Introduction to Geochemical Modeling

SD biotite

8.00 7.57 7.15 6.75 6.37 6.00 4.33 2.92 1.71 0.67

0.000 0.500 0.667 0.750 0.800 0.833 0.909 0.938 0.952 0.962

-65.0 -112.5 -128.3 -136.3 -141.0 -144.2 -151.4 -154.1 -155.5 -156.3

0.000 0.020 0.038 0.057 0.074 0.091 0.167 0.231 0.286 0.333

Likewise

Introducing the rock-water fractionation, we get

/wCwH(5Dr + 40) + / r Cr H 6D r = /WCWH5DW° + /rCrH5Dr° which can be rearranged as f C of818O ° + 2^-l- fC °8 18 O ° JWCW + J r C r

In the last equation, (p^H bears the usual meaning of the fraction of total hydrogen held by water, with similar definitions for oxygen and rock. The mass unit water/rock ratio widely used by economic geologists is simply Q = fw//r. We can write O _

/wC w ° + / r C r ° fC

H

e(c w H /c r H )

^W "/wC w H +/ r c r H = With the values given for CJCr° and Cj*/CTH9 a few values of 5 1 8 O r o c k and 5D b i o t i t e have been calculated for a range of Q (Table 1.9) and plotted in Figure 1.8.
Mass balance, mixing, and fractionation

26

Unaltered isotope composition of the granite

-60

.2-100 Q CO

-140

-180 -4

0

+4

+8

18

6 Orock Figure 1.8 Closed-system equilibration between biotite, rock and water for oxygen and hydrogen isotopes (from Taylor, 1978, modified). Data from Tablel.9. 5 1 8 O r o c k -5 1 8 O w a t e r =+2 and 5D b i o t i t e -5D w a t e r =-40.

1.3.4 Mixing hyperbola: the inverse problem The simplest inverse approach consists, given the end-member coordinates, in finding the mixing parameters of any individual mixture, i.e., finding the parameter q defined by equation (1.3.1). We will treat this case with an example. <& An andesite (and) with 87 Sr/ 86 Sr = 0.706 and 5 1 8 O = 8 results from the hybridization of a basalt (bas) with 87 Sr/ 86 Sr = 0.703 and 5 1 8 O = 6 and sediments (sed) with 87 Sr/ 86 Sr = 0.713 and 5 1 8 O = 16. Effects of radioactive decay on isotopic compositions are neglected. Estimate the ratio q such that (Sr/O)sed (Sr/O)bas Let us express 87 Sr/ 86 Sr in the mixture as a function of the the 87 Sr/ 86 Sr ratio of each end-member

86

Sr conservation requires

86

Sr allotments cp and

1.3 Working with ratios

27

and therefore


87 86 87 86 8 6 S r _( Sr/ Sr) a n d -( Sr/ Sr) b a s — ' 87 86 87 86

( Sr/ Sr)sed -( Sr/ Sr)bas

Since effects of radioactive decay on the molar mass can be neglected, cpbas86Sr can be replaced with an excellent precision by (pbasSr. Moreover, the definition of q> expressed through equation (1.3.2) as a function of mass fractions / and concentrations CSr requires Sr

f r Sr _

Jsed^sed

Vsed

f

Sr

Jbas^-bas

f

r

Sr

"+" ised^sed

with the mass conservation condition /bas + ./sed = 1

The last two equations can be recombined as L(l ~./sed)Cbas r + /sedQed T^sed * = /sedQed *

or (1 - / s e d ^ s e d ^ J * = /sed(l "
and therefore

which shows that/values cannot be uniquely computed unless a Sr sediment-basalt concentration ratio is assumed. Likewise, since 1 6 O makes most of the oxygen present in rocks, we get

(S 18 O) sed -(5 18 O) bas and

Cbas° The final expression for q is ^sed

/^sed Sr

C b a s /C b a s °

V^ r / U )sed

(Sr/O) bas

^sed \ l ~ ^sed J

28

Mass balance, mixing, and fractionation

Inserting numbers into the equations we get < 0.706-0.703 8-6 ^ J o (pSed = = 0.03 and (psed° = = 0.2 0.713-0.703 16-6 and therefore _(Sr/O) sed _ 0.3(1 -0.2) _ ^~(Sr/O) bas ~ 0.2(1-0.3)" ' Spectacular examples of hyperbolic mixing exist in the geochemical literature but one may wonder whether these plots can be used in a quantitative way. Although the positions of end-members of hyperbolic mixing arrays is underdetermined, these arrays offer an inestimable advantage over linear mixing arrays. Because the geochemical data must stay finite, we know that end-members of mixtures must be located on the mixing array between the extreme point of the array and the asymptote, which may eventually provide a tight constraint on their position. In addition, the curvature factor carries valuable information on the relative concentrations of denominator species in the end-members. Mixing hyperbolic trends with a negative slope for a maximum spread in coordinate ratios, and a strong curvature for better positioning of the asymptotes, are the most informative. An example of least-square estimate aimed at optimizing the knowledge of the mixing parameters from a hyperbolic data array will be presented in Section 5.1.

7.5.5 Ratio-ratio relationships in ternary mixing Ratio-ratio relationships in mixtures involving three or more end-members are no more difficult to handle than binary mixtures. Curved triangles are obtained in binary plots and curious surfaces may be drawn in a 3-D space using 3-D plotting programs. Different concentration ratios may produce widely different triangles and singular points may be created for a particular range of parameters. & In the 143 Nd/ 144 Nd vs 87Sr/ 86 Sr plot, draw the mixing triangle at the 20 percent mesh size for a mixture of three mantle components, the depleted mantle (DM), the enriched mantle I (EM I) and the enriched mantle II (EM II). The isotopic ratios and relative concentrations of each component are listed in Table 1.10. We will illustrate the calculation at one particular point, say/ DM = 0.5,/ EMI = 0.3, / EMII = 0.2. From equation (1.1.1), the concentrations in the mixture are CmixSr = 40 x 0.5 + 400 x 0.3 + 20 x 0.2 = 144

1.3 Working with ratios

29

Table 1.10. Isotope ratios and elemental concentrations (ppm) used for the three-component mixing calculation shown in Figure 1.9. 87

Comp.y

Sr/ 86 Sr

C/ r

0.703 0.705 0.710

40 400 20

DM EMI EM II

143

Nd/ 144 Nd 0.5131 0.5118 0.5121

C

Nd

5 10 10

and CmixNd = 5 x 0.5 + 10 x 0.3 + 10 x 0.2 = 7.5 Units are immaterial to the result since we will work with concentration ratios. We can therefore calculate the allotment of Sr and Nd to each end-member through the (p factors and compute the isotopic ratios of the mixture from equation (1.3.1) as

86

Sr7 mi¥

'

144

144

144

or 87o

\

—-I =0.703x0.139 + 0.705x0.833 + 0.710x0.028 = 0.704 87 86 Sr/ mix We identify 0.139 as the fraction (pDM86Sr«(pDMSr of 8 6 Sr contributed by the component DM, and the like for other end-members. Likewise

4

. Nd/ mix

=0.5131

5 x 0.5 7.5

+0.5118

10 x 0.3 7.5

+0.5121

10 x 0.2 7.5

i.e., /143Nd\

=0.5131x0.3333 + 0.5118x0.4000 + 0.5121x0.2667 = 0.51231

Figure 1.9 shows a mixing hyperbolic triangle drawn for / values at 20 percent intervals. Interesting conclusions may be drawn from this plot (a) If analytical data concentrate in only one of the triangles, they will look as fanning out of the intersection. The intersection of two of the hyperbolic triangle edges therefore may hint at a virtual end-member which actually does not exist. (b) The points representing mixtures are more likely to fall where the density is larger: for 20 percent fraction lines there are 5 + 4 + 3 + 2 + 1 = 15 small triangles. If the probability of

0.5130

0.5125

0.5120

0.5115 0.704

0.706 87

0.708

0.710

Sr/ 86 Si

Figure 1.9 A hypothetical hyperbolic triangle for the mixing of three mantle components in the sense of Zindler and Hart (1986). Data from Table 1.10.

