABLE

TAU

SER I ES

NS

0

5 • 531 E-04

1

3

5,279E-02

12

6

2.224E-01

9

IV

:

ERS

SLS(iv) T=30 PADE'

NP

5,531E-04

0

7, E-07

5,279E-02

5

5.E-05

5.279E-02

14

9, E-06

2,224E-01

6

5, E-05

2,224E-01

4. 054E-01

21

3.E-06

4.054E-01

10

2.E-09

4,054E-01

12

4. 638E-01

25

I.E-05

4.63gE-01

12

2,E-lO

4,638E-01

15

3,972E-01

25

6.E-04

3,972E-01

12

1,E-08

3.972E-01

18

3.01 OE-01

25

2, E-O1

2,867E-01

12

3.E-08

2.867E-01

21

2,240E+01

25

2,E+O0

1.902E-01

12

1,E-07

1,902E-01

24

3 , 1 0 9 E+04

25

2, E+O0

1,223E-01

12

7. E-06

1,223E-01

27

1. 018E+07

25

2. E+O0

7. 808E-02

12

9. E-06

7,808E-02

PADE'

NF'

ERP

CONVOLUTI ON

5,531E-04

0

.0

TABLE

V

:

ERP .0

CONVOLUTI ON

5.531E-04

SLS(v) T=.~O

SER I ES

NS

0

5,531E-04

1

3

4,835E-02

12

4, E-07

4. $35E-02

5

2, E-04

4.835E-02

6

t . 597E-01

t6

8, E-06

1,597E-01

7

4, E-06

1,597E-01

9

1.575E-01

21

4.E-06

1,575E-01

10

5,E-08

1.575E-01

12

-1.583E-02

25

3, E-05

-1,583E-02

12

5, E-06

-1.583E-02

15

-1,400E-01

25

2. E-03

-1.399E-01

12

2. E-05

-1,399E-01

18

-6.051E-02

25

6. E-01

-7,035E-02

12

2, E-04

-7,035E-02

21

2,030E+O1

25

2,E+O0

5,592E-02

12

4.E-03

5,590E-02

24

2,749E+04

25

2, E+O0

6,375E-02

12

5, E-03

6,556E-02

27

8,939E+06

25

2, E+O0 - 2 . 870E-02

12

1, E+O0

-1.381E-02

TAtl

ERS

.0

,0

5,531 E-04.

204 TABLE Vl

: THERMAL WAVES T= PADE"

ERP

LONG TI ME

X

SERIES

5,24

-5,358E-02

O.

-5.358E-02

O.

4.81

-4.851E-02

1, E-06

- 4 . 851 E-02

3. E-06

-2,56E-02

4.18

-3,807E-02

7, E-06

- 3 . 807 E-02

I , E-05

-2.39E-02

4.18

-2,276E-02

I,E-05

-2,276E-02

2. E-05

-2,39E-02

3.54

-2,011E-02

4, E-06

-2,011E-02

8, E-06

-2.17E-02

2.90

-1,708E-02

5. E-06

- I , 708E-02

3, E-07

-1,89E-02

2,27

-1,373E-02

7.E-05

-I,373E-02

6. E-07

-1,55E-02

1.63

-1,010E-02

4. E-03

- 1 . 009 E-02

9 • E-06

-1.16E-02

.99

-6.707E-03

2, E-01

- 6 . 241 E-03

1, E-04

-7.30E-03

.35

-7.372E-03

4. E+O0

-2,263 E-03

3, E-03

-2.67E-03

ERS

w ~

TABLE VII : ELASTIC WAVES T = ~ X

SERIES

ERS

PADE'

ERP

LONG TIME

5.24

5.369E-01

O.

5.369E-01

O.

4.81

7. 709E-01

4. E-06

7. 709E-01

6. E-05

9,74E-01

4.18

9.276E-01

4. E-06

9,276E-01

1.E-05

9.76E-01

4.18

9,785E-01

4. E-06

9. 785E-01

1. E-05

9.76E-01

3.54

9,811E-01

9.E-06

9.811E-01

1.E-05

9,78E-01

2.90

9 • 840E-01

4. E-06

9. 840E-01

8. E-07

9.81E-01

2.27

9.872E-01

4;E-06

9.872E-01

7, E-09

9.84E-01

1.63

9.906E-01

3. E-04

9.906E-01

8.E-07

9.88E-01

.99

9,950E-01

1.E-02

9,942E-01

9, E-06

9.93E-01

.35

1. O08E+O0

3. E-01

9,979E-01

7, E-05

9,97E-01

205

FIGURE CAPTIONS I - 2

The response o f a S . L . S .

with

a = .5

For i n p u t ( i i )

and

T=l,3,5;10,30w50.

3- 4

The same as F i g u r e s

1 - 2 for

S- 6

Thermal and e l a s t i c

waves w i t h

Eo(t)=Z, 7- 8

go(t)=O

The same as F i g u r e s

and 5- 6

input

(iii)

s = .03 ,

~= 1 . 3

T=2.5. For

~= .03 ,

~ = .9

for

input

206

R, 1

a=O,5

,

I

i 2

SLS .

i 4

2o

Fig. l

Fig.2

a=0,5

40

R~ 1 SLS a=0,5

SLS a=0,5

t.50

2

4

Fig.3

Malnardl~

Turc

20 Fig.4

het

tl

40

207

E

=0,03 (3 = 1 , 3

= 0,03 (3 = 1,3

t=5

-0,1

I

0

4

t~5

t=2

I

6

I 4

x

Fig. 5

I 8

E

F\ = 0,03 /3=0.9

C = 0,03 /3 = 0 , 9

t=2

I 4

Fig. 7

Mainardi-Turchetti

x)~

Fig. 6

I 0

x~

t=5

,I,

4

Fig. 8

0

x

APPLICATION OF METHODS FOR ACCELERATION OF CONVERGENCE TO THE CALCULATION OF SINGULARITIES OF TRANSONIC FLOWS* Andrew H. Van Tuyl Naval Surface Weapons Center White Oak Laboratory Silver Spring, Maryland 20910 USA

i.

Introduction Initial value problems in gas dynamics which lead to transonic flows include

the inverse blunt body problem, given and the body w h i c h w o u l d

in which a bow shock wave in a uniform flow is produce it is calculated,

and the inverse calculation

of nozzle flows starting from data given on the centerline.

Each of these problems

can be expressed as an initial value problem for a second order quasi-linear differential

equation satisfied by the stream function.

When the initial curve and initial data are such that the initial curve is noncharacteristic,

it follows from the Cauchy-Kowalewskl

theorem [i, page 39] that

the initial value problem can be solved in terms of power series in the neighborhood of a given point of the initial curve.

However,

the region of convergence

series obtained may be too small for practical use, due to the occurrence singularities,

either real or complex, near the initial curve.

of the of

This was found by

Van Dyke [2] in the case of the inverse blunt body problem, where a limiting line** (envelope of characteristics) flow.

occurs in the upstream analytic continuation

of the

This limiting line lies closer to the shock than the distance between the

latter and the body, and hence, a power series solution in the neighborhood

of a

point of the shock diverges at the body and cannot be used directly to calculate the flow there. In [3], Leavitt has calculated

the shape and position of this limiting

near the axis of symmetry by a modification

line

of a method due to Domb [4], starting

* This work was supported by the Naval Surface Weapons Center Independent Fund. **Also called limit line.

Research

210

from the power solution of the inverse blunt body problem in the neighborhood the nose of the shock.

The location of the limiting line was then used to trans-

form the series so that convergence was obtained at the body. Schwartz

More recently,

[5] has used Domb's method to calculate limiting lines in the flows

produced by parabolic and paraboloidal number.

of

Various modifications

shocks in a free stream of infinite Mach

and extensions

of Domb's method have been applied to

problems of statistical mechanics by Domb, Sykes, Fisher, and others

([6], for

example). Limiting lines in solutions of the inverse blunt body problem have also been calculated by Garabedian and his students

([7] and [8]) by use of Garabedian's

method of complex characteristics. Limiting lines may also occur in nozzle flows obtained from given centerline distributions

of velocity or Math number.

region of convergence

As in the case of blunt body flows, the

of a power series solution may be restricted by a limiting

line even though the point about which the solution is obtained lles in the subsonic region.

In nozzle design,

given centerline distribution

it is of practical interest to know if a

leads to a limiting line which lies between a

desired streamline and the axis of symmetry. A procedure for calculation of limiting lines will be described, from a power series solution, are used.

in which methods for acceleration

This procedure involves the ratio of successive

power series, as in Domb's method,

of convergence

coefficients

and a necessary requirement

the extent of the region of convergence

starting

of a

is therefore that

in the direction of at least one of the

coordinate axes should be determined by a limiting line.

Sequences are constructed

which converge to points on a limiting line and to its order k ~i.

With the

assumption that the single power series used in this calculation has only one singularity on its circle of convergence, sequence transformations,

including

it is proved that certain nonlinear

the e 1(s) transformation

defined by Shanks

([9], page 39) accelerate the convergence of these sequences.

211

The results obtained hold also for analytic initial value problems for other equations or systems of equations in two independent variables, when the given equation or system of equations can be replaced by a characteristic system in two independent variables.

In particular, limiting lines can be calculated by the

present method in the one-dimensional unsteady flow produced by a given piston motion.

Finally, the present method is also applicable to some of the series

occurring in [6].

2.

Limiting Lines of Order k In both the inverse blunt body problem and the inverse calculation of nozzle

flows, the stream function @ satisfies a quasi-linear second order partial differential equation of the form a~xx + b~xy + C~yy + d = 0, where the coefficients are analytic functions of their arguments.

(2.1) The independent

variables denote cartesian coordinates in the two-dimensional case and cylindrical coordinates in the axially symmetric case.

As in [i], pp. 491-493,

(2.1) can be

replaced by the system of characteristic equations y~ = h I x Y8 = h2 x8 d pe + h2q e + ~ x

= 0

(2.2)

P8 + hlq8 + ~a x8 = 0 ~

- px

- q y~

=

0

where h I and h 2 are the roots of the equation ah 2 - bh + c = 0.

(2.3)

Real values of = and 8 correspond to values of x and y for which (2.1) is hyperbolic.

It follows from the Cauchy-Kowalewski theorem that the solution of an

analytic initial value problem for (2.2) in the real eB-plane is analytic.

212

Given a solution of (2.2) which is analytic in a domain D of the real ~B-plane, the functions x(~,B) and Y(~,B) define a mapping which is one-to-one in any portion of D in which the Jacobian J = x yB-xBy ~ does not vanish.

Let k ~ l

be an integer, and let J and its derivatives of order up to and including k-i vanish along a curve C in D.

Then the image of C in the xy-plane is defined to be

a limiting line of order k. The well-known result ([i0], for example) that a regular arc of a limiting line of the first order is an envelope of one of the families of characteristics can also be shown to hold for limiting lines of order k > l .

As in the case of

limiting lines of first order, characteristics of the second family have infinite curvature at the limiting line for k>l. Finally, we can prove also that the behavior of flow quantities in the neighborhood of a limiting line of order k ~ l Theorem i.

is given by the following theorem:

Let a solution of (2.1) have a limiting line of order k ~ l

with

the equation x = Xo(Y), where Xo(Y) is analytic for yl
Let F(x,y) he any one of the flow variables.

Then F(x,y) has an expansion of the form

x ]n/(k+l) F(x,y) = n=o ~ an(Y) [i - x--~J for yl
3.

An Asymptotic Result An asymptotic result on which the present method of calculation is based is

given by the following theorem:

213 Theorem 2.

Let f(z) be analytic for

fzl

i except at z = I, with

anzn , Izl < 1

f(z) = ~ n=o

=~

bn(l-z)n/m,

Ii-zl < n, larg(l-z)I < ~,

n=o

where m=2,3,.., is an integer.

Then as n->~, we have

m~ a =(i/ b n \n/[o

N-I + j~_Icjn-J/m+ .=

0(n-N/m)}

for any N>_2, where cj = o when J + I is divisible by m.

When N + I is divisible

by m, the error is 0(n-(N+l)/m). Proof.

Let f(z) =

N+m-i ~ bj(l-z) j/m + fl(z). j=o

Then fl(z) is analytic for Izl __
Writing r=[N/m]+l, we see

that derivatives of fl(z) up to and including the rth are continuous on Izl=l, while the (r+l)st is discontinuous on IzI=l at z=l, but integrable there. Then with fl (z) = ~

a(1)n zn' Izl < i,

11=O

it is e a s i l y shown by u s e of C a u c h y ' s theorem t h a t

a (1) I=O(n - r - l ) n

as n--~.

The

stronger result fan(I) l=o(n -r-l) can be shown to hold by use of the Riemann-Lebesgue le~,na, b u t t h e weaker r e s u l t

Let N be such that bN#O.

is sufficient

for the present purpose.

Then

N-I an

=

where N+m-i J =N

n

"

214

It follows from Stirling's approximation that

lim nl+l(~l< ~ n->~o for I>o.

Hence, since [N/m] + 1 > N/m, we have

lira a (1) /{N/m~ = O. n.--~ n \n/ It follows that when bN~O , Rn,N=0(n-N/m-l).

Finally, by use of Stirling's

approximation, we obtain the result stated. Corollary i.

When bo#o , we have

an+z/a n =

=

Corollary 2.

_

-

3,'2

~

l+i/m

.

+. ~ d.n -j-1 j=l 3

+ 0(n -N-l)

m = 2

" + ~ d.n -j/m-I + 0(n -N/m-l) j=l 3

m > 2.

When bo=0 but bl~O ,

an+l/a n =

=

a - --~]{

_

l+2/m

+ j--liE e.n -~-lJ

+ O(n - N - l )

, m ~ 2

1 + E e.n-J/m-i + 0(n-N/m-l) j=l 3

m > 2.

A similar but more complicated theorem can be proved when there are several singularities on IzI=l, with a different value of m associated with each.

With

more than one singularity, however, the transformations used here become less effective.

4.

Sequences for Calculation of Limitin Z Lines Given an analytic initial value problem for an equation of the form of (2.1),

let the origin be taken at a point of the initial curve, and let (Xo,0) be a point on a limiting line of order k ~i.

Let F(x,y) be any one of the dependent

variables, such as the density or pressure in the inverse blunt body problem, and

215

as described in the introduction, let F(z,0) be analytic for Izl ~ x ° except at z = x . This assumption, that only one singularity lies on the circle of o convergence, appears to be verified in special cases which have been calculated. We have F(z,o) = ~

(4.1)

anzn , Izl < x

n= 0

o '

and by Theorem i,

(4.2)

F(z,o) = n=O~ b n ( l - xZ--) n/(k+l)o

in a cut neighborhood of z = x O .

It follows from Corollary 1 of Theorem 2

that when bo#0 in (4.2), (i

l+i/(k+l)) an = x n+l an+ I o

i + N-I ~ c.n-j-I + O(n -N-I) 1 J=l 3

k = I (4.3)

=

X

0

i + ~ c.n-j/(k+l)-I + O(n-N/(k+l)-l) , k > 2 j=l 3

From (4.3), we obtain rn

=

(Sn-l)(n+l)(n+2) = ~3

I

1

i + ~ d.n -3" + 0(n-N) j=l 3

k = i

= (l+i/(k+l)) I + N-I ~ d.n-j/(k+l) + 0 (n_N/(k+l ) )} , k ~ 2 j=l 3 2 when bo~0, where Sn=an+2an/an+ 1. N o t i n g t h a t ( n + 2 ) S n - ( n + l ) t e n d s t o u n i t y

(4.4)

as

n-~o, we see that the sequence t

= n

(Sn-l)(n+l)(n+2) (n+2)Sn_(n+l)

(4.5)

tends to the same limit as the left hand side of (4.4), and we can show that it has the same asymptotic form as (4.4) as n-~.

Denoting the coefficients of

the asymptotic expression for tn by ej, we find that e I = d I when k ~ 2 . When k = i, however, we have d I = e I + 3/2. Thus, while tn and rn have the same rate of convergence, one may be asymptotically more accurate than the other.

216

When bo=0 but bl#0 , sequences similar to the preceding can be obtained by use of Corollary 2 of Theorem 2.

By consideration of both the limit and the rate of

convergence of tn, it is possible to determine k and to decide whether or not

b

o

5.

vanishes.

Nonlinear Sequence Transformations Let A

be a given sequence, and let

n

B

= n

An+lAn 1-An

.