1.4 Normalized variables

31

combining any mass fraction of each component were uniform in the mixing process, each triangle would contain 1/15 of the data points. Mixtures would therefore concentrate where the triangles are smaller, i.e., between EM I and the intersection point.

1.4 Normalized variables

Similar methods apply when the concentration data are normalized or reduced. In particular, the simple properties relating ratios with identical denominators apply when the denominator is a sum. It is common practice in petrology and geochemistry to plot points in triangular diagrams which are a special case of a diagram with normalized coordinates. The AFM diagram (e.g., Cox et al, 1979; McBirney, 1984; Hall, 1987) and the Ti-Y-Zr plot of Pearce and Cann (1973) are enlightening examples of descriptive diagrams that have considerable conceptual importance but have received minimal quantitative treatment. The normalized concentration of the species i in the end-member j is defined as

where m is the number of species that enter the normalization. This definition results in the closure property

d-4.2)

The normalized concentration in a mixture of n end-members is defined in a similar way c c

i_

,

tc/fj

^mix

bmix

VC

„

_ J=l

k

_

yi

. ^j

i k=l

y r *

fc=i fc=i

hence m Cmix = ZJ *j ~

Jj

k=l

Finally n

t«J= I Sfr?

(1-4-3)

32

Mass balance, mixing, and fractionation Table 1.11a. Major-element compositions (%) used for the example of olivine fractionation in the AFM diagram of Figure 1.10.

FeO MgO Na 2 O K2O

bas

ol

10.0 12.0 2.5 1.0

14.35 45.64 0.00 0.00

where q>^ is defined as

c and represents the mass fraction of the summed concentrations that was contributed by the component j to the mixture. This situation is entirely analogous to the cases of elemental or isotopic ratios which have been discussed above, except that we have artificially created a new 'species' by summing the concentrations of m different species. Actually, any sort of linear combination of elemental concentrations would show the same properties. These properties have fundamental consequences on mixing relationships. If points on a plane are ascribed to different end-members, like making three end-members the apices of a triangle, mixture coordinates will be represented as linear combinations of end-member coordinates. For instance, binary mixing will be represented by linear segments, ternary mixing by triangles, and the usual mass balance relationships apply including the lever rule. <& Draw on the AFM diagram the curve representing the removal of up to 20 percent olivine fo85 from a basalt, assuming that olivine composition stays the same during the process. Chemical compositions in percent are listed in Table 1.11a. Let us subscript the residual liquid 'res'. The calculation is slightly complicated by our dealing with mineral removal which is opposite to mixing. Zconc is the sum FeO + MgO + Na 2 O + K 2O. In the AFM triangle, the usage is to label the three normalized coordinates £ as X = £(1) = (Na2O + K2O)/Zconc, F = £(2) = FeO/5;conc, and M = {(3) = MgO/Econc. Given a table of residual liquid fractions / res, we first calculate in the second column the change in Sconc from /y

v

_ (^conc)bas~(l ~/res)(^conc)ol Jres

1.4 Normalized variables

33

Table 1.11b. Evolution of A, F, and M parameters as afunction ofolivine removalfrom basalt. Compositions are listed in Table 1.11a.

basalt -> olivine —•

v^conc/res

A

F

M

25.5 60

0.137 0.000

0.392 0.239

0.471 0.761

0.137 0.156 0.179 0.212 0.259

0.392 0.413 0.439 0.476 0.528

0.471 0.432 0.381 0.312 0.213

/res

1

1.00 0.95 0.90 0.85 0.80

25.50 23.68 21.67 19.41 16.87

1.00 0.88 0.76 0.65 0.53

Next, we calculate (p res z from £

V^conc/res /res V^conc'bas

and finally each coordinate £res* can be calculated from

As an example, assume fres = O.S. We first calculate (£ conc ) res as 25.5-(l -0.8)60 (^conJres =

ZTZ

= 16.875

U.o

then (p resz from equation (1.4.4) as

es

25.5

and, finally, from equation (1.4.3), we compute the AFM coordinates 0.137-(1-0.529)0 0.529

= 0.259

0.392-(l-0.529)0.239 "

0.529 0.471-(1-0.529)0.761 0.529

= 0.528

= 0.213

The complete results are listed in Table 1.11b and displayed in Figure 1.10.

34

Mass balance, mixing, and fractionation F

A

5 percent increments

M

Figure 1.10 The liquid line of descent upon olivine removal from basalt in the AFM diagram using equations (1.4.3) and (1.4.4). Olivine has a fixed composition (data from Table 1.1 la). Each open circle indicates removal of 5 percent olivine.

1.5 Incremental processes (distillation)

1.5.1 Introduction Although this topic will be largely covered with respect to trace elements in magmatic processes in Chapter 9, incremental processes affect elemental and isotope distributions even when the species considered is not a trace element and therefore deserve a treatment in which dilution is not critical. During a distillation process, a condition of chemical equilibrium is applied to a state of the system which changes while the process progresses. Everyone is familiar with the principles of thermal distillation of alcoholic solutions in a still. Here, the chemical equilibrium condition is represented by the pressure of alcohol in the vapor phase being proportional to the concentration of alcohol in the liquid, i.e., Raoult's law. Upon progressive condensation of the vapor in a refrigerated side-flask, the residual liquid becomes more and more depleted in alcohol. Similar fractionation processes take place in the Earth whenever phases separate progressively from each other. A familiar case is the fractional crystallization of basalts. Progressive removal of mineral phases from a parent magma produces a sequence of residual (= differentiated) liquids which altogether form the liquid line of descent (Bowen, 1956). The opposite process, in which fractionating minerals accumulate in specific parts of magmatic systems, creates chemical trends known as control lines. Liquid lines of descent and control lines account for most of the chemical variability in igneous rocks. However difficult the distinction can be, the two sorts of trends should not be confused with each other. For instance, the olivine control lines observed in many basaltic series produce chemical variations that can be confused with the liquid line of descent of picritic magmas. Likewise, vapor expelled from

1.5 Incremental processes (distillation)

35

magmas also induces chemical distillation processes which can lead to a large geochemical variability (e.g., Candela, 1986). Another incremental process consists in the progressive flushing of a porous rock by a fluid. Solid-liquid exchange leads to chemical changes when more fluid is allowed to percolate, which is extremely efficient in producing strong elemental or isotopic fractionation. The mass balance aspects of distillation and percolation processes are often confused with their kinetics. Transfer of successive mass increments usually takes place over successive time increments but how the progress variable varies with time is often unknown in natural processes. Fortunately, knowledge of time-dependence is by no mean compulsory to achieve the description of the process in terms of mass balance.