(5.1)

An+ l+An- i- 2An

The sequence Bn is equal to the first order transform el(An ) defined by Shanks [9] and to the sequence e~ n) o b t a i n e d by t h e e - a l g o r i t h m of Wynn [11], referred to as the Aitken 6 2 process. sequence A

n

Under certain conditions, a convergent

is transformed to a sequence B

converges more r a p i d l y .

and i s a l s o

In particular,

n

which has the same limit as A

the latter

n

and

holds when lim IAAn+I/AAnI#I. n..+~

When lim ' n'IAAn+l/AAI=l but lim AAn/ABn=S@I, the transformed sequence B converges n_>~ ~ n with the same rapidity as A . For this case, Shanks ([9], page 39) has defined n the transform

e

~S)(A n

)=

sB -A n n s-l"

(5.2)

A transformation equivalent to this, called Un, was also introduced by Lubkin [12]. Finally, Lubkin ([12], page 229) has introduced a more general transformation which is expressed in the present notation by AA BnA--~

- An n

Wn =

AA n AB n

(5.3) 1

With the preceding definitions, we can verify the following theorems by direct calculation:

217

Theorem 3.

Let a given sequence have the form

A

as n-~o for all N ~ I .

n

N-I = A + E a.n-J-~ + O(n-N-=) . j 3=o

Then

N-I e I(c~+l) (An) = A + ~ b.n -j-~-2 + 0(n-N-~-2). j=o 3

Corollary.

With the notation e~Sl)(e~S2)(An)) = e(Sl)e~S2)(An ), we have

e(~+2r+l)^(~+2r-l) ... e(e+l)(An ) = A + 0(n -=-2r-2) =i

Theorem 4.

Let a given sequence have the form

A n

as n-~o for all N ~ 2

N-I = A + ~ a.n -j/m-~ + 0(n -N/m-~) j=l 3

where m = 2,3, . . . .

,. , e(l+e+i/m) i ~'~n)

Then

. = A + N-I ~ 5.n -3/m-~ + 0(n_N/m_~) j=2 3

for N = 3,4, . . . .

Corollary.

(lq~+r/m) (l+e+(r-l))/m e (l+~+I/m) ( A ) el el "'" I n

= A + 0(n-(r+l)/m-=).

Theorems 3 and 4 have direct application to the sequences of Section 4. In particular, with a

An

(i=

3/2~

n__ n---~" an+ I

218

in the case k = i, we see that

el(2r+l)el(2r-l) ... e ~ 3 ) ( A )

= x

+ 0(n -2r-2) o

We see that the sequences A respectively,

n

in Theorems 3 and 4 must have e > 0 and e > -i/m,

in order to be convergent.

However,

the theorems do not specify the

sign of ~, and the corollaries show that the iterated transformations converge to A for sufficiently large values of r whether or not A Finally, the following theorems for W Theorem 5.

= A + n

as n ->~ for N = 1,2, . . . .

Then

W

= A + n

are nearly identical with the preceding:

N-I ~ a.n -j-~ + O(n -N-e) j=o

N-I ~ c.n -j-e-2 + O(n-N-~-2), j=o J

Let a given sequence have the form

A

= A + n

N-1 ~ a.n -j/m-~ + 0(n-N/m-a). j=l 3

as n ->~ for N = 2,3,... where m = 2,3, . . . .

W

= A + n

for N = 3,4 . . . .

converges.

Let a given sequence have the form

A

Theorem 6.

n

n

Then

N-I 0 (n-N/m-a) ~ c.n -j/m-~ + j=2 3

Corollaries which are exactly parallel to the corollaries of

Theorems 3 and 4 clearly hold.

219

6.

Numerical Examples The sequences of Section 4, together with the sequence transformations of

Section 5, have been applied to the calculation of limiting lines of first order in the inverse blunt body problem and in the calculation of nozzle flows.

The

calculation of the axial point on the upstream limiting line in the case of a paraboloidal shock at a free stream Mach number of 2 is shown in Table I, which was calculated by use of 39 coefficients of the power series for the density. Calculations were carried out on the CDC 6500 in double precision, using the method of [13]. The sequences r and t converge to 3/2 and are accelerated by n n the transformation e~ , 2) -

showing that the upstream limiting line is of first

order, and that b #0 in the expansion of the density in the neighborhood of the o limiting line.

Values of rn, tn, and e~2)(tn ) are given in Table 2, and show that

t is preferable to r in this case. n n

When the same calculations are repeated

using the power series for the stream function, it is found that rn and tn approach the limit 5/2.

Thus, it appears that bo=0 while bl#0 in the expansion of the

stream function in the neighborhood of the axial point of the limiting line. The rates of convergence shown in Section 5 cannot usually be realized beyond the transformation e1(3) when using single precision, because of loss of significance due to subtraction.

However, the accuracy obtained by use of e~ 3)- in

single precision has been found to be sufficient in all examples calculated. Agreement of successive terms of the sequence to 5 or 6 significant figures is usually obtained. Finally, calculations of limiting lines in nozzle flows have been carried out in single precision on the CDC 6500 using e~ 3).

The result of such a calculation

is shown in Figure 1 in the axially symmetric case, where u = 1 + (%/3/2)x on the centerline.

This centerline velocity distribution is the same as that of case (f)

of [14] after a change of reference quantities.

The sonic line, limiting

characteristic, and streamlines were calculated by the method of [15]. All

220

calculations

were carried out using 25 terms of the series in y, and convergence

to 5 or more figures was found except near the limiting line.

With the use of

double precision, the more rapidly convergent transformation e(5)e(3)(A ~ would i i n" give additional aceuracy. The portion of the streamline ~ = 0.2 to the left of the limiting characteristic can be taken as the wall of a nozzle in the subsonic and transonic regions, since the flow is analytic on the limiting characteristic up to and on the streamline.

However, the streamline ~ = 0.45 meets the limiting line above its

point of tangency with the limiting characteristic, and cannot be used as part of a nozzle contour.

The dashed portion of the limiting characteristic belongs to a

second solution of the flow equations having the limiting line shown.

The two

solutions correspond to the same analytic solution of equations (2.2) in the •

i

~8-plane, but on opposite sides of the curve J =, 0 which corresponds to the limiting line.

7.

References i.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, Interscience Publishers, New York, 1962.

2.

M. D. Van Dyke, "A Model of Supersonic Flow Past Blunt Axisymmetric Bodies with Application to Chester's Solution," Journal of Fluid Mechanics, Vol. 3, (1958), pp. 515-522.

3.

J. A. Leavitt, "Computational Aspects of the Detached Shock Problem," AIAA Journal, Vol. 6 (1968), pp. 1084-1088.

4.

C. Domb, "Order-Disorder Statistics, II. A Two Dimensional Model," Proceedings of the Royal Society of London, Series A, Vol. 199 (1949), pp. 199-221.

5.

L . W . Schwartz, "Hypersonic Flows Generated by Parabolic and Paraboloidal Shock Waves," Physics of Fluids, Vol. 17 (1974), pp. 1816-1821.

6.

M. E. Fisher, "The Nature of Critical Points," Lectures in Theoretical Physics, Vol. VlIC, University of Colorado Press, Boulder, 1964, pp. 1-159.

7.

P. R. Garabedian, "Global Structure of Solutions of the Inverse Detached Shock Problem," Problems of Hydrodynamics and Continuum Mechanics, published by the Society for Industrial and Applied Mathematics, 1969, pp. 318-322.

221

8.

E. V. Swenson, "Geometry of the Complex Characteristics in Transonic Flow," Comm. Pure and App. Math., Vol. 21 (1968), pp. 175-185.

9.

D. Shanks~ "Non-Linear Transformations of Divergent and Slowly Convergent Sequences," Journal of Mathematics and Physics, Vol. 34 (1955), pp. 1-42.

10.

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, New York, 1948.

Ii.

P. Wynn, "On a Device for Computing the em(Sn) Transform," MTAC, Vol. i0 (1956), pp. 91-96.

12.

S. Lubkin, "A Method of Summing Infinite Series," Journal of Research of the Bureau of Standards, Vol. 48 (1952), pp. 228-254.

13.

A. H. Van Tuyl, "Use of Pad~ Fractions in the Calculation of Blunt Body Flows," AIAA Journal, Vol. 9 (1971), pp. 1431-1433.

14.

G. V. R. Rao and B. Jaffe, '~ Numerical Solution of Transonic Flow in a Convergent-Divergent ~ozzle," Final Rept., Contract NAS7-635, March 1969, NASA.

15.

A. H. Van Tuyl, "Calculation of Nozzle Flows Using Pad~ Fractions," AIAA Journal, Vol. II (1973), pp. 537-541.

222

Table i.

Distance of Upstream Limiting Line from Nose of Shock for Paraboloidal Shock Wave r 2 = 2x with Free Stream Maeh Number = 2.

n

34

An = -(I

n+l'an+ 1312~an

-

0.083546753454

e~3)(An)

el(5)e(3)(An)l

0.0835451981166

0.0835451972856

35

.0835466622853

.0835451980201

.0835451972406

36

.0835465789026

.0835451979365

.0835451972874

37

.0835465024447

.0835451978019

.0835451972615

Table 2.

Sequences r

n

and t Corresponding to Axial Point on Upstream Limiting n

Line for Paraboloidal Shock Wave r 2 = 2x with Free Stream Mach Number = 2.

n

r

n

t

n

e~2)(tn )

34

1.5685409

1.5012612

1.5000020

35

1.5665483

1.5012223

1.5000018

36

1.5646684

1.5011858

1.5000017

37

1.5628918

1.5011514

1.5000016

223

1.1 = 0.45

STREAMLINES

0.9

I.

I

0,8

0.7~'~,~

~ = 0.2 LIMITING LINE

0,6

--

0,5

--

0.4--

0.3

~LIMITING CHARACTERISTIC

--

0.2--

0.1

--

0 -0.6

I

1

I

I

I

-0.5

-0.4

-0.3

-0.2

-0.1

0

I

I

0.1

0.2

0.3

X

Figure i.

Limiting line in the nozzle flow with centerline velocity distribution u = 1 +

(/'3/2)x.

THE USE OF PADE FRACTIONS IN THE CALCULATION OF NOZZLE FLOWS* Andrew H. Van Tuyl Naval Surface Weapons Center White Oak Laboratory Silver Spring, Maryland 20910 USA

i.

Introduction In nozzle design, flow calculations are usually carried out by an inverse

procedure in which either the Mach number or velocity is given on the centerline. Calculations in the supersonic region can be carried out accurately by the method of characteristics when the solution in the transonic region is known.

However, the

inverse problem is improperly-posed in the subsonic region, and it is therefore not possible to calculate the subsonic and transonic portions of the flow to arbitrary accuracy by means of finite-difference marching procedures without the use of complex extension.

Methods for calculating the flow in the transonic region

([i], for example) are mainly applicable to the design of nozzles with large throat radius of curvature, such as wind tunnel nozzles.

These methods are not

sufficiently accurate for the design of short nozzles, since rapid changes then occur near the throat.

Improved accuracy in the case of short nozzles is given by

[2], but the accuracy which can be obtained is limited by the fixed number of terms of the power series used.

Short nozzles with a uniform exit flow are of interest

in connection with gas dynamic lasers [3], where two-dimensional nozzles have been used, and with chemical lasers [4], where both two-dimensional and axially symmetric nozzles are applicable.

More accurate subsonic calculations have been carried out

in the axially symmetric case by Armitage [5] and Rao and Jaffe [6] by means of Garabedian's method of complex characteristics

[7, 8].

Calculations by the method

of complex characteristics have also been carried out by Solomon [9] in the twodimensional and axially symmetric cases.

*This work was supported by the Naval Surface Weapons Center Independent Research Fund.

226

When the flow quantity given on the centerline is analytic, the Cauchy-Kowalewski

theorem

([i0], page 39) that the inverse problem can be

solved in terms of power series in the neighborhood eenterline.

it follows from

of a given point of the

While these series are known to converge near the given point, the

region of convergence may be such that the series are either divergent or too slowly convergent at points of physical interest.

It was noted by Van Dyke [ii]

that the former occurs in the case of the inverse blunt body problem, presence of a limiting line (envelope of characteristics) continuation

of the flow.

due to the

in the upstream analytic

This limiting line lies closer to the shock than the

distance between the shock and the body, and hence, the body does not lie within the region of convergence

of the series.

However,

it was found in [12],

[13],

and [14] that Pade" fractions formed from these series give accurate results at the body, and can be used to compute the flow there.

A similar procedure has been

used in [15], where Pade" fractions are formed from certain power series in the neighborhood body problem,

of points of the centerline.

examples indicate that the use of Pade" fractions leads to convergence

when the series diverge, converge.

As in the case of the inverse blunt

and that convergence

is accelerated when the latter

The region of convergence of the series may be restricted by limiting

lines, as in the blunt body problem, calculations

or by complex singularities.

Accurate

can also be carried out in the supersonic region by use of Pade"

fractions when limiting lines do not occur near the centerline.

Near a limiting

line, convergence of both the series and the sequences of Pade" fractions

is found

to be slow. The present paper extends

[15] by investigating

fractions can be used in the calculation

additional ways in which Pade"

of nozzle flows.

In particular,

Pade"

fractions are formed from power series along given curves which intersect the centerline. expansions

By use of these results, more efficient use of the double Taylor obtained in [15] can be made.

227

2.

Power Series Solution of the Inverse Problem The coordinate system used is shown in Figure i, where x and y denote

Cartesian coordinates in the two-dimensional case and cylindrical coordinates in the axially symmetric case, and where the origin is at the sonic point on the centerline.

We will assume irrotational flow of a perfect gas with ratio of

specific heats y.

Let p* = c* = i, where p* is the critical density and c* is the

critical sound speed, and let the stream function satisfy u = (i/py°)~/~y, v = - (i/pyO)~/~x,

(2.1)

where u and v are the x and y components of velocity, respectively, p is the density, and ~ = 0 and i, respectively, in the two-dimensional and axially symmetric cases.

Substituting (2.1) into Bernoulli's equation and the condition

for irrotationality, we obtain (i/y2O)(~x 2 + ~y2) + [2/(y-1)]py+l _ [(y+l)/(y_l)]p2 = 0

(2.2)

and

P(~xx

+ ~yy _ oy-l~y) _ Px~x _ py~y = O,

(2.3)

respectively. Let the density, Mach number, and velocity on the centerline be denoted by Po(X), Mo(X) , and Uo(X) , respectively.

Then Po(X) is given in terms of Mo(X) by

the relation Po = {2/(y+l) + [ (y-i) / (y+l) ]Mo2} -I/(y-i),

(2.4)

and in terms of Uo(X), by Po = { (7+1)/2 - [(Y-I)/2]Uo2}I/(y-I)" Let M 0

(2.5)

(x) and Uo(X) have the expansions Mo(X) = ~ Mi(X-Xl)i-1 i=i

(2.6)

Uo(X ) ~ ui(x-xl )i-I i=l

(2.7)

and

228

in the neighborhood of x = x I. Given either (2.6) or (2.7), we can find the coefficients Pi of the expansion ¢o

Po (x) = .i__~lPi(X-Xl)i-I

(2.8)

by use of (2.4) or (2.5) respectively, and subroutines for power series manipulations.

We can then find the solution of the inverse problem in the

neighborhood of the point (Xl, 0) in the form 4"= ~ ~ ~ij y2i-l+~(X-Xl)J-l' i=l j=l P =

j--~lPij y 2i-2 (X-Xl)J-l" i:1 "=

(2.9)

(2.10)

We see that PlJ = PJ" To summarize the calculation of the remaining coefficients, let the left hand sides of (2.2) and (2.3) be denoted by A and B, respectively.

We have A = ~ ~ y2i-2(X-Xl )j-I i=l j=l ai~

(2.11)

B = ~ ~ bij y2i-l+O(x-xi)J-i i=l j=l

(2.12)

and

For J~2, we find that alj = 2(i+~)241141j + a~j,

(2.13)

411 = [i/(I+~)]{[ (y+l)/(y-1)]Pll2 - [2/(y-l) ]PllY+l}I/2

(2.14)

where

and where alj does not involve ~lj" For i>_2, we have aij = CI 4ij + C2 Pij + a~j,

(2.15) bij = C3 4ij + C4 Pij + b~j,

229

where C I = 2(i+0)(21-i+~)~ii C2 :

,

[2(y+l)/(y-l)](PllY-Pll

), (2.16)

C 3 = 2(2i-l+~)(i-l)Pll , C4 = -2(l+q)(i-l)~ll. and where a~j and b~j do not involve ~ij and Pij" coefficients can now be described as follows: alj = 0 in (2.13) for j = 2, 3, .... and calculating alj.

The calculation of the remaining

First, ~lj is calculated by setting

For each j, a~j is found by setting ~lj = 0

The coefficients ~iJ and Pij are then found for i ~ 2 by

setting aij and bij equal to zero in (2.15) and solving the resulting pair of equations simultaneously.

For given i and j, a~j and b~j are obtained by setting

~ij and PiJ equal to zero and calculating aij and bij.

Assuming that Pi is known

for i ~ i ~ 2K - i, we can find ~ij and Pij for i ~ i ~ K and i ~ J ~ 2K - 2i + i.

3.

Pade" Fractions Let f(z) = ~ c.z i i i=0

be a given power series with co @ O. are defined as follows:

(3.1)

Then Pade" fractions fk,n(Z), k ~ 0, n ~ O,

fk,n(Z) is a rational fraction with numerator and

denominator of degrees less than or equal to n and k, respectively, such that the Taylor expansion of fk,n(Z) in the neighborhood of the origin agrees with (3.1) to more terms than that of any other rational fraction with numerator of degree < n and denominator of degree ~ k ([16], page 377). exists and is unique.

This fraction always

Convenient methods of calculation include the QD algorithm

of Rutishauser [17] and the ~ - algorithm of Wynn [18]. The sequence fn,n(Z), which involves the first 2n + i terms of (4.1), is often found to converge much more rapidly than sequences in which either k or n

230

remains constant.

The convergence of fn,n(Z) has been proved only for special

classes of functions, and none of the available convergence results appears to be directly applicable to the series occurring in either the inverse blunt body problem or the inverse calculation of nozzle flows.

In both [14] and [15],

however, it was found that the sequence fn,n(Z) converges in most cases when the series diverges, and that it accelerates convergence when the latter converges.

4.

Power Series for Constant x Before Pade" fractions can be used, the solution of the inverse problem must

be expressed in terms of power series in one variable.

One approach is to write

(2.9) and (2.10) as single power series in y with coefficients which are functions of x.

We have @

= ~

' , , ~i~x;y

2i-i+~

(4.1)

i=l and oo

y2i-2,

(4.2)

~ij (X-Xl)J-i

(4.3)

Pi (x) = ~ Pij(X-Xl)J-i j=l

(4.4)

p =

Pi(X) i=l

where

~i (x) = ~ j=l and

when Ix - Xll and y are sufficiently small.

The coefficients ~i(x) and Pi(X) can

he calculated by use of Pade" fractions for a given value of x, after which Pade" fractions can be formed from (4.1) and (4.2).

Pade" fractions formed from (4.3)

and (4.4) are exact at x = Xl, but their accuracy decreases in general as Ix - Xll increases.

In a given case, however, it may be possible to find an

interval about x I throughout which acceptable accuracy is obtained.

It would be

more economical to use several such intervals with overlapping regions of validity than to solve the inverse problem for each value of x.

231

In nozzle calculations, expansions in powers of 4 may be more convenient than (4.1) and (4.2).

We know by the Cauchy-Kowalewski theorem that for x sufficiently

near Xl, (4.1) has a nonzero radius of convergence.

Assuming that ~l(X) is not

zero for a given x, we can invert (4.1) to obtain an expansion of the form

y = ~ Yi(X) 4 2i-1 i=l

(4.5)

in the two-dimensional case, and

Yi(x) ~i y2 = ~ i=l in the axially symmetric case for sufficiently small 4.

(4.6) By substituting (4.5) or

(4.6) in (4.2), we can obtain an expansion for 0 in powers of 4.

Similarly,

starting from the Taylor expansion of other flow quantities in the neighborhood of (Xl, 0), we can find their expansions in powers of 4 for constant x.

5.

Power Series Along Given Curves Another procedure for expressing the solution of the inverse problem in terms

of power series in one variable is to calculate ~ and 0 along given curves through (Xl, 0).

Families of such curves can be chosen which sweep out a neighborhood of

(Xl, 0).

Let the equation of a given curve be x - x I = g(y), where

g(y) = ~ gi yi i=l for sufficiently small y.

(5.1)

On substituting x - x 1 = g(y) in (2.9) and (2.10), we

obtain expansions of the forms

(5.2)

4 = ~ aiyi+° i=l and

0 = ~ biY i=l respectively.

i-i

,

(5.3)

With the equation of the given curve written in the alternate form

y = g(x - Xl) , y is replaced by x - x I in (5.1) through (5.3). case when g(y) is a function of y2, we have

In the special

232

= ~

aiy2i-l+a

(5.4)

i=l and 0 = ~ biY i=l along the given curve.

2i-2

(5.5)

We see that the coefficients a. and b. in (5.2) through I

1

(5.5) are known exactly, while the coefficients of (4.1) and (4.2) for x # x I are known only approximately.

In addition, ~i(x) and Pi(x) become less accurate for a

given x as i increases, since the number of terms of their Taylor expansions which are available then decreases.

However, the accuracy obtained in a particular

case is dependent on the choice of Xl, and can be determined only by trial. As in section 4, if a I @ 0, we can invert (5.2) to obtain y =

ci~

(5.6)

i=l in the two-dimensional case and y = ~ di~i/2 i=l in the axially symmetric case for sufficiently small ~.

(5.7) Similarly, if a I ~ 0 in

(5.4), we have y = ~

c

i=l

i*

2i-1

(5.8)

in the two-dimensional case, and y in the axially symmetric case.

2

" = ~ di~l i=l

(5.9)

By use of the preceding expansions for y, we can

obtain expansions for 0 and other flow quantities along the given curve in powers of ~ or ~i/2. A special case of an expansion along a curve of the form x - x I = g(y2) is given by the expansions at constant potential of [15].

The equation of the

equipotential through the point (Xl, 0) is of the form

x - xI =

piy i=l

2i

(5.10)

233

The calculation of the coefficients Pi' starting from the coefficients of (2.9) and (2.10), is described in detail in [15]. We see that the special choices g(y) = ~y and g(y) = ~y2 correspond to Moran's procedure in [13].

In [13], a double power series in x and y is written

so that terms of the same degree in x and y jointly are grouped together.

The

given series is then expressed as a power series in one of the variables with coefficients which are polynomials in the ratio of the variables, is held constant in a given calculation.

and this ratio

The use of the above choices for g(y),

after making the substitution y = z I/2 in the second, is seen to be equivalent to Moran's procedure. In the preceding two cases, it is possible to obtain the coefficients a i and b i explicitly as polynomials in ~.

In the case x - x I = ey (or y = a(x - Xl)),

a more symmetrical approach is to transform (2.9) and (2.10) to polar coordinates r and 8 with origin at (Xl, 0). y

2

On substituting x - x I = r cos e and

2 = r (I - cos28) in (2.9) and (2.10), we obtain power series in r for ~/yl+O

and p with coefficients which are polynomials in cos 8.

6.

The Equation of the Sonic Line The equation of the sonic line has an expansion of the form

x

=

i= I siY

(6.1)

in the neighborhood of the origin both in the two-dimensional and axially symmetric cases.

We can calculate the coefficients s. successively by substituting 1

(6.1) in the left hand side of the equation

oijy2i-2x j - l o 1

(6.2)

i=1 j=l and equating the coefficients of powers of y2 past the constant term to zero. This calculation can be carried out on a computer by use of subroutines for power

234

series manipulations.

Finally, we can obtain power series for flow quantities

along the sonic line by substituting (6.1) or its inversion in their Taylor expansions in the neighborhood of the origin.

7.

The Equation of the Limitin$ Characteristic As shown in Figure i, the limiting characteristic is the right-running

characteristic which passes through the origin.

In both the two-dimensional and

axially symmetric cases, the two families of characteristics satisfy the equations dy/dx = tan (8±~),

(7.1)

where tan8 = - ~x/~y and tans = (M2 - 1) -1/2 . Making these substitutions, we obtain [-~x(M2-1) I/2 ± ~y]dx/dy - ~y(M2-1) I/2 ~ ~x = 0.

(7.2)

The equation of the limiting characteristic has an expansion of the form x = ~

i=l

for sufficiently

s m a l l y.

r i Y 2i

Denoting the left

hand s i d e of ( 7 . 2 ) by C, we f i n d t h a t

C= £ ei y2i-l+e i=l On s e t t i n g

(7.3)

(7.4)

c 1 = 0 and p r o c e e d i n g as i n [ 1 5 ] , we o b t a i n r 1 = (~+1)P12/4

(7.5)

r I = (/5-i) (y+l)P12/8

(7.6)

i n t h e t w o - d i m e n s i o n a l c a s e , and

in the axially symmetric case. P12 is negative.

We note that both values are negative, since

We can verify that c.i = - [2i - (y+l)Pl2/2dl]ri + c~,

where cf does not involve r.. 1 1

(7.7)

Finally, in order to determine r i for i ~ 2, we

start from (7.5) in the two-dimensional case and (7.6) in the axially symmetric case, and calculate

235

r i = c~/[2i - (y+l)Pl2/2dl] , for i = 2, 3, . . . .

For each i, the calculation of cf is carried out by setting l

r° = 0 and calculating 1

c.. 1

After the coefficients

r i have been determined,

flow quantities along the limiting characteristic inversion in their Taylor expansions

8.

(7.8)

we can find power series for

by substituting

in the neighborhood

(7.1) or its

of the origin.

Numerical Results and Discussion As in [15], calculations have been carried out on the CDC 6400 in the axially

symmetric case with y = 1.4, using the velocity on the centerline given by example

(c) of [6].

Coefficients

of the Taylor expansions

computed for i ~ 25 in these calculations, equal to 49.

and hence, with the maximum value of j

The centerline velocity of example

present coordinates

(2.9) and (2.10) are

(c) is given in terms of the

and reference quantities by u = A I + A2/[A 3 + (x-A4)2]2 ,

where A 1 = 0.0657267,

A 2 = 3.1224458,

A 3 = 1.7125400,

centerline velocity is shown in Figure 2.

(8.1) and A 4 = 0.34000641.

In the present units,

This

the value of the

stream function which defines the nozzle contour in [6] becomes ~ = 0.31104. Figure 3 shows the calculated nozzle contour, characteristic

sonic line, and limiting

for the eenterline velocity of example

(c).

Equipotential ~urves

are also shown, as calculated in [15] by means of Pade" fractions formed from expansions

of x - x I and y2 in powers of 4.

calculated by use of Pade" fractions for x = x I.

Points on the nozzle contour were

formed from expansions

The sonic line was calculated

in powers of y2 or

in two ways, by solving the equation

p = 1 by Newton's method with the left hand side replaced by a Pade" fraction, and by forming a Pade" fraction from (6.1).

In the present example,

method was found to be more accurate than the latter for y > 0.6.

the former

Finally,

the

limiting characteristic was calculated in [15] by means of a Pade" fraction formed from the right hand side of (7.3).

Comparison is made with the calculations

[6] and [9] by the method of complex characteristics.

of

236

Similarly, calculations

Figure 4 compares the flow angle calculated in [15] with the

of [6] and [9].

of Pade ~ fractions

The calculations

of [15~ were carried out by means

formed from power series in y2 for x = x I.

In Tables 1 and 2, Pade" fractions of the form fn,n(Z) are compared with the corresponding partial sums of the series y.

(4.1) for x = x I = -I and for two values of

The tables indicate that the sequence of Pade ~ fractions converges

for both

values of y, while the series converges for the smaller value of y and diverges for the larger.

Tables 3 and 4 compare Pade" fractions and partial sums at the points

(-i, I.i) and (-I, 1.6) for expansions form x - x I = ~y2

with x I = -3.

of ~ in powers of y2 along parabolas of the

Finally, Tables 5 and 6 give the same comparison

for expansions of ~ in powers of y along the rays from (-3,0). along rays through hence,

The expansions

(-3, 0) contain all powers of y up to and including y48, and

49 coefficients

are found in the present calculations.

expansions use all 625 of the coefficients

We note that these

~ij obtained in the solution of the

inverse problem, while only 325 are used by the expansions

along parabolas.

The

series converges for both values of y in Tables 3 through 6, but more slowly than the sequence of Pade" fractions. Comparison through

of Tables 4 and 6 shows that the expansion along a straight line

(-3, 0) leads to more rapid convergence of the sequence of Pade" fractions

at (-i, 1.6) than the expansion along a parabola, while both tables show more rapid convergence than Table 2.

We see that the expansion along x = -i used in Tables 1

and 2 is a special case both of an expansion along a straight line through and of an expansion along a parabola. expansion

(xl, 0) in a particular

at which the flow is c~iculated.

(-i, 0)

It follows that the most suitable center of

case is not necessarily Further calculations

the one nearest the point

show that the portion of the

nozzle contour shown in Figure 3 can be calculated to 4 figures or more for x < -0.5 by means of Pade" fractions formed from power series along rays through

(-3, 0).

The remaining portion of the nozzle contour in Figure 3 can be calculated by use of Pade" fractions formed from expansions along rays through the origin and through the point

(-0.25, 0).

This method is found to be more economical than the procedure of

237

section 4, in which ~i(x) and Pi(X) are calculated by means of Pade

fractions,

since the range of x for which the latter are sufficiently accurate becomes small for i ~ 15 in the present calculations.

The time required to find expansions along

a given ray through (Xl, 0) when ~ij and PiJ are known is much less than that for solution of the inverse problem.

9.

References i.

J. R. Baron, "Analytic Design of a Family of Supersonic Nozzles by the Friedrichs Method," WADC Report 54-279, June 1954, Naval Supersonic Lab., MIT, Cambridge, Mass.

2.

D. F. Hopkins and D. E. Hill, "Effect of Small Radius of Curvature on Transonic Flow in Axisymmetric Nozzles," AIAA Journal, Vol. 4 (1966), pp. 1337-1343.

3.

J. D. Anderson, Jr. and E. L. Harris, "Modern Advances in the Physics of Gasdynamic Lasers," AIAA Paper 72-143, San Diego, Calif., 1972.

4.

T. A. Cool, "A Summary of Recent Research on Continuous Wave Chemical Lasers," Modern Optical Methods in Gas Dynamics Research, edited by D. S. Dosanjh, Plenum Press, New York, 1971, pp. 197-220.

5.

J. V. Armitage, "Flow in a deLaval Nozzle by the Garabedian Method," ARL 66-0012, Jan. 1966, Aerospace Research Labs., Dayton, Ohio.

6.

G. V. R. Rao and B. Jaffe, "A Numerical Solution of Transonic Flow in a Convergent-Divergent Nozzle," Final Report, Contract NAS7-635, March 1969, NASA.

7.

P. R. Garabedian, "Numerical Construction of Detached Shock Waves," Journal of Mathematics and Physics, Vol. 36 (1957), pp. 192-205.

8.

P. R. Garabedian, Partial Differential Equations, Wiley, New York, 1964, Chapter 16.

9.

J . M . Solomon, private communication, Jan. 1971, Naval Surface Weapons Center, White Oak, Silver Spring, Md.

i0.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, Interscience Publishers, New York, 1962.

ii.

M. D. Van Dyke, "A Model of Supersonic Flow Past Blunt Axisymmetric Bodies with Application to Chester's Solution," Journal of Fluid Mechanics, Vol. 3 (1958), pp. 515-522.

12.

A. H. Van Tuyl, "The Use of Rational Approximations in the Calculation of Flows With Detached Shocks," Journal of the Aero/Space Sciences, Vol. 27 (1960), pp. 559-560.

238

13.

J. P. Moran, "Initial Stages of Axisymmetric Shock-on-Shock Interaction for Blunt Bodies," Physics of Fluids, Vol. 13 (1970), pp. 237-248.

14.

A. H. Van Tuyl, "Use of Pad~Fraction in the Calculation of Blunt Body Flows," AIAA Journal, Vol. 9 (1971), pp. 1431-1433.

15.

A. H. Van Tuyl, "Calculation of Nozzle Flows Using Pad~ Fractions," AIAA Journal, Vol. Ii (1973), pp. 537-541.

16.

H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand, New York, 1948.

17.