1.5.2 Concentration changes upon closed-system crystallization A homogeneous mass M liq of liquid contains the amount mliq* of species i. The concentration Cliq* of species i in the liquid is defined as

The fraction F of residual liquid is relative to that of parent magma M o F = MnJM0 Taking the differential of In Cliq\ we get dC l i q f _ dm liq f

dF

The closed-system condition requires

=0

q

(1.5.1)

Let Di be the ratio of the concentration dmsoi ydM sol of species i in the last increment of precipitated solid to the concentration C liq l in the liquid, i.e.,

H=Dinh}l dM8ol

(1.5.2)

MIiq

with no implication that Dt is constant. Applying the mass balance relationships, the last equation also reads

dM l i q

"M liq

36

Mass balance, mixing, and fractionation

or, equivalently

M

liq

Expressing the concentration leads to

liq

*

or dlnCliqI' = (D I --l)dlnF

(1.5.3)

This is the fundamental distillation equation, often referred to as the Rayleigh law when in its integrated form (Rayleigh, 1896). As far as Dt is considered to be a function of F, this equation applies to the change of any species concentration in the course of phase separation. Liquid-vapor or solid-solid fractionations are liable to the same formulation.

1.53 Changes in element and isotope ratios upon closed-system crystallization If we are dealing with major elements, partition coefficients Dt may be expected to vary with many parameters, including temperature or liquid chemistry. For some elemental pairs and particularly for isotopes for which activity coefficients are correlated, there is a better chance, however, that the ratio of two partition coefficients Dn and Di2 shows lesser variations. We therefore subtract the distillation equation (1.5.3) for element or isotope i\ from that for il as

liq

or d \n(Ci2/Cn)Uq = (Dni2 - l)Dn d In F

(1.5.4)

where we note Dni2 the ratio of the elemental partition coefficients. These equations are known collectively as the Doerner-Hoskins law (Doerner and Hoskins, 1925). Petrologists would commonly label Dni2 with the symbol KD, while stable isotope geochemists could label it a. Eliminating Dn with the distillation equation (1.5.3) for il, we get the two expressions d \n(Ci2/Cn)Uq = (Dni2 - l)(dIn C liq jl + d In F)

(1.5.5)

1.5 Incremental processes (distillation)

37

and (1.5.6)

We sometimes want to track the change in the concentration of a species *3 as a function of an index ratio i'2/il. Dividing equation (1.5.3) for i = i3 by equation (1.5.4), the appropriate equation is

i2

(1.5.7)

{Dni2-\)Dn

ICn)^

The new variable z is defined as z=

FCUqil/Coil

where the subscript 0 in the denominator refers to the parent magma. Taking the log-differential, we get = dlnF + dlnCliql1 which, inserted into equation (1.5.5) gives dln(C2/Cll)liq = (A/ 2 -l)cilnz

(1.5.8)

For constant Dni29 equation (1.5.8) is integrated into

' Equation (1.5.9) has been derived by Irvine (1977) and Pearce (1978) in the case of olivine fractionation from magmas and is also found in fluid-magma systems (Candela, 1986). Dividing equation (1.5.8) for the pair i3-il by the equation for the pair i2-il gives

In logarithmic plots, such as In Ci3/Cn vs In Ci2/Cn9 the slope of the curve is given by the right-hand side of equation (1.5.10). When Dni2 and Dni3 are constant, so is the slope and these plots must be linear arrays with the equation

\c 1 liq

\

C

/0

n U

il

~

L

1 \r

I

\ ^

/O

n U

i\

i ~V

\r

I

\ ^

/liq

(1.5.11)

where the subscript 0 refers to the parent magma. Some related equations were derived by Irvine (1977) for the particular case where f3 is an element excluded by olivine,

38

Mass balance, mixing, and fractionation

and by Pearce (1978) for i\ being constant. Equation (1.5.11) is actually fairly general. The reader can easily check that, in the same diagram, the solid (cumulate) defines an array parallel to the liquid array and displaced by l n D a l 2 in x and lnD^ 1 3 in y. & 18 O/ 16 O fractionation during closed-system crystallization. A magma body precipitates a cumulate whose solid-liquid fractionation factor a for 1 8 O/ 1 6 O fractionation is 1.0005. What magma fraction should crystallize with distillation effect in order to lower the melt 5 1 8 O from 7 to 6? The fractionation coefficient a is defined as

( 18 O/ 16 O) liq

When inserted into equation (1.5.5), we get dln( 1 8 O/ 1 8 O) l i q = (a-l)(dlnC l i q 1 6 ° + d l n F )

(1.5.12)

Since we do not know the solid-liquid oxygen partition coefficient, we must resort to an approximation. Most oxygen atoms are 1 6 O and crystallization rarely changes the total oxygen concentration of the residual silicate melt very significantly. We can assume d In Cliq'6° % d In Cliq° « d In F which enables equation (1.5.12) to be integrated into

r where the subscript 0 refers to the liquid when crystallization starts (F= 1). Turning to the 8 notation, we can reformulate the distillation equation (1.5.13) as

1000

-=F -

(1.5.14)

1000 Inserting the numerical values, we get 1.0005-1 _

1.006 1.007

giving F = 0.14 (86 percent crystallization). This equation also applies to the H 2 O liquid-vapor system, o

7.5 Incremental processes (distillation)

39

<& Sr/Ca fractionation during anhydrite precipitation from brines. A solution of

calcium sulfate contains 20mmol/kg Ca and 0.2mmol/kg Sr. Calculate the Sr concentration in the residual brine (res) when 50 percent Ca is removed. Use The distillation equation (1.5.6) reads ^

C

a

(1.5.15)

Since Ca is a major element in anhydrite while it is still relatively dilute in the solution, we can assume £>Ca»l , hence, upon integration Ca\DcaS

where the subscript 0 refers to the initial brine. Inserting the numerical values into equation (1.5.16) gives

and the requested result CresSr = (20 x 0.5) x 0.0157 = 0.157 mmol/kg

1.5.4 FeO-MgO fractionation during olivine crystallization in basalts The Fe/Mg fractionation between olivine and basaltic liquids is independent of temperature (Roeder and Emslie, 1970) while its pressure-dependence is quite small (Ulmer, 1989). This is also the case, although to a lesser extent, between other mafic minerals and basaltic liquids (see a compilation in Warren, 1986). This remarkable property makes these two elements invaluable tools for understanding basalt genesis. The approach taken here is that of Albarede (1992). KD is defined by solid-liquid equilibrium A^

FeO

Q"*sol

(-< FeO

Cliq

M S 17

which, inserted into equation (1.5.8) gives dln(C Hq FeO /C liq M8 °)=(K D -1) dlnz

(1.5.18)

40

Mass balance, mixing, and fractionation

where z = FC liq Mg0 /Co Mg0

(1.5.19)

For constant KD, equation (1.5.9) reads

Olivine is assumed to be the only mineral precipitating and the fa and fo subscripts refer to pure fayalite and forsterite. Upon crystallization of the incremental amount dMsol of cumulate, the incremental amount dMfa of fayalite is precipitated, and, since the iron content of fayalite is constant r

J^, FeO FeO_ d m sol

Therefore, conservation of FeO and MgO requires

Factoring the numerator of the last term on the right-hand side gives

We note that, from mass balance, /m

M

8°

M liq

which we insert into equation (1.5.21). Dividing by Mo, making concentrations appear in thefirstterm of the parenthesis, and rearranging the last differential term, we obtain 1 \ /A/I *M MgO\ dm s o l F e O /dM s o l1 \ J l i q ^liq \ Mg Qo ~ °// U o Mliq / sol ~ / sol~i Wo \ 0 liq l

^0

\Wa

um

UJVi

M

JVi

which is recast into MgO ' q

Using the definition (1.5.19) of z, we get FeO

r

MgO

MgO

JVi

7.5 Incremental processes (distillation)

41

Up to this point, the only assumption is that olivine is the only phase precipitating. Taking the further step that KD is constant, equation (1.5.21) can be integrated into M

8°

which is an implicit equation with no solution in closed form. We can now compute exactly the FeO and MgO contents of residual magma corresponding to a given extent of olivine fractionation. Selecting a pair of parent magma values C 0 FeO and C 0 MgO and an olivine fraction crystallized 1 - F , z can be calculated iteratively from equation (1.5.23). Then, using equation (1.5.19) gives CliqMgO while equation (1.5.20) gives C liq FeO The same equations can be used for reversing olivine precipitation: the parent magma values are to be replaced by those of the residual magmas, F values being higher than unity. Because solving the transcendental equation (1.5.23) requires iterative methods of root finding, the numerical solution to this equation is postponed to Section 3.1. o

7.5.5 Elemental fractionation during basalt differentiation From equation (1.5.7), change in the concentration of an element i with the differentiation index (FeO/MgO) liq , written in lieu of (CFeO/C MgO) liq , is calculated as

d ln(FeO/MgO)liq

D F e O - DMgO

DMgO(KD

-

(1.5.24)

Even for a constant KD, calculating the evolution of a species i in the residual melt requires that both Dt and DMgo a r e known for each value of (FeO/MgO) liq . <& The Baffin Bay picrites (Francis, 1985) show a very good covariation of FeO, MgO, and Ni. Defined from the twelve XRF data listed in Table 1.12, the variables x = ln(FeO/MgO) and y = In (Ni/MgO) have been fitted by the parabolic equation >; = 2.92-1.60*-0.670* 2

Assess the variations of DMgONi by assuming that D MgO FeO = X D = 0.3. We should remember that the Ni/MgO ratio unit is in ppm Ni per percent MgO. When plotted in Figure 1.11, we see that the covariation between the two ratios is smooth but not linear. Given the assumption made of a constant X D , we conclude that DMgoNi varies with differentiation, which is a well-documented observation of experimental petrology (Arndt, 1977; Hart and Davis, 1978; Kinzler et a/., 1990). Taking the derivative dy/dx, we obtain the slope of the curve and, from equation (1.5.10), we write MgO

42

Mass balance, mixing, and fractionation Table 1.12. FeO, MgO (%) and Ni (ppm) concentrations in some Baffin Bay picrites (Francis, 1985).

Sample # 14 1 64 15 24 56 3 21 4 29 13 19

1

F e CVy O

MgO

Ni

10.39 10.68 10.25 10.77 10.82 10.91 10.77 10.24 10.49 10.40 10.44 10.32

11.97 13.58 14.40 14.91 14.97 15.68 16.13 17.91 20.03 21.83 22.13 21.93

278 361 407 410 435 498 517 687 775 930 943 906

FeO

Ni x 104

MgO

MgO

0.868 0.786 0.712 0.722 0.723 0.696 0.668 0.572 0.524 0.476 0.472 0.471

23.22 26.58 28.26 27.50 29.06 31.76 32.05 38.36 38.69 42.60 42.61 41.31

4.00

-0.8

-0.6

-0.4

-0.2

In FeO/MgO Figure 1.11 Ni/MgO vs FeO/MgO relationship for the Bay of Island basalts and picrites (Table 1.12 data from Francis, 1985). The curve is a least-square parabolic fit through the data. Since K D = D MgO FeO remains constant, the DMgONi partition coefficient increases with increasing FeO/MgO [equation (1.5.10)].

1.5 Incremental processes (distillation)

43

or ^M g o Ni = 1 + ( 0 . 3 - 1 ) ( - 1 . 6 0 - 1 . 3 4 I n ^ ^ - ) = 2.12 + 0 . 9 4 I n ^ ^ ~ MgO/ MgO \

Z)MgONi increases from 1.5 for a picrite with FeO/MgO = 0.5, to 3.2 for a differentiated basalt with FeO/MgO = 3, values that are consistent with the range found by Arndt (1977) and Hart and Davis (1978). <*

1.5.6 Fractional melting The approach just used for fractional crystallization can be transposed immediately to fractional melting, a process by which each packet of melt is withdrawn from the source thereby prevented from equilibration with the solid. Again, these equations will be developed in Chapter 9, but the present section emphasizes a representation which does not require constant Berthelot-Nernst partition coefficients, and therefore is more useful for major elements. As a parallel to equation (1.5.2), we define Dt as the ratio of the concentration msoiyMsol of species i in the residual solid to the concentration dmliq7dMliq in the last increment of liquid ^

=» i ^

Msol

(1-5-25)

dM H q

which, again, carries no implication on Dt being constant. Combining with mass balance conditions (1.5.1) gives = U:

*so.

dM sol

M sol

mso,'

or

The logarithmic differential of concentration is, as before dC5Oi' _ dmj

Cj

mj

dM 5ol

Msol

which, inserting equation (1.5.26), becomes dCj Cj Introducing F as the molten fraction

_l-DidMsol D, Mso,

(1.5.27)

44

Mass balance, mixing, and fractionation

we get —^dln(l-F)

(1.5.28)

which is the equivalent for fractional melting of equation (1.5.3). Subtracting equation (1.5.28) for species i\ from the same equation for species il gives

'VC1).* = (J- - J - W l -F) \Ui2

(1.5.29)

"il/

or 1

dln(l - F )

(1.5.30)

From equation (1.5.28), we obtain 1/Dn as

Dn

dln(l-F)

which, inserted into equation (1.5.30), gives - 1 Ydln Csol11 + d ln(l - F ) ]

(1.5.31)

If we consider three elements in a ln(Ci3/Cn)sol vs \n(Ci2/Cn)sol plot and divide equation (1.5.31) for the pair il — B by the same equation for the pair il — i2, we get the slope s of the graph as

d\n(C2/Cl)sol

(1.5.32)

_1 D

i2

If Dni2 and Dni3 are constant, the two ratios define a straight-line in a logarithmic plot and equation (1.5.32) can be integrated as

o J Da' 2

Vc'Vo J V

2

where the 0 subscript now refers to the so/W source [compare with equation (1.5.11)]. Because each phase, mineral or liquid, differs from the other by a partition coefficient, they all should follow trends parallel to that defined by equation (1.5.33).