P. Henrici, "The Quotient-Difference Algorithm," Further Contributions to the Solution of Simultaneous Linear Equations and the Determination of Eisenvalues, National Bureau of Standards Applied Mathematics Series, No. 49, 1958, pp. 23-46.

18.

P. Wynn, "On a Device for Computing the em(Sn) Transform," Math. Tables and Other Aids to Computation, Vol. I0 (1956), pp. 91-96.

Table I.

Pade" fractions for stream function at (-i, i.i) from power series on the line x = -i.

No. of terms

ii 13 15 17 19 21 23 25

Table 2.

Series

Pade" fractions

0.22136847 0.22139025 0.22137986 0.22138595 0.22138235 0.22138439 0.22138328 0.22138386

0.22138164 0.22138385 0.22138366 0.22138367 0.22138366 0.22138367 0.22138367 0.22138367

Pade" fractions for stream function at (-I, 1.6) from power series on the line x = -i.

No. of terms

ii 13 15 17 19 21 23 25

Series

0.86601815 0.72548523 -0.44164134 1.9872070 -3.3585190 8.5084977 -17.484076 38.076299

Pade" fractions

0.32739899 0.32960800 0.32908139 0.32913508 0.32907155 0.32911728 0.32912434 0.32911631

239

Table 3.

Pade ~ fractions for stream function at (-i, i.i) from power series along parabola through (-3, 0).

No. of terms

ii 13 15 17 19 21 23 25

Table 4.

Pade" fractions

0.22082834 0.22121951 0.22135508 0.22138460 0.22138868 0.22138819 0.22138670 0.22138537

0.22139959 0.22139261 0.22140720 0.22138378 0.22138298 0.22138362 0.22138367 0.22138366

Pade" fractions for stream function at (-i. 1.6) from power series along parabola through (-3, 0).

No. of terms

ii 13 15 17 19 21 23 25

Table 5.

Series

Series

Pade ~ fractions

0.33051685 0.32947504 0.32910476 0.32904116 0.32906254 0.32909083 0.32910815 0.32911519

0.32899479 0.32910273 0.32913379 0.32911671 0.32911675 0.32912158 0.32911528 0.32911678

Pade" fractions for stream function at (-i, i.i) from power series along straight line through (-3, 0).

No. of terms

35 37 39 41 43 45 47 49

Series

Pade" fractions

0.22138349 0.22138358 0.22138358 0.22138363 0.22138366 0.22138366 0.22138366 0.22138367

0.22138363 0.22138366 0.22138367 0.22138367 0.22138367 0.22138367 0.22138367 0.22138367

240 Table 6.

Pade" fractions for stream function at (-i, 1.6) from power series along straight line through (-3, 0).

No. of terms

35 37 39 41 43 45 47 49

"~

Series

Pade" fractions

0.32910043 0.32910691 0.32910959 0.32911685 0.32911775 0.32911594 0.32911575 0.32911506

0.32913355 0.32911442 0.32911469 0.32911464 0.32911467 0.32911474 0.32911480 0.32911470

"~k

/

~1/

SONlCUne~

LIMITING

J

\,{,' 0

Figure I.

\

/ -/~

/ ', I CHARACTERISTIC

~/ X

Schematic diagram of nozzle flow.

241

U 0.5

I

I

-3.0

Figure 2.

-2.0

I

I

-I. 0

X

0

0.5

Centerline velocity distribution of example (c) of [6].

2.5

o,

~ REF. 9

NOZZLE CONTOUR

11"0'I041

Y

-3.0

Figure 3.

-2.5

-2.0

-1.5 x

-1.0

-0.5

Subsonic and transonic portions of nozzle in example (c).

242

/

20

e,-

elm

PRESENTMETHOD I; 15 ......... REF.6 I/ o REF. 9 //~i 10

I

,

,~.~>....7.,I"

0.2 Figure 4.

0.4

y

,

,

,

0.6 Ii 0.8 !

Flow angle e along sonic line in example (c).

A B I B L I O G R A P H Y ON PADE A P P R O X I M A T I O N AND SOME RELATED MATTERS Claude BREZINSKI University

of

Lille

The aim of this paper is to give a bibliography on Pad6 approximants, related matters and applications.

some

For several years, Pad6 approximants had become more and more important in mathematics, numerical analysis and various fields in physics and engineering. They are closely related to many subjects in mathematics as analytic function theory, difference equations, the theory of moments, approximation, analytic continuation, continued fractions, etc. Then a whole bibliography should be a huge one to include the corresponding references of these disciplines. I have divided the references given in this paper into three sections. The first one deals with Pad6 approximation and I hope it is quite complete. The second one is devoted to continued fractions and includes only some historical references and most of the recent papers on this subject. The thrid section contains some applications of Pad6 approximants with a special emphasis on mechanics ; I have also included references on numerical analysis and methods to accelerate the convergence of sequences. Miscellaneous references end the paper. It is obvious that this bibliography is far to be complete because of the limited number of pages of this volume. It is, in fact, less than half of a bigger bibliography on this subject and on all the related matters that I hope to publish in the future. I apologize in advance for any errors and omissions and I thank everybody who would send me any new reference on this subject.

I - PADE approximants i - R.J. ARMS, A. EDREI - The Pad6 tables and continued fractions generated by totally positive sequences - in "Mathematical essays dedicated to A.J. Macintyre" (1970) 1-21. 2 - G.A. BAKER Jr.- The Pad6 approximant method and some related generalizations in "The Pad6 approximant in theoretical physics", G.A. Baker Jr. and J.L. Gammel eds., Academic Press, New York, 1970. 3 - G.A. BAKER Jr.- The theory and application of the Pad~ approximant method J. Adv. Theor. Phys., i (1965) 1-56. 4 - G.A. BAKER Jr.- Reeursive calculation of Pad6 approximants - in "Pad~ approximants and their applications", P.R. Graves - Morris ed., Academic Press, New-York, 1973. 5 - G.A. BAKER Jr.- Best error bounds for Pad~ approximants to convergent series of Stieltjes - J. Math. Phys., i0 (1969) 814-820. 6 - G.A. BAKER Jr.- Certain invariance and convergence properties of Pad~ approximants - Rocky Mountains J. Math., 4 (1974) 141-150. 7 - G.A. BAKER Jr.- The existence and convergence of subsequences of Pad6 approximants - J. Math. Anal. Appl., 43 (1973) 498-528.

246

8

- G.A. 1975.

BAKER Jr.- E s s e n t i a l

of Pads a p p r o x i m a n t s

9 - G.A. B A K E R Jr., J.L. G A M M E L eds. A c a d e m i c Press, New York, 1972.

- Academic

- The Pads a p p r o x i m a n t

Press,

New-York,

in t h e o r e t i c a l

physics-

i0 - G.A. BAKER Jr., J.L. GAMMER, J.G. WILLS - An i n v e s t i g a t i o n of the applicability of the Pads a p p r o x i m a n t m e t h o d - J. Math. Anal. Appl., 2 (1961) 405-418. ii - M. B A R N S L E Y - The b o u n d i n g p r o p e r t i e s of the m u l t i p o i n t Pads a p p r o x i m a n t series of Stieltjes - Rocky M o u n t a i n s J. Math., 4 (1974) 331-334.

to.a

12 - M. B A R N S L E Y - The b o u n d i n g p r o p e r t i e s of the m u l t i p o i n t Pads a p p r o x i m a n t series of Stieltjes on the r e a l line - J. Math. Phys., 14 (1973) 299-313.

to a

13 - M. BARNSLEY, P.D. R O B I N S O N - Dual v a r i a t i o n a l p r i n c i p l e s x i m a n t s - J. Inst. Math. Appl., 14 (1974) 229-250. 14

-

M.

BARNSLEY,

P.D.

tions for Kirkwood (1974) 251-265.

ROBINSON

- Riseman

and P a d S - t y p e

- Pads a p p r o x i m a n t b o u n d s and a p p r o x i m a t e integral e q u a t i o n s - J. Inst. Math. Appl.,

15 - J.L. BASDEVANT - Pads a p p r o x i m a n t s - in "Methods VoI. IV, Gordon and Breach, London, 1970. 16 - J.L. B A S D E V A N T - The Pads a p p r o x i m a t i o n der Physik, 20 (1972) 283-331. 17 - A.F. B E A R D O N - On the c o n v e r g e n c e Appl., 21 (1968) 344-346.

in s u b n u c l e a r

and its p h y s i c a l

of Pads a p p r o x i m a n t s

appro-

solu14

physics",

applications

- J. Math.

- Fort.

Anal.

18 - D° BESSIS - Topics in the theory of Pads a p p r o x i m a n t s - in "Pads approximants", P.R. G r a v e s - M o r r i s ed., The institute of physics, London, 1973. 19 - D. BESSIS, J.D. T A L M A N - V a r i a t i o n a l a p p r o a c h to the t h e o r y of o p e r a t o r Pads a p p r o x i m a n t s - Rocky M o u n t a i n s J. Math., 4 (1974) 151-158. 20 - C. BREZINSKI - C o n v e r g e n c e of Pads a p p r o x i m a n t s for some special sequences Thrid C o l l o q u i u m on a d v a n c e d c o m p u t i n g m e t h o d s in t h e o r e t i c a l physics, Marseille, 1973. 21 - C. BREZINSKI - Rhombus f r a c t i o n s - to appear. 22 - C. BREZINSKI to appear.

algorithms

- Computation

of Pads

23 - C. BREZINSKI - SSries de Stieltjes m e c h 58, Toulon, 12-14 m a i 1975.

connected

-

to the Pads table and c o n t i n u e d

approxlmants

and c o n t i n u e d

et a p p r o x i m a n t s

de Pads

24 - C. BREZINSKI Linear Algebra,

- Some r e s u l t s in the t h e o r y of the v e c t o r 8 (1974) 77-86.

25 - C. BREZINSKI Rocky M o u n t a i n s

- Some r e s u l t s and a p p l i c a t i o n s J. Math., 4 (1974) 335-338.

about

26 ~ C. BREZINSKI - G ~ n S r a l i s a t i o n s de la t r a n s f o r m a t i o n de Pads et de l ' e - a l g o r i t h m e - C a l c o l o (to appear).

fractions

- Colloque

~-algorithm

the vector

de Shanks,

Euro-

-

~-algorlthm

de la table

-

247

27 - J.S.R. ximants",

CHISHOLM - Mathematical P.R. G r a v e s - M o r r i s ed.,

t h e o r y of P a d 6 a p p r o x i m a n t s - in "Pad6 a p p r o The i n s t i t u t e of P h y s i c s , L o n d o n , 1973.

28 - J.S.R. C H I S H O L M - R a t i o n a l a p p r o x i m a n t s Math. Comp., 27 (1973) 841-848.

defined

29 - J.S.R. C H I S H O L M - A p p r o x i m a t i o n by sequences of m e r o m o r p h y - J. Math. Phys., 7 (1966) 39-44,

from

of P a d 6

double

power

approximants

series

-

in r e g i o n s

30 - J.S.R. C H I S H O L M - C o n v e r g e n c e p r o p e r t i e s of P a d 6 a p p r o x i m a n t s - in "Pad6 approximants and their applications", P.R. G r a v e s - M o r r i s ed., A c a d e m i c P r e s s , N e w - Y o r k , 1973. 31 - C.K. CHUI, 0. S H I S H A , P.W. S M I T H - P a d 6 rational approximants - J. A p p r o x . T h e o r y ,

approximants as l i m i t s 12 (1974) 2 0 1 - 2 0 4 .

32 - A.K. C O M M O N - P a d 6 a p p r o x i m a n t s P h y s . , 9 (1968) 32-38.

and bounds

33 - A.K. C O M M O N , P.R. G R A V E S - M O R R I S J. Inst. Math. A p p l i c s . , 13 (1974)

- Some properties 229-232.

34 - J.D.P. D O N N E L L Y - The P a d 6 t a b l e D.C. H a n d s c o m b ed., P e r g a m o n P r e s s ,

to s e r i e s

of b e s t

of S t i e l ~ e s

of C h i s h o l m

- in " M e t h o d s of n u m e r i c a l N e w - Y o r k , 1966.

- J. Math.

approximants

-

approximation",

35 - A. E D R E I - The P a d 6 t a b l e of m e r o m o r p h i c f u n c t i o n s of s m a l l o r d e r w i t h n e g a t i v e z e r o s a n d p o s i t i v e p o l e s - R o c k y M o u n t a i n s J. M a t h . , 4 (1974) 1 7 5 - 1 8 0 . 36 - A. E D R E I - C o n v e r g e n c e de la m 6 t h o d e m o r p h e s - C o l l o q u e E u r o m e c h 58, T o u l o n ,

de P a d 6 a p p l i q u 6 e s 1 2 - 1 4 m a i 1975.

aux fonctions

37 - D. E L L I O T - T r u n c a t i o n e r r o r s in P a d 6 a p p r o x i m a t i o n s to c e r t a i n an a l t e r n a t i v e a p p r o a c h - Math. Comp., 21 (1967) 3 9 8 - 4 0 6 .

functions

38 - S.T. E P S T E I N , M. B A R N S L E Y - A v a r i a t i o n a l a p p r o a c h to the t h e o r y point Pad6 approximants - J. Math. P h y s . , 14 (1973) 3 1 4 - 3 2 5 . 39 - W. F A I R - P a d 6 a p p r o x i m a t i o n Math. Comp., 18 (1964) 627-634.

to the

solution

40 - W. F A I R , Y.L. L U K E - P a d 6 a p p r o x i m a t i o n s Numer. M a t h . , 14 (1970) 3 7 9 - 3 8 2 . 41 - J. F L E I S C H E R - N o n l i n e a r P h y s . , 14 (1973) 246-248.

Pad6

of the R i c c a t i

to t h e o p e r a t o r

approximants

for

Legendre

m6ro-

of m u l t i -

equation

-

exponential

series

:

-

- J. Math.

42 - J. F L E I S C H E R - N o n l i n e a r P a d 6 a p p r o x i m a n t s f o r L e g e n d r e s e r i e s - in " P a d 6 approximants and their applications", P.R. G r a v e s - M o r r i s ed., A c a d e m i c P r e s s , N e w - Y o r k , 1973. 43 - J. F L E I S C H E R - G e n e r a l i z a t i o n s of P a d 6 a p p r o x i m a n t s - in " P a d 6 P.R. G r a v e s - M o r r i s ed., The i n s t i t u t e of p h y s i c s , L o n d o n , 1973. 44 - N.R. Theory,

F R A N Z E N - Some c o n v e r g e n c e 6 (1972) 254-263.

results

for Pad6

approximants

45 - N.R. F R A N Z E N - C o n v e r g e n c e o f P a d 6 a p p r o x i m a n t s for a certain m e r o m o r p h i c f u n c t i o n s - J. A p p r o x . T h e o r y , 6 (1972) 264-271.

approximants",

- J. A p p r o x .

class

of

248

46 - J.L. GAMMEL - Review of two recent g e n e r a l i z a t i o n s of the Pad6 approximant in "Pad6 approximants and their applications", P.R. G r a v e s - M o r r i s ed., Academic Press, New-York, 1973. 47 - J.L. GAMMEL - Effect of r a n d o m errors (noise) in the terms of a power series on the convergence of the Pad~ a p p r o x i m a n t s - in "Pad~ approximants", P.R. G r a v e s - M o r r i s ed., The institute o f physics, London, 1973. 48 - J.L. GAMMEL, J. N U T T A L L - C o n v e r g e n c e of Pad~ approximants to quasi-analytic functions beyond n a t u r a l b o u n d a r i e s - J. Math. Anal. Appl. 49 - J. G I L E W I C Z - N u m e r i c a l detection of the best Pad~ a p p r o x i m a n t and determination of the Fourier coefficients of the i n s u f f i c i e n t l y sampled functions - in "Pad~ approximants and their applications", P.R. G r a v e s - M o r r i s ed., Academic Press, New-York, 1973. 50 - J. GILEWICZ - Totally monotonic and totally positive sequences for the Pad~ approximants m e t h o d - Colloque E u r o m e c h 58, Toulon, 12-14 mai 1975. 51 - J.E. GOLDEN, J.H. McGUIRE, J. N U T T A L L - C a l c u l a t i n g Bessel functions w i t h Pad~ approximants - J. Math. Anal. Appl., 43 (1973) 754-767. 52 - W.B. GRAGG - The Pad~ table and its r e l a t i o n to certain algorithms of numer i c a l analysis - SIAM Rev., 14 (1972) 1-62. 53 - W.B. GRAGG - On Hadamar's theory of polar singularities - in "Pad~ approxim a n t s and their applications", P.R. G r a v e s - M o r r i s ed., A c a d e m i c Press, New-York, 1973. 54 - W.B. GRAGG, G.D. J O H N S O N - The Laurent - Pad~ table - P r o c e e d i n g s IFIP Congress, North-Holland, 1974. 55 ~ P.R. G R A V E S - M O R R I S ed. - Pad~ approximants and their applicationsA c a d e m i c Press, New-York, 1973. 56 - P.R. G R A V E S - M O R R I S ed. - Pad~ approximants - The institute of physics, London, 1973. 57 - T.N.E. GREVILLE - On some conjectures of P. Wynn concerning the E-algorithm MRC Technical summary report 877, Madison, 1968. 58 - A.S. H O U S E H O L D E R - The Pad~ table, the Frobenius identities and the q-d a l g o r i t h m - Linear Algebra, 4 (1971) 161-174. 59 - A.S. HOUSEHOLDER, G.W. STEWART - Bigradients, Hankel determinants and the Pad~ table - in "Constructive aspects of the f u n d a m e n t a l t h e o r e m of algebra", B. D e j o n and P. Henrici eds., A c a d e m i c Press, New-York, 1969. 60 - R.C. JOHNSON - A l t e r n a t i v e a p p r o a c h to Pad~ approximants - in "Pad~ appreximants and their application", P.R. G r a v e s - M o r r i s ed., Academic Press, NewYork, 1973. 61 - W.B. JONES - T r u n c a t i o n error b o u n d for continued fractions and Pad~ approximants - in "Pad~ approximants and their applications", P.R. G r a v e s - M o r r i s ed., A c a d e m i c Press, New-York, 1973. 62 - W.B. JONES - Analysis of t r u n c a t i o n error of a p p r o x i m a t i o n s b a s e d on the Pad~ table and continued fractions - R o c k y Mountains J. Math., 4 (1974) 241-250.