1.5 Incremental processes (distillation)

45

Table 1.13. Major-element data (%) on clinopyroxenes from the mid-ocean ridge peridotites analyzed by Johnson et al. (1990).

FZ stands for 'fracture zone'.

Vulcan FZ

Bullard FZ Bouvet FZ

A12O3

FeO

MgO

8.20 5.68 5.30 6.40 4.98 3.68 4.58 3.39 3.68 4.27

3.01 2.73 2.75 2.88 2.71 2.39 2.58 2.23 2.39 2.35

14.95 15.76 16.43 15.67 16.47 16.90 17.46 17.11 17.23 16.89

FeO MgO

A12O3 MgO

0.201 0.173 0.167 0.184 0.165 0.141 0.148 0.130 0.139 0.139

0.548 0.360 0.323 0.408 0.302 0.218 0.262 0.198 0.214 0.253

& Ion-probe data for trace elements in clinopyroxene from abyssal peridotites collected along America-Antarctica and South-West Indian ridges led Johnson et al. (1990) to suggest that the peridotites are residues left by a process of fractional melting of the asthenosphere. Use the electron microprobe data for A12O3, FeO and MgO given in Table 1.13 on peridotites from three fracture zones around the AntarcticaAfrica-America triple junction to test the fractional melting hypothesis and, assuming #MgoFeO = 0.3, calculate the DMgoAl2°3 fractionation coefficient. When plotted in a In (Al 2 O 3 /MgO) vs ln(FeO/MgO) (Figure 1.12), the clinopyroxene data define a unique trend which is reasonably linear, thereby supporting the hypothesis of fractional melting made by Johnson et al (1990). In addition, the linear array supports a homogeneous source and rather constant DMgOFeO = 0.3 and DMgOAh°l fractionation coefficients. Standard regression suggests the following expression

and therefore, from equation (1.5.32) 1

\ //

AI2O3 O

//

1

FeO \ n V'MgO

-1

or 1

>- = 1 + 1 2 2 Ur

=2.22

46

Mass balance, mixing, and fractionation

0.0

-2.0

-2.0 -1.9

-1.8

-1.7

-1.6

-1.5

In FeO/MgO Figure 1.12 Al 2 O 3 /MgO vs FeO/MgO plot of peridotite clinopyroxenes from near the Antarctica-Africa-America triple junction (Table 1.13 data from Johnson et al, 1990). The linear array [equation (1.5.32)] supports the assumption that a homogeneous peridotitic source has experienced fractional melting with D MgO Al2 ° 3 ^0.18.

and finally sO.18

Although less compatible than Mg, Al apparently is still well retained by the residue during melting, which supports the presence of an aluminous phase in the residue as suggested by Johnson et al (1990). If we knew the Al2O3/MgO ratio of the whole rock prior to melting, it would be possible to calculate that ratio in the last liquid extracted and compare its value with mid-ocean ridge basalt (MORB) values. <= 1.5.7 Fractional condensation

Distillation processes explain (Dansgaard, 1964) the correlation between 818O and 5D values of meteoric water found by Craig (1961). Most water vapor forms near the equator by evaporation of warm surface seawater, i.e., from a reservoir which can be assumed to have nearly homogeneous isotope composition of oxygen and hydrogen. When air masses move poleward, progressive cooling of the water vapor and condensation takes place which induces water/vapor isotopic fractionation. Liquid water is enriched in the heavy isotopes (18O and D) relative to the light isotopes (16O and H) relative to the residual vapor. With increasing latitude, atmospheric moisture therefore becomes more and more depleted in 18O and D by rainfalls. Fractional condensation can be handled like fractional crystallization with rain water (w) instead of solid, and vapor (v) instead of liquid. Then, the Dni2 fractionation

7.5 Incremental processes (distillation)

47

coefficients are related to the isotopic fractionation factors a commonly used in this field through n is w/v ( 1 8 O/ 1 6 O) W 18 D 1 6 = a o ' =—z—Ti 16

°

an

(18O/16O)V

d

^ H =aH

(D/H) w =

(D/H)v

Dividing equation (1.5.4) for D/H by the same equation for

18

O/ 1 6 O produces

dln(D/H)v _ a H w / v - l DR _ a H w / v -l ( H / 1 6 O ) W dln(18O/16O)v " a o w / v -l D 16o a o w / v -l (H/16O)V The last term in the right-hand side of the last equality is obviously very close to unity, therefore dln(D/H)w dln(18O/16O) w

a H w / v -l a o w / v -l

The 5 notation is more convenient and, since SMOW values, by definition, are constant, we get

a H w / v -l V 1000/ 18 ,1 A 5 Ow\~aow/v-l din 1 + V 1000 / The linear expansion of the natural logarithm in a Taylor series \n(l+s)&s holds for any small number e. For small values of 5 18 O and 8D, we can use the approximation d5Dw d518Ow

a H w / v -l a o w / v -l

Craig (1961) found that the slope of the correlation between 8 18 O and 5D in precipitation is constant and equal to 8 which is reasonably consistent with atmospheric temperatures.

1.5.8 Open-system isotopic exchanges In hydrothermal systems, a simple distillation equation for an isotope ratio i2/il can be worked out if only il is easily exchangeable. This is the case of the 1 8 O/ 1 6 O ratio because the bulk of oxygen is the isotope 16 and its concentrations are determined

48

Mass balance, mixing, and fractionation

by stoichiometry (Taylor, 1978). This may also be the case of Sr when the bulk concentrations are not changed drastically and isotopic exchange is the most visible effect (Albarede et al., 1981; McCulloch et al, 1981). We consider a system of total mass M o made of porous rock (r) wetted by water (w) with respective mass fractions / w and fT which we assume to be constant. The isotope i\ is assumed to be unaffected by water-rock interaction. Change in the amount of isotope il held by the whole system takes place by circulation of a fluid incremental amount dM w which enters the system with a concentration Cini2 and leaves it with the il concentration Cj2 of the interstitial liquid d[M0(/ wCwi2 + /r Cri2)] = Cin'2 dMw - Cj2 dMw

(1.5.34)

Defining the incremental water/rock ratio dQ as dQ = dMJM0 and dividing equation (1.5.34) by dM w dC i2 dC i2 / w r 4 + d / w ) 4 = Cin dQ dQ

(1.5.35)