249

63 - W.B. JONES, W.J. THRON - On c o n v e r g e n c e Anal., 6 (1975) 9-16. 64 - I.M. LONGMAN (1971) 53-64.

- Computation

Of Pad6 a p p r o x i m a n t s

of the Pad6 table

65 - Y.L. LUKE - E v a l u a t i o n of the g a m m a f u n c t i o n SIAM J. Math. Anal., i (1970) 266-281. 66 - Y.L. LUKE - The Pad6 table and the T - m e t h o d 110-127. 67 - A. MAGNUS - Certain c o n t i n u e d Math. Z., 78 (1960) 361-374. 68 - A. MAGNUS - P - f r a c t i o n s 4 (1974) 257-260.

fractions

- Intern.

by means

and the Pad6 table

J. comp.

Math.,

3B

of Pad6 a p p r o x i m a t i o n s -

- J. Math.

associated

- SIAM J. Math.

Phys.,

37 (1958)

w i t h the Pad6 table -

- Rocky M o u n t a i n s

J. Math.,

69 - D. M A S S O N - Hilbert space and Pad~ a p p r o x i m a n t - in "The Pad~ a p p r o x i m a n t in t h e o r e t i c a l physics", G.A. Baker Jr. and J.L. G a m m e l eds., A c a d e m i c Press, New-York, 1970. 70 - D. M A S S O N - Pad6 a p p r o x i m a n t s and Hilbert spaces - in "Pad6 a p p r o x i m a n t s and their a p p l i c a t i o n s " , P.R. G r a v e s - M o r r l s ed., A c a d e m i c Press, 1973. 71 - J.H. M c C A B E Pad6 q u o t i e n t s

- A formal e x t e n s i o n of the Pad6 table to include - J. Inst. Math. Applies., 15 (1975) 363-372.

two point

72 - J.B.

- A note on the e - a l g o r i t h m

17-24.

McLEOD

- Computing,

7 (1971)

73 - G. MERZ - Pad6sche N [ h e r u n g s b r [ c k e und I t e r a t i o n s v e r f a h r e n h ~ h e r e n D o c t o r a l thesis, T e c h n i s c h e Hochschule Clausthal, Germany, 1967.

Ordnung

-

74 - J. N U T T A L L - The c o n n e c t i o n of Pad6 a p p r o x i m a n t s w i t h s t a t i o n a r y v a r i a t i o n a l p r i n c i p l e s and the c o n v e r g e n c e of c e r t a i n Pad6 a p p r o x i m a n t s - in "The Pad6 a p p r o x i m a n t in t h e o r e t i c a l physics", G.A. Baker Jr. and J.L. G a m m e l eds., A c a d e m i c Press, New-York, 1970. 75 - J. N U T T A L L - The c o n v e r g e n c e J. Math. Anal. Appl., 31 (1970)

of Pad6 a p p r o x i m a n t s 147-153.

of m e r o m o r p h i c

functions

-

76 - J. N U T T A L L - V a r i a t i o n a l p r i n c i p l e s and Pad6 a p p r o x i m a n t s - in "Pad6 approximants and their applications", P.R. G r a v e s - M o r r i s ed., A c a d e m i c Press, New-York, 1973. 77 - J. N U T T A L L - The c o n v e r g e n c e J. Math., 4 (1974) 269-272.

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104 - P. W Y N N - Z u r t h e o r i e d e r m i t g e w i s s e n s p e z i e l l e n P a d ~ s c h e n t a f e l n - Math. Z., 109 (1969) 66-70.

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105 - P. W Y N N - U p o n the P a d 6 t a b l e d e r i v e d J. Numer. A n a l . , 5 (1968) 8 0 5 - 8 3 4 .

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106 - P. W Y N N - U p o n s y s t e m s of r e c u r s i o n s w h i c h o b t a i n the P a d 6 t a b l e - N u m e r . M a t h . , 8 (1966) 2 6 4 - 2 6 9 . 107 - P. W Y N N - L ' s - a l g o r i t m o (1961) 4 0 3 - 4 0 8 .

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108 - P. W Y N N - U p o n a c o n j e c t u r e c o n c e r n i n g equations, and certain other matters - MRC M a d i s o n , 1966. 109 - P. W Y N N - G e n e r a l p u r p o s e M a t h . , 6 (1964) 22-36.

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116 - G. B L A N C H - N u m e r i c a l 7 (1964) 3 8 3 - 4 2 1 .

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115 - S. B E R N S T E I N , 0. S Z A S Z - U b e r m i t e i n e r A n w e n d u n g a u f die R e i h e

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119 - V.F. C O W L I N G , general continued

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120 - I.V. C Y G A N K O V - S o l u t i o n o f R i c c a t i e q u a t i o n s by c o n t i n u e d Perm. Gos. Univ. Ucen. Zap. M a t . , 17 (1960) 99-107. 121 - I.V. C Y G A N K O V f r a c t i o n s - Perm.

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122 - J.D.P. D O N N E L L Y - C o n t i n u e d f r a c t i o n s - in " M e t h o d s o f n u m e r i c a l m a t i o n " , D.C. H a n d s c o m b ed., P e r g a m o n P r e s s , O x f o r d , 1965. 123 - H.G. E L L I S - C o n t i n u e d f r a c t i o n s s o l u t i o n s o f the g e n e r a l t i a l e q u a t i o n - R o c k y M o u n t a i n s J. M a t h . , 4 (1974) 3 5 3 - 3 5 6 . 124 - L. E U L E R - De f r a c t i o n i b u s St P ~ t e r s b o u r g , ii (1739).

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FAIR - Noncommutative 226-232.

127 - W. F A I R - A c o n v e r g e n c e J. A p p r o x . T h e o r y , 5 (1972)

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FRANK - Corresponding 89-108.

133 - E. diques

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type

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Anal.,

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130 - D.A. F I E L D - A p r i o r i K ( i / b n) - R o c k y M o u n t a i n s

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139 - W.B. G R A G G - T r u n c a t i o n ii (1968) 3 7 0 - 3 7 9 .

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mit

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142 - T.L. 7 (1965)

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143 - T.L. H A Y D E N M a t h . , 4 (1974)

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145 - E. (1922)

HELLINGER 18-29.

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der

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144 - T.L. H A Y D E N - A c o n v e r g e n c e Math. Soc., 14 (1963) 546-552.

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Stieltjesschen

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Potenzreihe

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147 - P. H E N R I C I , P. P F L U G E R - T r u n c a t i o n e r r o r f r a c t i o n s - Numer. M a t h . , 9 (1966) 120-138.

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152 - T.H. J E F F E R S O N - T r u n c a t i o n Numer. A n a l . , 6 (1969) 3 5 9 - 3 6 4 . 153 - W.B. J O N E S Ph.D. thesis,

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86

fractions

149 - K.L. H I L L A M , W.J. T H R O N - A g e n e r a l c o n v e r g e n c e c r i t e r i o n f o r f r a c t i o n s K ( a n / b n) - Proc. Amer. Math. Soc., 16 (1965) 1 2 5 6 - 1 2 6 2 . 150 - A. H U R W I T Z GrSssen - Acta

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158 - W.B. J O N E S , W.J. continued fractions

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163 - A.N. K H O V A N S K Z I - The a p p l i c a t i o n o f c o n t i n u e d f r a c t i o n s a n d t h e i r g e n e r a l i z a t i o n s to p r o b l e m s in a p p r o x i m a t i o n t h e o r y - P. N o o r d h o f f , G r o n i n g e n , 1963. 164 - E. L A G U E R R E - Sur la r @ d u c t i o n & t e n d u e de f o n c t i o n s - C.R. Acad.

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d'une

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167 - R.E. L A N E - The c o n v e r g e n c e a n d v a l u e s of p e r i o d i c Bull. Amer. Math. Soc., 51 (1945) 2 4 6 - 2 5 0 .

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fractions

168

region

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169 - L.J. regions 170 - W. Soc.,

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L A N G E , W.J. T H R O N - A t w o - p a r a m e t e r f a m i l y of b e s t t w i n c o n v e r g e n c e f o r c o n t i n u e d f r a c t i o n s - Math. Z e i t . , 73 (1969) 2 7 7 - 2 8 2 .

LEIGHTON - A test-ratio 45 (1939) 97-100.

171 - W. L E I G H T O N - C o n v e r g e n c e 5 (1939) 2 9 8 - 3 0 8 .

test for

theorems

continued

fractions

for continued

- Bull.

fractions

172 - W. L E I G H T O N - S u f f i c i e n t c o n d i t i o n s f o r t h e c o n v e r g e n c e f r a c t i o n - D u k e Math. J., 4 (1938) 7 7 5 - 7 7 8 . 173 - W. L E I G H T O N , W.T. S C O T T - A g e n e r a l Amer. Math. Soc., 45 (1939) 5 9 6 - 6 0 5 . 174 - W. L E I G H T O N , W.J. T H R O N - On v a l u e Amer. Math. Soc., 48 (1942) 9 1 7 - 9 2 0 . 175 - W. L E I G H T O N , W.J. T H R O N - C o n t i n u e d Math. J., 9 (1942) 7 6 3 - 7 7 2 . 176 - W. L E I G H T O N ,

W.J.

THRON

continued

regions

fraction

for

fractions

- On the c o n v e r g e n c e

Math.

- D u k e Math.

J.,

of a c o n t i n u e d

expansion

continued

with

Amer.

fractions

complex

of continued

- Bull.

elements

fractions

- Bull.

- Duke

to

255

meromorphic

functions

- Ann.

Math.,

(2)

44

(1943)

177 - J.S. M A C N E R M E Y - I n v e s t i g a t i o n s c o n c e r n i n g f r a c t i o n s - D u k e Math. J., 26 (1959) 6 6 3 - 6 7 8 . 178 - A. M A G N U S - E x p a n s i o n (1962) 2 0 9 - 2 1 6 .

of power

179 - A. M A G N U S - On P - e x p a n s i o n s S e l s k a b s S k r i f t e r (1964), n°3.

series

of p o w e r

80-89.

positive

into P-fractions

series

- Det.

180 - A. M A G N U S - The c o n n e c t i o n b e t w e e n P - f r a c t i o n s Prec. Amer. Math. $oc., 25 (1970) 6 7 6 - 6 7 9 .

- Math.

Kgl.

Norske

and associated

181 - A. M A R K O V - D e u x d ~ m o n s t r a t i o n s de la c o n v e r g e n c e c o n t i n u e s - A c t a M a t h . , 19 (1895) 93 - 104.

de c e r t a i n e s

182 - A. M A R K O V - N o t e sur les f r a c t i o n s c o n t i n u e s - Bull. Acad. Imp. Sc. St P ~ t e r s b o u r g , 5 (2) (1895) 9-13. 183 - D.F. M A Y E R S - E c o n o m i z a t i o n n u m e r i c a l a p p r o x i m a t i o n " , D.C.

definite

classe

continued

Z.,

Videnskabers

fractions

- On t r u n c a t i o n A n a l . , 3 (1966)

-

fractions

Physico-Math.

of c o n t i n u e d f r a c t i o n s - in " M e t h o d s of H a n d s c o m b ed., P e r g a m o n P r e s s , O x f o r d , 1965.

1 8 4 - J.H. Mc C A B E - A c o n t i n u e d f r a c t i o n e x p a n s i o n , w i t h a t r u n c a t i o n e s t i m a t e , f o r D a w s o n ' s i n t e g r a l - M a t h . C o m p . , 28 (1974) 8 1 1 - 8 1 6 . 185 - E.P. M E R K E S S I A M J. N u m e r .

80

errors for 486-496.

continued

fraction

186 - E.P. M E R K E S , W.T. S C O T T - C o n t i n u e d f r a c t i o n s o l u t i o n s e q u a t i o n - J. Math. Anal. A p p l . , 4 (1962) 3 0 9 - 3 2 7 .

error

computations

of the Riccati

187 - R. DE M O N T E S S U S DE B A L L O R E - Sur les f r a c t i o n s Bull. Soc. Math. de F r a n c e , 30 (1902) 28-36.

continues

alg~briques

-

188 - R. DE M O N T E S S U S DE B A L L O R E - Sur les f r a c t i o n s C.R. Acad. Sc. P a r i s , 134 (1902) 1 4 8 9 - 1 4 9 1 .

continues

alg~briques

-

189 - R. DE M O N T E S S U S DE B A L L O R E - Sur les f r a c t i o n s c o n t i n u e s L a g u e r r e - C.R. Acad. Sc. P a r i s , 140 (1905) 1 4 3 8 - 1 4 4 0 .

alg~briques

de

190 - R. DE M O N T E S S U S DE B A L L O R E - Les f r a c t i o n s A c t a M a t h . , 32 (1909) 2 5 7 - 2 8 1 . 191 - T. M U I R - A t h e o r e m in c o n t i n u a n t s w i t h an i m p o r t a n t a p p l i c a t i o n - Phil.

continues

-

alg~briques

- E x t e n s i o n of a t h e o r e m H a g . , (5) 3 (1877).

-

in c o n t i n u a n t s

192 - T. M U I R - On the p h e n o m e n o n of g r e a t e s t m i d d l e in the c y c l e of a c l a s s of a c l a s s of p e r i o d i c c o n t i n u e d f r a c t i o n - Prec. Roy. Soc. E d i n b u r g h , 12 (1884) 578-592. 193

- H. P A D E - M 6 m o i r e sur les d 6 v e l o p p e m e n t s en f r a c t i o n s c o n t i n u e s de la f o n c t i e n e x p o n e n t i e l l e p o u v a n t s e r v i r d ' i n t r o d u c t i o n ~ la t h 6 o r i e d e s f r a c t i o n s c o n t i n u e s a l g 6 b r i q u e s - Ann. Fac. Sci. de l'Ec. Norm. Sup., 16 ( 1 8 9 9 ) 395-436.

194 - H. P A D E C.R. Acad.

- Sur la d i s t r i b u t i o n des r 6 d u i t e s Sc. P a r i s , 130 (1900).

anormales

d'une

fonction

-

256

195 - H. P A D E - Sur le d 6 v e l o p p e m e n t en f r a c t i o n c o n t i n u e de la f o n c t i o n F ( h , l , h ' , u ) et la g 6 n 6 r a l i s a t i o n de la t h ~ o r i e des f o n c t i o n s s p h 6 r i q u e s C.R. Acad. Sc. P a r i s , 141 ( 1 9 0 5 ) 8 1 9 - 8 2 1 . 196 - O. P E R R O N - 0 h e r z w e i K e t t e n b r [ c h e n Math. - Nat. KI. S. - B., (1957) 1-13. 197 - O. P E R R O N 1950.

- Die

Lehre yon dem

198 - O. P E R R O N - U b e r P a l e r m o , 29 (1910). 199 - S. P I N C H E R L E Norm. Sup., (3)

eine

Kettenbr[chen

spezielle

Klasse

- Sur les f r a c t i o n s 6 (18-89) 1 4 5 - 1 5 2 .

200 - A. P R I N G S H E I M - U b e r G l i e d e r n - Sb. M ~ n c h e n ,

v o n H.S.

von

continues

Wall

- Bayer,

- Chelsea

Pub.

Kettenbr~chen

alg~briques

ein K o n v e r g e n z k r i t e r i u m 29 ( 1 8 9 9 ) .

fir

Akad.

Co.,

-

Wiss.

New-York,

- Rend. Circ. Mat.

- Ann.

Sci.

Kettenbr[che

Ec.

mit positiven

201 - E. R O U C H E - M 6 m o i r e sur le d 6 v e l o p p e m e n t d e s f o n c t i o n s en s ~ r i e s O r d o n n ~ e s s u i v a n t les d ~ n o m i n a t e u r s des r ~ d u i t e s d ' u n e f r a c t i o n c o n t i n u e - J. Ee. P o l y t e c h n i q u e , 37 (1858). 202 - H. R U T I S H A U S E R - 0 b e r e i n e V e r a l l g e m e i n e r u n g Math. M e c h . , 38 (1958) 2 7 8 - 2 7 9 . 203 - H. R U T I S H A U S E R - B e s o h l e u n i g u n g d e r K e t t e n b r [ c h e n - ZAMM, 38 (1958) 187.

der

Konvergenz

Kettenbr[che

einer gewisse

204 - F.T. S C H U B E R T - De t r a n s f o r m a t i o n e s e r i e s in f r a c t i o n e m Acad. So. Imp. St P ~ t e r s b o u r g , 7 ( 1 8 1 5 - 1 8 1 6 ) . 205 - W.T. S C O T T - The c o r r e s p o n d i n g M a t h . , (2) 51 (1950) 56-67.

continued

206 - H. S I E B E C K - 0 b e r p e r i o d i s c h e r m a t h . , 33 (1846) 71-77.

Kettenbr6che

207 - V. S I N G H , W . J . T H R O N n u e d f r a c t i o n s - Proc. 208

- J. S H E R M A N nued fraction

fraction

Klasse

von

continuam

- Mem.

of a J - f r a c t i o n

- Ann.

- J. f ~ r die r e i n e

- A f a m i l y of b e s t t w i n c o n v e r g e n c e Amer. M a t h . Soc., 7 (1956) 2 7 7 - 2 8 2 .

- On t h e n u m e r a t o r s - Tran. Amer. Math.

- Z. A n g e w .

regions

u. a n g e w .

for conti-

of the c o n v e r g e n t s of the S t i e l t j e s S o c . , 35 (1933) 64-87.

conti-

209 - T.J. S T I E L T J E S - Sur la r ~ d u c t i o n en f r a c t i o n c o n t i n u e d ' u n e s ~ r i e p r ~ c ~ d a n t s u i v a n t les p u i s s a n c e s d e s c e n d a n t e s de la v a r i a b l e - Ann. Fac. Sci. T o u l o u s e , 3 (1889) 1-17. 210 - T.J. S T I E L T J E S - N o t e appl. m a t h . , 25 (1891).

sur q u e l q u e s

fractions

211 - T.J. S T I E L T J E S - R e c h e r c h e s sur les f r a c t i o n s U n i v . T o u l o u s e , 8 (1894) 1-122. 212 - T.J. S T I E L T J E S - S u r un d ~ v e l o p p e m e n t Sc. P a r i s , 99 (1884) 5 0 8 - 5 0 9 . 213 - T.J.

STIELTJES

- Recherches

continues

continues

en f r a c t i o n s

sur les f r a c t i o n s

- Quart.

- Ann.

continues

continues

J. p u r e

Fac.

- C.R.

- M6moires

and

Sci.

Acad.

pr6sen-

257

t6s p a r divers savants ~ l'acad6mie des Sciences Sciences et M a t h ~ m a t i q u e s , (2) 32 (2) (1892).

de l ' i n s t i t u t

de France,

214 - W.B. SWEEZY, W.J. THRON - E s t i m a t e s of the speed of c o n v e r g e n c e c o n t i n u e d fractions - SIAM J. Numer. Anal., 4 (1967) 254-270. 215 - W.J. THRON - On p a r a b o l i c c o n v e r g e n c e Math. Zeit., 69 (1958) 172-182.

regions

for c o n t i n u e d

of certain

fractions

-

216 - W.J. THRON - Recent approaches to c o n v e r g e n c e t h e o r y of c o n t i n u e d f r a c t i o n s in "Pad~ a p p r o x i m a n t s and their a p p l i c a t i o n s " , P.R. G r a v e s - M o r r i s ed., A c a d e m i c Press, New-York, 1973. 217 - W.J. THRON - A survey of recent Rocky M o u n t a i n s J. Math., 4 (1974)

convergence 273-282.

results

218 - W.J. THRON - Two families of twin c o n v e r g e n c e f r a c t i o n s - Duke Math. J., i0 (1943) 677-685. 219 - W.J. THRON - Twin convergence Math., 66 (1944) 428-439. 220 - W.J. THRON - A family Duke Math. J., ii (1944)

regions

of simple 779-791.

for c o n t i n u e d

regions

for c o n t i n u e d

convergence

221 - W.J. THRON - C o n v e r g e n c e regions for the g e n e r a l Amer. Math. Soc., 49 (1943) 913-916. 222 - W.J. THRON - Some p r o p e r t i e s of the c o n t i n u e d Bull. Amer. Math. Soc., 54 (1948) 206-218.

for c o n t i n u e d

fractions

regions

fractions-

- Amer.

for c o n t i n u e d

continued

fraction

J.

fractions-

fraction

- Bull.

(l+doz)+K(z/(l+dnZ))-

223 - W.J. THRON - Z w i l l i n ~ s k o n v e r g e n z g e b l e t e f~r K e t t e n b r ~ c h e l+K(a /i), deren eines die K r e i s s c h e i b e |a2n_l 1 < 02 ist. - Math. Zeit., 70 (1959) n 310-344. 224 - W.J. THRON - Convergence regions for c o n t i n u e d f r a c t i o n s processes - Amer. Math. Monthly, 68 (1961) 734-750.

and other infinite

225 - W.J. THRON - Convergence of sequences of linear f r a c t i o n a l t r a n s f o r m a t i o n s and of c o n t i n u e d fractions - J. Indian Math. Soc., 27 (1963) 103-127. 226 - W.J. THRON - Some results and p r o b l e m s in the a n a l y t i c fractions - Math. Student, 32 (1964) 61-73. 227 - W.J. THRON - On the c o n v e r g e n c e fractions - Math. Zeit., 85 (1964)

t h e o r y of c o n t i n u e d

of the even part of c e r t a i n 268-273.

228 - J. T R E M B L E Y - Recherches sur les f r a c t i o n s Belles-Let., Berlin (1794-1795)

continues

229 - K.T. V A H L E N - 0her N [ h e r u n g s w e r t e angew, math,, 115 (1895) 221-233.

und K e t t e n b r [ c h e

230 - E.B. VAN V L E C K - On an e x t e n s i o n Amer. Math. Soc., 4 (1908) 297-332.

Of the 1894 m e m o i r

- Mem.

V A N V L E C K - On the c o n v e r g e n c e

and c h a r a c t e r

Acad.

Roy.

Sc.

- J. fur die reine u.

of Stieltjes

231 - E.B. VAN V L E C K - On the c o n v e r g e n c e of c o n t i n u e d f r a c t i o n s elements - Trans. Amer. Math. Soc., 2 (1901) 215-233. 232 - E.B.

continued

- Trans.

with c o m p l e x

of the c o n t i n u e d

fraction

258

alZ/l+... 233

- Trans.

- E.B. V A N V L E C K coefficients have

Amer.

Math.

Soc.,

476-483.

- On t h e c o n v e r g e n c e of a l g e b r a i c c o n t i n u e d f r a c t i o n s w h o s e l i m i t i n g v a l u e s - Trans. Amer. Math. Soc., 5 (1904) 253-262.

234 - E.B. V A N V L E C K - S e l e c t e d t o p i c s t i n u e d f r a c t i o n s - Amer. Math. Soc. 1903. 235

2 (1901)

in t h e t h e o r y of d i v e r g e n t C o l l o q u i u m Pub., i, B o s t o n

- E.B. V A N V L E C K - On t h e c o n v e r g e n c e of t h e c o n t i n u e d o t h e r c o n t i n u e d f r a c t i o n s - Ann. M a t h . , 3 (1901) 1-18.

s e r i e s and conColloquium,

fraction

236 - H. V O N K O C H - Q w e l q u e s t h & o r ~ m e s c o n c e r n a n t la t h ~ o r i e t i o n s c o n t i n u e s - O f v e r s i g t af. Kongl. V e t e n s k a p s - Akad.

of G a u s s

and

g ~ n ~ r a l e des f r a c F6rhandlingen, 52

(1895). 237 - B. V I S C O V A T O F F - De la m ~ t h o d e g ~ n ~ r a l e p o u r r ~ d u i r e t o u t e s s o r t e s de q u a n t i t ~ s en f r a c t i o n s c o n t i n u e s - Mem. Acad. Sc. Imp. St P & t e r s b o u r g , 1 (1803-1804). 238 - H. W A A D E L A N D - T - f r a c t i o n s J. M a t h . , 4 (1974) 3 9 1 - 3 9 4 .

from

a different

point

of v i e w

- Rocky

Mountains

239 - H. W A A D E L A N D - On T - f r a c t i o n s of f u n c t i o n s h o l o m o r p h i c a n d b o u n d e d c i r c u l a r d i s c - N o r s k e Vid. Selsk. S k r ( T r o n d h e i m ) 1964, n°8. 240 - H. W A A D E L A N D - A c o n v e r g e n c e p r o p e r t y N o r s k e Vid. Selsk. Skr ( T r o n d h e i m ) 1966,

of certain n°9.

T-fraction

241 - H.S, W A L L - On some c r i t e r i a of C a r l e m a n for t h e c o m p l e t e J - f r a c t i o n - Bull. Amer. Math. Soc., 54 (1948) 528-532. 242 - H.S. W A L L - C o n v e r g e n c e of c o n t i n u e d f r a c t i o n s Bull. Amer. Math. Soc., 55 (1949) 3 9 1 - 3 9 4 . 243

- H.S. W A L L - N o t e 56 (1949) 96-97.

244 - H.S. W A L L of S t i e l t j e s 73-84.

on a p e r i o d i c

expansions

convergence

in p a r a b o l i c

fraction

- Amer.

domains

Math.

-

of a

-

Monthly,

- Concerning continuous continued fractions and certain systems i n t e g r a l e q u a t i o n s - Rend. Circ. Mat. di P a l e r m o , 11,2 (1953)

245 - H.S. W A L L - P a r t i a l l y 7 (1956) 1 0 9 0 - 1 0 9 3 .

bounded

246 - H.S. W A L L - S o m e c o n v e r g e n c e M o n t h l y , 54 (1957) 95-103. 247 - H.S. W A L L - N o t e 51 (1945) 930-934. 248 - H.S. W A L L t i o n - Bull.

continued

in a

on a c e r t a i n

continued

problems

continued

fractions

for

- Proc.

continued

fraction

fractions

- Bull.

- N o t e on the e x p a n s i o n of a p o w e r s e r i e s Amer. Math. Soc., 51 (1945) 97-105.

on a r b i t r a r y

J-fractions

Soc.,

- Amer.

Math.

Soc.,

into a continued

frac-

- Bull,

Amer.

Math.

Math.

249 - H.S. W A L L - C o n t i n u e d f r a c t i o n e x p a n s i o n s f o r f u n c t i o n s p a r t s - Bull. Amer. Math. Soc., 52 (1946) 138-143. 250 - H.S. W A L L - T h e o r e m s (1946) 671-679.

Amer.

with positive

Amer.

Math.

real

Soc.,

52

259

251 - H.S.

WALL - Bounded

J-fractions

- Bull.

Amer.

Math.

252 - H.S. W A L L - S o m e r e c e n t d e v e l o p m e n t s in the t h e o r y Bull. Amer. Math. Soc., 47 (1941) 4 0 5 - 4 2 3 . 253

Soc.,

52

of continued

- H.S. W A L L - A c o n t i n u e d f r a c t i o n r e l a t e d to s o m e p a r t i t i o n E u l e r - Amer. Math. M o n t h l y , 48 (1941) 1 0 2 - 1 0 8 .

254 - H.S. W A L L - The b e h a v i o u r of c e r t a i n s i n g u l a r l i n e - Bull. Amer. Math. S o c . , 255 - H.S. W A L L - C o n t i n u e d f r a c t i o n s Math. Soc., 50 (1944) 1 1 0 - 1 1 9 . 256 - H.S. W A L L - C o n v e r g e n c e Soc., 17 (1931) 5 7 5 - 5 7 9 .

criteria

257 - H.S. W A L L - On the c o n t i n u e d t i o n s - Bull. Amer. Math. Soc., 258 - H.S. W A L L - C o n t i n u e d f o r m a t i o n s - Bull. Amer.

Stieltjes 48 (1942)

and bounded

for

fractions 39 (1933)

fractions and cross-ratios groups Math. Soc., 40 (1934) 5 7 8 - 5 9 2 . of the form

260 - H.S. W A L L - On c o n t i n u e d f r a c t i o n s Math. Soc., 44 (1938) 94-99.

representing

261 - H.S. W A L L - C o n t i n u e d f r a c t i o n s a n d t o t a l l y Amer. Math. Soc., 48 (1940) 1 6 5 - 1 8 4 . - The a n a l y t i c

268 - H.S. W A L L , J.J. D E N N I S f r a c t i o n - D u k e Math. J., 264 - H.S. W A L L , H.L. f r a c t i o n s - Trans.

theory

- The l i m i t c i r c l e 12 (1945) 2 5 5 - 2 7 3 .

Amer.

meromorphic

of C r e m o n a

Math.

func-

trans-

Amer.

functions

- Bull.

Amer.

sequences

- Trans.

- Van Nostrand,

for a positive

G A R A B E D I A N - H a u s d o r f f m e t h o d s of s u m m a t i o n Amer. Math. Soc., 48 ( 1 9 4 0 ) 1 8 5 - 2 0 7 .

the

Amer.

- Bull.

fractions

case

near

- Bull.

- Bull.

-

of

l+Kl(bZ/l)

monotone

of continued

fractions

fractions

686-693.

fractions

formulas

functions

which represent 942-952.

259 - H.S. W A L L - On c o n t i n u e d f r a c t i o n s Math. Soc., 41 ( 1 9 3 5 ) 7 9 7 - 7 3 6 .

262 - H.S. W A L L Y o r k , 1948.

continued 427-431.

analytic

continued

(1946)

New-

definite

J-

and continued

265 - H.S. W A L L , E, H E L L I N G E R - C o n t r i b u t i o n s to the a n a l y t i c t h e o r y f r a c t i o n s a n d i n f i n i t e m a t r i c e s - Ann. M a t h . , 44 ( 1 9 4 3 ) 1 0 3 - 1 2 7 .

of c o n t i n u e d

266 - H.S. W A L L , W. L E I G H T O N - On the t r a n s f o r m a t i o n f r a c t i o n s - Amer. J. M a t h . , 58 (1936) 2 6 7 - 2 8 1 .

of c o n t i n u e d

267 - H.S. W A L L , J.F. P A Y D O N - The c o n t i n u e d t r a n s f o r m a t i o n s - D u k e Math. J., 9 (1942) 268 - H.S. W A L L , and odd parts

fraction 360-372.

R.E. L A N E - C o n t i n u e d f r a c t i o n s w i t h - Trans. Amer. Math. S o c . , 67 (1949)

and

convergence

as a s e q u e n c e

absolutely 368-380.