We finally divide both sides by Cj1 = Cinil to obtain

- W \C W 1 V

dQ

d.5.36)

dQ

<& Oxygen isotope exchange during open-system meteoric-hydrothermal alteration of a granitic body (Taylor, 1978). Meteoric water (8 1 8 O i n = —16) percolates through a granitic body with 5 1 8 O = + 8. Assuming that water (w)-rock (r) oxygen fractionation can be approximated by the water-feldspar value (5 18 O feldspar —8 1 8 O w a t e r = + 2), draw the locus of possible 5 1 8 O r o c k for variable proportions of interstitial water up to a water/rock ratio of 0.8. Assume the oxygen concentration ratio C w °/C r ° = 2. Letting il = 1 6 O and i'2= 1 8 O, equation (1.5.36) becomes ,d( 1 8 O/ 1 6 O) w /w

dQ

i n

nCr°d(

+(1 / J

^7

18

O/16O)r

dQ

/18O\ _/18O\

*{-^5)in

[*o)w

where the « sign emphasizes that 1 6 O is considered as immobile. Dividing each equation by ( 18 O/ 16 O) SMOW , subtracting 1 and multiplying by 1000 enables 5 values to be introduced C.5.,7, Since the difference in 8 values for water and rock is constant, their incremental

1.5 Incremental processes (distillation)

49

variation d 5 1 8 O is identical and — »5 1 8 O- —518O

_l_ M— /*)—?—I

this equation can be rearranged for easier integration as d(5 1 8 O w -5 1 8 O i n ) 5 1 8 O w -5 1 8 O i n

dg

(1.5.38)

which can be integrated into (1.5.39)

6 1 8 O w °-5 1 8 O i n where eo=/w+(i-/w)~ 518Orock values can now be calculated from 518Or = 2 + 5 18 O in -f(5 18 O r °-2-5 18 O in )e- Q/Qo

(1.5.40)

a relationship drawn in Figure 1.13. o

+10

-10 0.2

0.4

0.6

(2 = M w a t e r /M 0 Figure 1.13 Incremental water/rock interaction: change of the whole rock 5 18 O with the integrated water/rock ratio Q as given by equation (1.5.40). Water-rock fractionation is 5 1 8 O r o c k -5 1 8 O w a l e r =+2.

50

Mass balance, mixing, and fractionation

& Strontium isotope exchange during open-system hydrothermal alteration of basalts (Albarede et aL 1981). Seawater (sw) with 8ppm Sr and ( 87 Sr/ 86 Sr) sw = 0.709 percolates through mid-ocean ridge basalts (bas) with 120 ppm Sr and 87 Sr/ 86 Sr = 0.703, initially. Seawater and basalt react to form a hydrothermal solution (hw) and a hydrothermally altered rock (hr). Sr concentrations are not modified in the process. Calculate the water/rock ratio Q for a hydrothermal solution with ( 87 Sr/ 86 Sr) hw = 0.704, assuming an extremely low rock porosity and complete rock-fluid isotopic equilibrium. The assumption of low rock porosity is equivalent to / w « 0, hence, from equation (1.5.36) Crn\d(Ci2/Cil)r_/Ci2\ AT 1 /

dg

~\C"/in"

We assign the superscripts i\ to 86 Sr, il to 87 Sr, the subscripts r to the hydrothermal rock (hr), w to the hydrothermal solution (hw), and in to inflowing seawater (sw). Since 86 Sr and Sr concentrations are nearly exactly proportional quantities, we get C hr Sr d( 87 Sr/ 86 Sr) hr _/ 87 Sr\ Chw

V86Sr/sw

dg

/ 8 7 Sr\ \86SrAw

The Sr isotope composition of the hydrothermally altered basalts (hr) being identical to that of the hydrothermal fluid (hw), we prepare for integration rewriting

C 86

Sr

Sr

which holds true because ( 87 Sr/ 86 Sr) sw is constant. If the fluid Sr concentration stays constant during the water-rock interaction process, so does rock Sr concentration, hence, upon integration 7

Sr

Rearranging

Mbas

^ ,

( 8 7 Sr/ 8 6 Sr) s w -( 8 7 Sr/ 8 6 Sr) b a s

ChwSr

87 86 8 787 6 (( 87 S /86 S )sw( -( SSr/ / 8 86 SSr) ) hw Sr/ Sr)

and inserting numerical values 120 l n 0.709-0.703 = 2 7 8 0.709-0.704

7.5 Incremental processes (distillation)

51

How does the system behave for small Q values? Equation (1.5.43) can be rewritten

( 87Sr/86Sr)sw-(87Sr/86Sr)hv

(1.5.44)

Using once more the linear expansion of the natural logarithm in a Taylor series for the second term between brackets allows us to find an approximation valid for

C h r S r ( 8 7 Sr/ 8 6 Sr) h w -( 8 7 Sr/ 8 6 Sr) b a s ^^ChwSr(87Sr/86Sr)sw-(87Sr/86Sr)hw

{

"

)

Inserting the appropriate numerical values, we obtain Q = 3.0. The reader is urged to check that equation (1.5.45) is equivalent to assuming closed-system fluid-rock equilibrium. Unless water-rock ratio is quite high, the closed-system approximation is sufficient for practical purposes (Albarede et al, 1981). <=

2 Linear algebra

The purpose of this Chapter is not to present an exhaustive theory of linear algebra that would take more than a volume by itself to be presented adequately. It is rather to introduce some fundamental aspects of vectors, matrices and orthogonal functions together with the most common difficulties that the reader most probably has encountered in scientific readings, and to provide some simple definitions and examples with geochemical connotations. Many excellent textbooks exist which can complement this introductory chapter, in particular that of Strang (1976).

2.1 A matrix refresher

2.1.1 Definitions • An n-tuple is an ordered set of n objects. • A vector space W of dimension n is the set of all the n-tuples, called vectors, of real numbers or scalars. • In the space 9?n, m vectors (xx • • • •• xm) are independent if, given the m scalars af, the relationship m

I oyc,=0

(2.1.1)

i= 1

requires that the m a/s are zero. • A base of 9?" is a set of n vectors such that any vector in 9T can be represented by a linear combination of the base vectors. • A matrix is an array, either square or rectangular, of numbers. It will be noted subsequently with bold-face, upper-case symbols. The real matrix Amxn has m rows and n columns. A matrix Amxn also represents either a set of n column-vectors in SRm, or a set of m row-vectors in 9T. A scalar element of the matrix A will be referred to by its row and column index and noted atJ (ith row, jih column), so the array form of the matrix A is

A=

52

2.1 A matrix refresher

53

Example: "1 0

-r 2

1

3_

A vector may be considered as a special matrix, such that n= 1, and will be noted with bold-face, lower-case symbols. Example: V b=

0 L-U

• A square matrix has the same number of rows and columns. A square matrix is symmetric if atj = ajt. A diagonal matrix is a square matrix with all off-diagonal entries equal to zero. A diagonal matrix is necessarily symmetric. Example of a diagonal matrix: 1

0 0"

A= 0 - 1 0 L0

0

3.