269 - H.S. W A L L , W.T. S C O T T - On the c o n v e r g e n c e a n d d i v e r g e n c e f r a c t i o n s - Amer. J. M a t h . , 69 (1947) 5 5 1 - 5 6 1 . 270 - H,S. W A L L , s e r i e s - Ann.

W.T. S C O T T - C o n t i n u e d f r a c t i o n M a t h . , 41 (1940) 3 2 5 - 3 4 9 .

expansion

of linear

convergent

even

of c o n t i n u e d

for arbitrary

power

260

271 - H.S. W A L L , W.T. S C O T T - V a l u e r e g i o n s Math. Soc., 47 (1941) 5 8 0 - 5 8 5 . 272 - H.S. 1-18.

WALL,

for c o n t i n u e d

W.T.

SCOTT

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fractions

273 - H.S. W A L L , W.T. Trans. Amer. Math.

SCOTT Soc.,

- A convergence theorem 47 (1940) 1 5 5 - 1 7 2 .

274 - H.S. W A L L , M. W E T Z E L - Q u a d r a t i c f o r m s n u e d f r a c t i o n s - D u k e M a t h . J.~ ii ( 1 9 4 4 )

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Math.

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275 - H.S. W A L L , M. W E T Z E L - C o n t r i b u t i o n s to the a n a l y t i c T r a n s . Amer. Math. Soc., 55 (1944) 3 7 3 - 3 9 7 .

- Bull.

13

Amer.

(1939)

fractions

regions

theory

for

-

conti-

of J - f r a c t i o n s

276 - R. W I L S O N - D i v e r g e n t c o n t i n u e d f r a c t i o n s a n d p o l a r s i n g u l a r i t i e s - Proc. Lond. Math. Soc., 26 (1927) 1 5 9 - 1 6 8 / 27 (1928) 4 9 7 - 5 1 2 / 28 (1928) 1 2 8 - 1 4 4 . 277 - R. W I L S O N - D i v e r g e n t c o n t i n u e d f r a c t i o n s Proc. Lond. Math. Soc., 30 (1928) 38-57.

and nonpolar

singularities

-

278 - A. W I N T N E R - E i n q u a l i t a t i v e s K r i t e r i u m f i r dis K o n v e r g e n z K e t t e n h r ~ c h e - M a t h . Z e i t . , 30 (1929) 2 8 5 - 2 8 9 .

des a s s o z i i e r t e n

279 - L. W U Y T A C K - E x t r a p o l a t i o n to t h e p o l a t i o n - R o c k y M o u n t a i n s J. M a t h . ,

fraction

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inter-

280 - P. W Y N N - C o n t i n u e d f r a c t i o n s w h o s e c o e f f i c i e n t s o b e y a n o n c o m m u t a t i v e of m u l t i p l i c a t i o n - Arch. Rat. M e c h . A n a l . , 12 ( 1 9 6 3 ) 2 7 3 - 3 1 2 . 281 - P. W Y N N - A n o t e on t h e c o n v e r g e n c e of c e r t a i n n o n c o m m u t a t i v e f r a c t i o n s - M R C t e c h n i c a l s u m m a r y r e p o r t 750, M a d i s o n , 1967. 282 - P. W Y N N - V e c t o r

continued

fractions

- Linear

Algehra,

283 - P. W Y N N - U p o n t h e d e f i n i t i o n o f an i n t e g r a l as t h e f r a c t i o n - Arch. Rat. M e c h . A n a l . , 28 (1968) 8 3 - 1 4 8 .

continued

i (1968)

limit

law

357-395.

of a c o n t i n u e d

284 - P. W Y N N - An a r s e n a l of A l g o l p r o c e d u r e s for the e v a l u a t i o n of c o n t i n u e d f r a c t i o n s a n d f o r e f f e c t i n g the e p s i l o n a l g o r i t h m - C h i f f r e s , 9 (1966) 327-362. 285 - P. W Y N N - F o u r l e c t u r e s tions - CIME summer school

on t h e n u m e r i c a l l e c t u r e s , 1965.

286 - P. W Y N N - C o n v e r g i n g f a c t o r s i (1959) 2 7 2 - 3 0 7 a n d 3 0 8 - 3 2 0 .

for

application

continued

fractions

of continued

- Numer.

frac-

Math.,

287 - P. W Y N N - A n o t e on a m e t h o d of B r a d s h a w f o r t r a n s f o r m i n g s l o w l y c o n v e r g e n t s e r i e s a n d c o n t i n u e d f r a c t i o n s - Amer. M a t h . M o n t h l y , 69 (1962) 8 8 3 - 8 8 9 . 288 - P. W Y N N - A n u m e r i c a l 175-196.

s t u d y of a r e s u l t

of S t i e l t j e s

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6 (1963)

289 - P. W Y N N - On s o m e r e c e n t d e v e l o p m e n t s in the t h e o r y a n d a p p l i c a t i o n c o n t i n u e d f r a c t i o n s - S I A M J. N u m e r . A n a l . , i (1964) 1 7 7 - 1 9 7 , 290 - P. W Y N N - N o t e on a c o n v e r g i n g N u m e r , M a t h . , 5 (1963] 3 3 2 - 3 5 2 .

factor

for a certain

continued

of

fraction

-

-

261

291 - P. W Y N N - The n u m e r i c a l e f f i c i e n c y o f c e r t a i n c o n t i n u e d K o n i n k l . N e d e r l . Akad. Wet., 65 A (1962) 1 2 7 - 1 4 8 . 292 - P. W Y N N - C o m p l e x n u m b e r s on a p p l i c a t i o n to the t h e o r y r e p o r t 646, M a d i s o n , 1966.

fraction

expansions

a n d o t h e r e x t e n s i o n s to the C l i f f o r d a l g e b r a w i t h of c o n t i n u e d f r a c t i o n s , M R C t e c h n i c a l s u m m a r y

llI-

Applications

293 - A.C. A I T K E N - On B e r n o u l l i ' s n u m e r i c a l s o l u t i o n Proc. Roy. Soc. E d i n b u r g , 46 (1926) 2 8 9 - 3 0 5 .

of a l g e b r a i c

294 - A.C. A I T K E N - On i n t e r p o l a t i o n b y p r o p o r t i o n a l parts, d i f f e r e n c e s - Proc. Edin. Math. Soc., 3 (1932) 56-76. 295 - R. A L T - M 6 t h o d e s A - s t a b l e s p o u r l ' i n t 6 g r a t i o n mal conditionn6s - T h ~ s e 3~me c y c l e , P a r i s , 1971.

des

296 - F.L. B A U E R - C o n n e c t i o n s b e t w e e n t h e q - d a l g o r i t h m e - a l g o r i t h m of W y n n - D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t

equations

without

syst~mes

use

of

diff6rentiels

of R u t i s h a u s e r a n d the Teeh. Rep. B a / 1 0 6 , 1957.

297 - F.L. B A U E R - N o n l i n e a r s e q u e n c e t r a n s f o r m a t i o n s - in " A p p r o x i m a t i o n f u n c t i o n s " , G a r a b e d i a n ed., E l s e v i e r , N e w - Y o r k , 1965. 298 - F.L.

BAUER

- The g - a l g o r i t h m

- SIAM

J.,

8 (1960)

-

of

1-17.

300 - M. B A U S S E T - U n e d 6 t e r m i n a t i o n des s u r f a c e s de c h o c p a r a e c 6 1 6 r a t i o n c o n v e r g e n c e - C o l l o q u e E u r o m e e h 58, T o u l o n , 1 2 - 1 4 m a i 1975.

de

301 - L.C. B R E A U X - A n u m e r i c a l s t u d y o f t h e a p p l i c a t i o n of a c c e l e r a t i o n techniq u e s a n d p r e d i c t i o n a l g o r i t h m s to n u m e r i c a l i n t e g r a t i o n - M. Sc. T h e s i s , Louisiana State Univ., New-Orleans, 1971. 302 - C. B R E Z I N S K I - C o n v e r g e n c e d ' u n e f o r m e e o n f l u e n t e C.R. Acad. Sc. Paris, 273 A (1971) 5 8 2 - 5 8 5 .

de l ' E - a l g o r i t h m e

303 - C. B R E Z I N S K I - L ' E - a l g o r i t h m e et les s u i t e s t o t a l e m e n t o s c i l l a n t e s - C.R. Acad. Sc. P a r i s , 276 A (1973) 3 0 5 - 3 0 8 . 304 - C. rique

BREZINSKI - M6thodes d'acc616ration - T h ~ s e , Univ. de G r e n o b l e , 1971.

de la c o n v e r g e n c e

305 - C. B R E Z I N S K I - A p p l i c a t i o n du o - a l g o r i t h m e C.R. Aead. Sc. P a r i s , 270 A (1970) 1 2 5 2 - 1 2 5 3 . 306 - C. B R E Z I N S K I (1971) 1 5 3 - 1 6 2 .

- Etudes

sur les

307 - C. B R E Z I N S K I RIRO, R3 (1970)

- RSsultats 147-153.

get

309 - C. B R E Z I N S K I - R e v i e w of m e t h o d s Rend. Mat. Roma, 7 (1974) 3 0 3 - 3 1 6 .

- Numer.

de s o m m a t i o n

de l ' s - a l g o r i t h m e

to a c c e l e r a t e

the

et

en a n a l y s e

~ la q u a d r a t u r e

p-algorithmes

s u r les p r o c 6 d 6 s

308 - C. B R E Z I N S K I - F o r m e c o n f l u e n t e M a t h . , 23 (1975) 3 6 3 - 3 7 0 .

monotones

-

num6rique

Math.,

num6-

-

17

et l ' c - a l g o r i t h m e

topologique

convergence

-

- Numer.

of s e q u e n c e s -

-

262

310 - C. BREZINSKI - Acc~l~ration de la c o n v e r g e n c e en analyse num~rique P u b l i c a t i o n 41, labo. de Calcul, Univ. de Lille, 1973. 311 - C. BREZINSKI - Conditions d ' a p p l i c a t i o n et de convergence de p r o c ~ d ~ s d ' e x t r a p o l a t i o n - Numer. Math., 20 (1972) 64-79. 312 - C. BREZINSKI - A p p l i c a t i o n de l'e-algorithme ~ la r ~ s o l u t i o n des syst~mes non lin~aires - C.R. Acad. Sc. Paris, 271 A (1970) 1174-1177. 313 - C. BREZINSKI - Integration des syst~mes d i f f ~ r e n t i e l s ~ l'aide du 0-algorithme - C.R. Acad. Sc. Paris, 278 A (1974) 875-878. 314 - C. BREZINSKI - Sur un algorithme de r ~ s o l u t i o n des syst~mes non lin~aires C.R. Acad. Sc. Paris, 272 A (1971) 145-148. 315 - C. BREZINSKI - Numerical stability of a quadratic m e t h o d for solving systems of non linear equations - Computing, 14 (1975) 205-211. 316 - C. BREZINSKI - C o m p u t a t i o n of the eigenelements of a m a t r i x by the e-algorithm - Linear Algebra, ii (1975) 7-20. 317 - C. BREZINSKI, M. CROUZEIX - Remarques sur le proc~d~ A 2 d'Aitken - C.R. Acad. Sc. Paris, 270 A (1970) 896-898. 318 - C. BREZINSKI, A.C. RIEU - The solution of systems of equations using the e-algorithm, and an a p p l i c a t i o n to b o u n d a r y value p r o b l e m s - Math. Comp., 28 (1974) 731-741. 319 - H. CABANNES, M. BAUSSET - A p p l i c a t i o n of the m e t h o d of Pad~ to the determ i n a t i o n of shock wavres - in "Problems of h y d r o d y n a m i c s and continuum mechanics ", in honor of L.I. Sedov, English ed. publ. by SIAM (1968) 95-114. 320 - H.K. CHENG, M. HAFEZ - Convergence a c c e l e r a t i o n of iterative solutions and transonic flow computations - Colloque Euromech 58, Toulon, 12-14 mai 1975. 321 - J.S.R. CHISHOLM - Pad~ a p p r o x i m a n t s and linear integral equations - in "The Pad~ approximant in t h e o r e t i c a l physics", G.A. Baker Jr. and J.L. Gammel eds., Academic Press, New-York, 1970. 322 - J.S.R. CHISHOLM - A p p l i c a t i o n of Pad~ a p p r o x i m a t i o n to n u m e r i c a l integration - Rocky Mountains J. Math., 4 (1974) 159-168. 323 - J.S.R. CHISHOLM - Pad~ a p p r o x i m a t i o n of single v a r i a b l e integrals Colloquium on c o m p u t a t i o n a l methods in t h e o r e t i c a l physics, Marseille, 1970. 324 - J.S.R. CHISHOLM - A c c e l e r a t e d convergence of sequences of quadratur e approximants - second c o l l o q u i u m on c o m p u t a t i o n a l m e t h o d s in t h e o r e t i c a l physics, Marseille, 1971. 325 - J.S.R. CHISHOLM, A.C. GENZ, G.E. ROWLANDS - A c c e l e r a t e d convergence of sequences of quadrature a p p r o x i m a t i o n - J. Comp. Phys., i0 (1972) 284-307. 326 - J. COUNTS, J.E. AKIN - The a p p l i c a t i o n of continued fractions to wave p r o p a g a t i o n problems in a v i s c o e l a s t i c rod - DEMVPI R e s e a r c h Rep. i-i, Dept. of Eng. Mech., V i r g i n i a p o l y t e c h n i c institute, Blacksburg, 1968. 327 - J.D.P. D O N N E L L Y - Applications of the q-d and e-algorithms - in "Methods of numerical approximation", D.C. Handscomb ed., P e r g a m o n Press, Oxford, 1965.

263

328 - B.L. EHLE - A - s t a b l e m e t h o d s and Pad~ a p p r o x i m a t i o n s SIAM J. Math. Anal., 4 (1973) 671-680.

to the e x p o n e n t i a l

-

329 - B.L. EHLE - On Pad~ a p p r o x i m a t i o n s to the e x p o n e n t i a l f u n c t i o n and A - s t a b l e m e t h o d s for the n u m e r i c a l s o l u t i o n of initial value p r o b l e m s - R e s e a r c h rep. CSRR 2010, dept. of AACS, Univ. of Waterloo, Ontario, 1969. 330 - M. F R O I S S A R T - A p p l i c a t i o n s of the Pad~ m e t h o d to n u m e r i c a l analysis C o l l o q u i u m on c o m p u t a t i o n a l m e t h o d s in t h e o r e t i c a l physics, Marseille, 1970. 331 - E. G E K E L E R - U b e r den e - a l g o r i t h m u s 51 (1971) 53-54.

von W y n n - Z. Angew.

332 - E. G E K E L E R - On the s o l u t i o n of systems of e q u a t i o n s r i t h m of Wynn - Math. Comp., 26 (1972) 427-436.

Math.

Mech.,

by the e p s i l o n

algo-

333 - A. GENZ - The e - a l g o r i t h m and some other a p p l i c a t i o n s of Pad~ a p p r o x i m a n t s in n u m e r i c a l a n a l y s i s - in "Pad~ a p p r o x i m a n t s " , P.R. Graves- Morris ed., The institute of physics, London, 1973. 334 - A. G E N Z - A p p l i c a t i o n s of the e - a l g o r i t h m to q u a d r a t u r e p r o b l e m s - in "Pad~ a p p r o x i m a n t s and their a p p l i c a t i o n s ", P.R. G r a v e s - M o r r i s ed., A c a d e m i c Press, New-York, 1973. 335 - B. G E R M A I N

BONNE - T r a n s f o r m a t i o n s

de suites

- RAIRO,

R1

(1973)

84-90.

336 - M. HAFEZ, H.K. CHENG - On a c c e l e r a t i o n of c o n v e r g e n c e and s h o c k - f i t t i n g t r a n s o n i c f l o w c o m p u t a t i o n s - Univ. So. Calif. Memo., 1973. 337 - M, HAFEZ, H.K. CHENG - C o n v e r g e n c e a c c e l e r a t i o n and shock f i t t i n g t r a n s o n i c a e r o d y n a m i c s c o m p u t a t i o n s - AIAA paper, No. 75-51. 338 - P. HENRICI - Some a p p l i c a t i o n s of the q u o t i e n t - d i f f e r e n c e Symp. Appl. Math., 20 (1963) 159-183. algorithm

- NBS appl.