The identity matrix InXn (usually written for short /„) is a diagonal matrix with ones on the diagonal. Example:

/,=

o

2.1.2 A few rules for matrix manipulation The transpose of the matrix Amxn

is the matrix Bnxm

such that (2.1.2)

Example: '1

-1"

0

2

.1

3_

r * on L - l 2 3j

54

Linear algebra

• The addition of matrices Amxn

and Bmxn results in the matrix Cmxn, such that (2.1.3)

C=A+Bocij = ai

Example: "1

-1"

If A = 0

2

2

and B=

0

3_

.1

0 -2"

-1"

~-l

- 1 , then C=A + B= 2 1_

.1

1 4j

• The multiplication of the matrix Am x „ by the scalar a results in the matrix Cm x B, such that C=0^OCy=0Kly

(2.1.4)

Example: "1

-1"

If .4 = 0

2

.1

2 and a = 2, then

C=2A-

3.

• The multiplication of the matrix Amxn calculated from

by the matrix Bnxp

-2

0

4

.2

6.

results in the matrix Cmxp

(2.1.5)

Ca =

Example:

3 .-1

-1"

'1

1

0

andtf= 0

2

1.

2

• - 1

If A =

.1

'-2

-r

2"

2 , then C=AB=

3

3_

0

-1 8.

For instance, c32 = ^3 + ^32^22 + ^33^32 = ( ~ 1) x ( ~ 1) + 2 x 2 - ( - l x 3 = 8. • The product of the matrix /4 m x n by the vector xn produces the vector ym. A is therefore a linear operator which associates a vector from W" with a vector from $RW. Example:

1 If A =

-1 and x = \

, then y = Ax =

.0 •

Matrix product is in general not commutative, i.e.,

AB^BA

(2.1.6)

(AB)C=A(BQ

(2.1.7)

but is associative

2.1 A matrix refresher

55

A(B+Q = AB+AC

(2.1.8)

and distributive

Identity matrices represent the unitary elements of matrix multiplication for A I —I A —A s*m x nln Mm^tm x n ^mxn

(2.1.9)

It is left to the reader to show that, through equation (2.1.5) (2.1.10)

(AB)J = BTAT • The dot or inner product of two vectors xn and yn is the scalar a, such that tx = x-y = x y=y x= ^

x}y}

(2.1.11)

Because of its scalar nature, the order of vectors in the dot product can be exchanged. • The outer product of two vectors xm and yn is the matrix Am x „, such that

(2.1.12)

The outer product is not commutative. • Given xn and a symmetric matrix An x „, the scalar S, such that = xTAx

(2.1.13)

is called a quadratic form. Example:

• The squared-length || JC|| 2 of the vector x makes the vector space 9T Euclidian if it is given by ||JC||2 = A:TA:

(2.1.14)

The cosine of the angle 6 between vectors x and y is given by cos 6 = -

(2.1.15)

Example: Given the vectors x = [ l , 2 ] T and y = \_— 1,1]T, we compute their squared-lengths, dot

56

Linear algebra product, and angle as

d=1+1=2

COS0 =

10 • Two vectors x and y are orthogonal if their dot product is zero.

2.1.3 The common-dimension expansion of the matrix product Let us prove a useful alternative expression of the matrix product. Let ei ( i = l , ..., n) be the column vector whose n coordinates are zero except for the ith which is equal to 1. The n e,'s form a base of the Euclidian space 9T. Uet is the ith column of a matrix UmXn while e? V is the ith row of a matrix Vn x p. Outer products such as e{e? are nxn matrices. From the previous definitions

2>i«,-T = /,.

(2.1.16)

i=l

where /„ is the identity matrix. For a diagonal matrix DnXn with diagonal element du e?Dei = du

(2.1.17)

eiTDej = 0

(2.1.18)

while, for i / j

We now assume that the matrix ^ m X p can be expressed as the product of three matrices Umxn, DnXn, and VnXp An,xp=UmxnDnxnVnxp

(2.1.19)

Let us express the matrix A as (2.1.20)

2.1 A matrix refresher

57

or, using equation (2.1.16) and keeping the dummy index different for each sum

Using the distributive property of the matrix product, and since each summation index is independent, we get

i= 1

k= 1

All the terms under the second summation sign with i ^ k vanish and therefore (Uei)dii(eiTV)= £ dH(ith column of U)(ith row of V)

A=j] i= 1

(2.1.21)

i= 1

The common-dimension expansion shows that the matrix product can be viewed as a linear combination of the pairwise outer products of U columns and V rows. Example: 1

-2"

2

1

_0

1.

o % -a0

= 2 0

_0

-1 -2 +3 0.

-12

-8"

2

1 =

6

-1

2

1.

6

-4

_2

3j

a result, of course, which can also be obtained with the usual rule of matrix multiplication.

2.1.4 The subspaces of a matrix Most of the following definitions and theorems form the core of many textbooks in linear algebra. They have been given for reference, but the reader looking for completeness and rigor could refer to Strang (1976) or Leon (1990). • The span of a set of vectors in R" is the subspace of all the linear combinations of these vectors. For instance, the span of one vector is a straight-line, the span of two vectors is a plane, and so on. • The range, or column-space, of a matrix Amxn is the span in 9?m of the column-vectors of A. The row-space of A is the span in 9?" of the row-vectors of A. Given a vector xn, the product vector y = Ax belongs to the column-space of A. Using the notation of the current elements, the product of the matrix / 4 4 x 3 by the vector x 3 = [x 1 ,x 2 ,x 3 ] T gives the vector

Linear algebra

58 ]T

as

a linear combination of the column-vectors of A "fill

-a 1 2 a 22

yi

a21

y3

*31

^32

-y*-

-041-

-«42-

+ x2

"«13" «23

+ x3

«33 -«431

The column-space of ^ is identical to the row-space of A . The space of the column-vectors xn such as Ax = Om, where 0m is a m x m matrix of zeroes, is called the nullspace of the matrix A. Any vector from the nullspace is therefore orthogonal to any vector from the row-space. The left nullspace of A is the set of vectors ym, such as yTA = 0n. Any vector from the left nullspace is therefore orthogonal to any vector from the column-space. The left nullspace of A is identical to the nullspace of AT. The rank r of the matrix A is the dimension of its column-space, i.e., the maximum number of independent column-vectors. It can be demonstrated that the ranks of A and AT are equal. The dimension of the nullspace is n — r, that of the left nullspace m — r. Example: let us consider a mineralogical assemblage made of enstatite, forsterite, quartz similar to that discussed in Section 1.2. The matrix A2x3 such that en

fo

qz"

A = SiO 2

1/2

1/3

1

LMgO

1/2

2/3

0.

where, for clarity, headings have been added to rows and columns, is called the mineralogical matrix or component matrix of the assemblage. Any geochemical sample formed as a combination of silicon and magnesium oxides can be considered as a vector in either the oxide space or the mineral space. Both oxides and minerals can be actually present or simply be virtual. The normative composition (e.g., the CIPW norm) of a sample is the representation of its composition in terms of virtual minerals. Oxide or mineral components are independent if they form a base of the corresponding space. The number of independent components of the assemblage is therefore the rank of the mineralogical matrix. This is the celebrated Brinkley rule (Brinkley, 1946), quite straightforward for simple artificial assemblages, of considerable difficulty for complex natural rocks and solutions. If the number of phases exceeds the number of independent components, the mineral phases form a reactional assemblage. Examples of calculations of stoichiometric coefficients from the mineralogical matrix will be presented in Chapter 5. Additional examples will be discussed for solutions in Chapter 6. Each vector xn can be decomposed as the sum of a vector from the row-space and a vector in the nullspace. These two vectors are orthogonal. Each vector >>m can be decomposed as the sum of a vector from the column-space and a vector in the left nullspace. These two vectors are orthogonal. 2.2 Square matrices