340 - D . C . ' J O Y C E - Survey of e x t r a p o l a t i o n Rev., 13 (1971) 435-490.

processes

in n u m e r i c a l

341 - D.K. K A H A N E R - N u m e r i c a l (1972) 689-694.

quadrature

by the e - a l g o r i t h m

for

algorithm

339 - P. HENRICI - The q u o t i e n t - d i f f e r e n c e 49 (1958) 23-46.

Math.

- Proc.

series,

analysis

- Math.

in

Comp.,

342 - D. LEVIN - Development of n o n - l i n e a r t r a n s f o r m a t i o n s for i m p r o v i n g gence of sequences - Intern. J. Comp. Math., B3 (1973) 371-388.

- SIAM

26

conver-

343 - I.M. LONGMAN - Use of Pad~ table for a p p r o x i m a t e Laplace t r a n s f o r m inversion - in "Pad~ a p p r o x i m a n t s and their a p p l i c a t i o n s " , P.R. G r a v e s - M o r r i s ed., A c a d e m i c Press, New-York, 1973. 344 - E.D. MARTIN, H. LOMAX - Rapid finite d i f f e r e n c e c o m p u t a t i o n and t r a n s o n i c a e r o d y n a m i c flows - AIAA paper, No. 74-11. 345 - I. MARX - Remark c o n c e r n i n g a n o n l i n e a r tion - J. Math. Phys., 42 (1963) 834-335. 346 - K.J.

OVERHOLT

- Extended

sequence

Aitken acceleration

to sequence

- BIT,

5 (1965)

of subsonic

transforma-

122-132.

264

347

- R. P E N N A C C H I - S o m m a di s e r i e n u m e r i c h e t i c a T2, 2 - C a l c o l o , 5 (1968) 51-61.

348

- R. P E N N A C C H I 5 (1968) 37-50.

- La t r a s f o r m a z i o n i

mediante

razionali

la t r a s f o r m a z i o n e

di u n a

349 - A. R O N V E A U X - P a d 6 a p p r o x i m a n t a n d h o m o g r a p h i c p h a s e e q u a t i o n s - in " P a d 6 a p p r o x i m a n t s and thein M o r r i s ed., A c a d e m i c P r e s s , N e w - Y o r k , 1973.

successione

- Calcolo,

transformation applications",

350 - H. R U T I S H A U S E R - On a m o d i f i c a t i o n of the q - d a l g o r i t h m c o n v e r g e n c e - ZAMP, 13 (1962) 4 9 3 - 4 9 6 .

with

quadra-

of Riccati's P.R. G r a v e s -

Graeffe

type

351 - H. R U T I S H A U S E R - B e s t i m m u n g d e r E i g e n w e r t e u n d E i g e n v e k t o r e n einer Matrix m i t H i l f e des Q u o t i e n t e n - D i f f e r e n z e n A l g o r i t h m u s - ZAMP, 6 (1955) 387-401. 352

- H. R U T I S H A U S E R - E i n e F o r m e l y o n W r o n s k i u n d i h r e B e d e n t u n g t i e n t e n - D i f f e r e n z e n A l g o r i t h m u s - ZAMP, 7 (1956) 164-169.

353 - H. R U T I S H A U S E R - S t a b i l e S o n d e r f [ l l e des Q u e t i e n t e n r i t h m u s - Numer. M a t h . , 5 (1963) 95-112. 354 - H. R U T I S H A U S E R 233-251.

- Der Quotienten

355 - H. R U T I S H A U S E R - A n w e n d u n g e n ZAMP, 5 (1954) 496-508. 356 - J.R. S C H M I D T b y an i t e r a t i v e 357

des

Quotienten

- Differenzen

Algorithmus

SCHWARTZ - Series - Phys. Fluids.

solution

structure

- Differenzen

asymmetric

361 - R.E. S H A F E R - On q u a d r a t i c (1974) 447e460.

approximation

-

equations

conical

blunt-body

359 - L.W. S C H W A R T Z - S o l u t i o n s to t h e a s y m m e t r i c b l u n t - b o d y p r o b l e m approximants - C o l l o q u e E u r o m e c h 58, T o u l o n , 1 2 - 1 4 m a i 1975. 360 - L.W. S C H W A R T Z - H y p e r s o n i c f l o w s g e n e r a t e d b y p a r a b o l i c s h o c k w a v e s - Phys. F l u i d s , 17 (1974) 1 8 1 6 - 1 8 2 1 .

5 (1954)

Algorithmus

of t h e T a y l o r - M a c c a l l

for the planar

Algo-

- ZAMP,

- On the n u m e r i c a l s o l u t i o n of l i n e a r s i m u l t a n e o u s m e t h o d - Phil. M a g . , 7 (1951) 3 6 9 - 3 8 3 .

- L.W. S C H W A R T Z - On t h e a n a l y t i c f l o w s o l u t i o n - ZAMP.

358 - L.W. problem

- Differenzen

fir den Quo-

using

Pad6

and paraboloidal

- S I A M J. Numer.

Anal.,

ii

362 - D. S H A N K S - An a n a l o g y b e t w e e n t r a n s i e n t s a n d m a t h e m a t i c a l sequences and s o m e n o n l i n e a r s e q u e n c e to s e q u e n c e t r a n s f o r m s s u g g e s t e d by it - N a v a l O r d n a n c e Lab. Mem. 9994, W h i t e Oak, M d . , 1949. 363 - D. S H A N K S - N o n l i n e a r t r a n s f o r m a t i o n s s e r i e s - J. Math. P h y s . , 34 (1955) 1-42. 364 - M.D. V A N D Y K E N e w - Y o r k , 1964.

- Perturbation

365 - M.D. V A N D y K E - A n a l y s i s Mech. Appl. Math.

and

methods

of d i v e r g e n t

in f l u i d

improvement

and slowly

mechanics

of p e r b u r t a t i o n

convergent

- Academic

series

Press,

- Quart.

J.

265

366 - A.H. V A N T U Y L - C a l c u l a t i o n ii (1973) 537-541.

of n o z z l e

using

Pad~

fractions

- AIAA

J.,

367 - A.H. V A N T U Y L - A p p l i c a t i o n of m e t h o d s f o r a c c e l e r a t i o n of c o n v e r g e n c e to the c a l c u l a t i o n of s i n g u l a r i t i e s of t r a n s o n i c f l o w s - C o l l o q u e E u r o m e c h 58, T o u l o n , 1 2 - 1 4 m a i 1975. 368

- A.H. V A N T U Y L flows - Colloque

- The u s e of P a d ~ f r a c t i o n s in the c a l c u l a t i o n E u r o m e c h 58, T o u l o n , 1 2 - 1 4 m a i 1975.

369 - H.S. WALL, W.T. S C O T T - The t r a n s f o r m a t i o n Amer. Math. Soc., 51 (1942) 255-279.

of s e r i e s

and

of n o z z l e

sequences

- Trans.

370 - P.J.S. W A T S O N - A l g o r i t h m s f o r d i f f e r e n t i a t i o n a n d i n t e g r a t i o n - in " P a d s approx~mants and their applications", P.R. G r a v e s - M o r r i s ed., A c a d e m i c P r e s s , N e w - Y o r k , 1973. 371 - L. W U Y T A C K - The u s e of P a d ~ a p p r o x i m a t i o n C o l l o q u e E u r o m e c h 58, T o u l o n , 1 2 - 1 4 m a l 1975.

in n u m e r i c a l

integration

372 - P. W Y N N - On a c o n n e c t i o n b e t w e e n t h e f i r s t a n d the s e c o n d of the e - a l g o r i t h m - Nieuw. Arch. W i s k . , ii (1963) 19-21. 373 - P. W Y N N - U p o n a s e c o n d c o n f l u e n t Math. Soc., 5 (1962) 160-165.

form

of the

e-algorithm

-

confluent

- Proc.

form

Glasgow

374 - P. W Y N N - A c o m p a r i s o n b e t w e e n t h e n u m e r i c a l p e r f o r m a n c e s of the E u l e r transformation a n d the E - a l g o r i t h m - C h i f f r e s , i (1961) 23-29. 375 - P. W Y N N 19-22.

- On r e p e a t e d

application

o f the

e-algorithm

376 - P. w Y N N - The e p s i l o n a l g o r i t h m a n d o p e r a t i o n a l a n a l y s i s - Math. C o m p . , 15 (1961) 1 5 1 - 1 5 8 . 377

- P. W Y N N - A c c e l e r a t i o n t e c h n i q u e s in n u m e r i c a l r e f e r e n c ~ to p r o b l e m s in one i n d e p e n d a n t v a r i a b l e N o r t h H o l l a n d , (1962) 149-156.

378 - P. W Y N N - A n o t e on p r o g r a m m i n g C h i f f r e s , 8 (1965) 23-62. 379 - P. W Y N N - The r a t i o n a l defined by a power series 380 - P. W Y N N recherches

repeated

formulas

of n u m e r i c a l

analysis with particular - Proc. I F I P C o n g r e s s ,

application

of the

- The a b s t r a c t t h e o r y of t h e e p s i l o n a l g o r i t h m math~matiques n ° 74, Univ. de M o n t r e a l , 1971.

e-algorithm

arrays

- Centre

- Louisiana

State

- P. W Y N N - I n v a r i a n t s a s s o c i a t e d w i t h t h e e p s i l o n a l g o r i t h m a n d c o n f l u e n t f o r m - Rend. Circ. Mat. P a l e r m o , (2) 21 (1972) 31-41.

383 - P. W Y N N 175-195.

- Singular

384 - P. W Y N N

- A sufficient

4 (1961)

-

approximation of f u n c t i o n s w h i c h a r e f o r m a l l y e x p a n s i o n - Math. C o m p . , 14 (1960) 1 4 7 - 1 8 6 .

381 - P. W Y N N - U p o n a h i e r a r c h y of e p s i l o n N e w - O r l e a n s , techn, rep. 46, 1970. 382

- Chiffres,

rules

for certain

condition

nonlinear

for the

algorithms

instability

de

Univ.,

its f i r s t

- BIT,

of the

3 (1963)

e-algorithm

-

266

Nieuw.

Arch.

Wisk.,

3 (1961)

117-119.

385 - P. WYNN - On the p r o p a g a t i o n Numer. Math., i (1959) 142-149,

of error in certain n o n l i n e a r

386 - P. WYNN - On the c o n v e r g e n c e and s t a b i l i t y SIAM J. Numer. Anal., 3 (1966) 91-122.

of the e p s i l o n

algorithms

algorithm

-

-

387 - P. W Y N N - H i e r a r c h i e s of arrays and f u n c t i o n sequences a s s o c i a t e d w i t h the epsilon a l g o r i t h m and its first c o n f l u e n t f o r m - Rend. Mat. Roma, 5 (1972) 819-852. 388 - P. WYNN - A c c 6 1 6 r a t i o n de la c o n v e r g e n c e de s6ries d ' o p 6 r a t e u r s n u m 6 r i q u e - C.R. Acad. Se. Paris, 276 A (1973) 803-806. 389 - P. W Y N N - T r a n s f o r m a t i o n s de s6ries Sc. Paris, 275 A (1972) 1351-1353. 390 - P. WYNN - U p o n some c o n t i n u o u s 197-234 and 235-278. 391 - P. WYNN - Confluent ii (1960) 223-234.

forms

~ l'aide de l ' s - a l g o r i t h m e

prediction

of certain

392 - P. W Y N N - A note on a confluent (1960) 237-240.

en analyse

algorithms

nonlinear

- C.R. Aead.

- Calcolo,

algorithms

f o r m of the s - a l g o r i t h m

9 (1972)

- Arch.

- Arch.

Math.,

Math.,

ii

393 - P. WYNN - On a p r o c r u s t e a n t e c h n i q u e for the n u m e r i c a l t r a n s f o r m a t i o n of slowly convergent sequences and series - Proc. Camb. Phil. Soc., 52 (1956) 663-671. 394 - P. W Y N N - T r a n s f o r m a t i o n s to accelerate the c o n v e r g e n c e of F o u r i e r series G e r t r u d e Blanch a n n i v e r s a r y vol., Wright P a t t e r s o n Air Force Base, (1967) 339-379. 395 - P. WYNN - P a r t i a l d i f f e r e n t i a l equations algorithms - ZAMP, 15 (1964) 273-289.

associated

with certain nonlinear

396 - P. W Y N N - A n u m e r i c a l m e t h o d for e s t i m a t i n g p a r a m e t e r s in m a t h e m a t i c a l m o d e l s - Centre de r e c h e r c h e s m a t h 6 m a t i q u e s , Univ. de Montr6al, rep. CRM 443, 1974. 397 - P. WYNN - On a device i0 (1956) 91-96.

for c o m p u t i n g

the em(S n) t r a n s f o r m a t i o n

398 - P. W Y N N - A c o n v e r g e n c e t h e o r y of some m e t h o d s r e c h e r c h e s m a t h ~ m a t i q u e s , Univ. de Montr6al, rep. 399 - P. W Y N N - A sufficient c o n d i t i o n Numer. Math., i (1959) 203-207.

of i n t e g r a t i o n CRM-193, 1972.

for the i n s t a b i l i t y

400 - P. W Y N N - A c o m p a r i s o n t e c h n i q u e for the n u m e r i c a l slowly c o n v e r g e n t series b a s e d on the use of r a t i o n a l Math., 4 (1962) 8-14.

- MTAC,

- Centre de

of the q-d a l g o r i t h m

t r a n s f o r m a t i o n of f u n c t i o n s - Numer.

-

267

IV - M I S C E L L A N E O U S

401 ~ G.D. A L L E N , C.K. CHUI, W.R. M A D Y C H , Pad~ approximation of S t i e l t j e s s e r i e s

F.J. N A R C O W I C H , P.W. S M I T H - J. Appr. T h e o r y , 14 (1975) 3 0 2 - 3 1 6 .

402 - G.D. A L L E N , C.K. CHUI, W.R. M A D Y C H , F.J. N A R C O W I C H , P.W. S M I T H Pad~ approximant~, N u t t a l l ' s f o r m u l a a n d a m a x i m u m p r i n c i p l e - to a p p e a r . 403 - M.F. B A R N S L E Y - C o r r e c t i o n 16 (1975) 9 1 8 - 9 2 8 4 0 4 - C. B R E Z I N S K I to appear.

- Matrices

terms

for Pad~

semi-d~finies

approximants

positives

et s u i t e s

405 - F. C O R D E L L I E R - U n e n o u v e l l e f o r m e de l ' e - a l g o r i t h m e C o l l o q u e E u r o m e c h 58, T o u l o n , 1 2 - 1 4 m a i 1975. 406 - F. C O R D E L L I E R - R ~ g l e s p a r t i c u l i ~ r e s C o l l o q u e d ' A n a l y s e N u m ~ r i q u e , La G r a n d e

- J.

de m o m e n t s

~tape

vectoriel

407

- F. Publ.

408

~ F. C O R D E L L I E R - L ' e - a l g o r i t h m e vectoriel : interpretation r~gles singuli~res - Colloque d'analyse num~rique, Gourette,

409

- A.C. G E N Z - The a p p r o x i m a t e extrapolation methods - Ph.D.

- S.T. PENt, S I A M J. Math.

412 - J. T O D D

lemniscate

-

g~om~trique m a i 1974.

-

et

c a l c u l a t i o n of m u l t i d i m e n s i o n a l integrals using T h e s i s , Univ. of K e n t at C a n t e r b u r y , 1975.

A. H E S S E L - C o n v e r g e n c e o f n o n c o m m u t a t i v e A n a l . , 6 (1975) 7 2 4 - 7 2 7 .

- The

-

de l ' c - a l g o r i t h m e

410 - P. H I L L I O N - M ~ t h o d e d ' A i t k e n i t ~ r ~ e p o u r les s u i t e s o s c i l l a n t e s x i m a t i o n s - C.R. Acad. Sc. P a r i s , 280 A (1975) 1 7 0 1 - 1 7 0 4 . 411

Phys.,

et ses v a r i a n t e s

pour l'c-algorithme M o t t e , m a i 1975.

CORDELLIER - Interpretation g~om~trique d'une 40, Labo. de C a l c u l , Univ. de L i l l e , 1973.

Math.

constants

- Comm.

ACM,

18

continued

(1975)

d'appro-

fractions

14-19.

UER d'IEEA - Informatique BP 36 59650 - Villeneuve d'Ascq FRANCE

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