2.2.7 The determinant of a matrix • The determinant det A of the square matrix A recurrence formula

is most simply defined by the row-wise

(2.2.1)

2.2 Square matrices

59

Figure 2.1 Hyper-prism built on the column-vectors (a l5 a 2 , a3) of a non-singular matrix A3x3. The determinant is equal to the volume of the hyper-prism. When one of the vectors can be expressed as a linear combination of the others, both the volume and the determinant vanish and the matrix A is singular. In this formula, the choice of the row index i is indifferent and \Atj\ is the ij th minor, i.e., the determinant of the matrix obtained by deleting the ith row and the jth column of the matrix A. The recurrence relation requires a starting condition: the determinant of a scalar a, i.e., of a matrix of dimension 1 x 1, is a. The product (-1)' + i \A i } \ is the cofactor of rank ij. The development can be equally made column-wise, i.e., (-lY+>a,j\AtJ\

(2.2.2)

Example (row-wise): det

^11

^12

a2l

a22

Example (column-wise) 1 2 0 det 0

2

1 = (-l) 1

3

1

1

+1

det A represents the volume of the hyper-prism made of the column-vectors of the matrix A (Figure 2.1). If det A^O, the column-vectors of A are independent and the matrix is regular. Since the rank of the column-space equals that of the row-space, the row-vectors are also independent. If det A = 0, the column-vectors A are not linearly independent (nor are the column-rows) and the matrix is singular. At least one edge-vector of the hyper-prism made of the column-vectors is in the subspace of the remaining edge-vectors: the volume of the hyper-prism vanishes and det A = 0. A few useful properties of the determinants: det AB=det A detB

(2.2.3)

60

Linear algebra (2.2 A) (2.2.5) (2.2.6)

2.2.2 The inverse of a matrix • Given a square matrix Anxn, the matrices Bnxn and Cnxn such that AB=In and CA=In must be identical because matrix product is associative C(AB) = {CA)B

(2.2.7)

B and C are both noted A~l which is called the inverse matrix of A. If A has an inverse, it is said to be regular. If B does not exist, A is said to be singular. The demonstration of the following useful properties will be found in standard textbooks M1)"1^"1)1 1

(AB)

=B

1

A

(2.2.8) l

(2.2.9)

• The adjoint adj A of the matrix A is the transpose of the matrix of its cofactors. If A is regular (2.2.10) This property is rarely used to calculate inverses for matrices with a dimension in excess of three. Example: The matrix A = \

2 2

3 3

has the adjoint 3

-2

l

Its determinant is

and its inverse 3/5 -2/5

1/5 l/5_

2.2.3 Orthogonal matrices A square matrix O is orthogonal if its column-vectors form a set of orthogonal vectors and have unit length, i.e., OTO=I

(2.2.11)

2.2 Square matrices

61

SO

OT = O~ 1

(2.2.12)

OTO=OOT = I

(2.2.13)

hence

which shows that row-vectors are also orthogonal. An orthogonal matrix is always regular because its vectors (either row or column) are orthogonal. Using the properties of determinants and inverse matrices shows that det O=det 0 T = det O~l = 1/det O

(2.2.14)

The determinant of an orthogonal matrix is therefore equal to ± 1. Example:

r i/v/2 1/V21 1—1/^2 1/^2J

2.2.4 The trace of a matrix The trace of a square matrix AnXm

denoted trA, is the sum of its diagonal elements trA=tatt

(2.2.15)

Straightforward properties of the trace are tvAT = tvA

(2.2.16)

tr(^±J?) = t r ^ ± t r ^

(2.2.17)

while using equation (2.1.5) would show that XvAB=\rBA

(2.2.18)

The trace of a scalar is the scalar itself. Since the inner (dot) product of two vectors xn and yn is a scalar, we can write r

(2.2.19)

Replacing y by the product Az gives the useful form (2.2.20)

62

Linear algebra

Example:

Tl/2 1/2T3 O l r p / 2 1/21=3 + 1 = : |_l/2 1/2 JLO l j L3/2 l/2j 2 2

a

2.2.5 The fundamental geometric transformations The square matrix ^4wXw transforms the vector xn into a vector yn by the product y = Ax. Multiplication by the matrix A associates two vectors from the Euclidian space 9T and therefore corresponds to a geometric transformation in this space. A is a geometric operator. Non-square matrices would associate vectors from Euclidian spaces with different dimensions. The ordered combination of geometric transformations, such as multiple rotations and projections, can be carried out by multiplying in the right order the vector produced at each stage by the matrix associated with the next transformation. • If A is the orthogonal matrix O, the product Ox may be viewed as a rotation. The relationship (0JC)T (0JC) = xTOTOx = xTx

(2.2.21)

implies that length Ox — length x is preserved, which identifies the transformation as a rotation. Example: ^ 1 . •, * ^ fcosfl For plane rotation with angle 0, O = \ |_sin 6

— sin0~| cos 0 J

If 0 = 45°, the vector (x = 1, v = 0) becomes the vector (x = y=

Rotation matrices can be defined for an arbitrary number of dimensions. They are particularly useful to examine compositional data in three-dimensional spaces in search for regularities unsuspected in two-dimensional spaces. Commercial software (e.g., Systat™) exists that produces geometric transformations in a convenient way. & Table 2.1 gives SiO2, CaO and K2 O data for 2.1 Ga-old granitic rocks from West-Africa (Boher et a/., 1992). Plot the standardized data, i.e., the data from which the mean is first subtracted then divided by the standard deviation, in diagrams showing the second (y) vs the first (x) coordinate after rotation around the third coordinate (z) axis of —70 degrees, then after rotation around the first coordinate axis of + 30 degrees. Table 2.1 gives the data row-wise, so we will call this array XT. The mean vector (67.73, 2.93, 3.27) is first subtracted from the data and the result divided by the

2.2 Square matrices

63

Table 2.1. Major-element data (%) on some 2.1 Ga-old granites from (Boherei al., 1992).

West-Africa

Sample #

SiO 2

CaO

K2O

Sample #

SiO 2

CaO

K2O

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

65.40 63.90 60.71 61.69 59.53 62.55 66.43 64.67 72.60 70.40 71.83 67.43 65.90 64.96 71.95 68.17

3.30 3.10 4.45 3.44 4.97 4.35 2.76 2.90 0.75 1.90 1.27 2.95 4.25 4.59 1.28 3.84

3.90 3.75 3.86 3.59 2.87 3.40 3.93 4.41 5.30 4.93 4.53 3.54 0.83 1.50 5.12 0.58

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

63.46 72.90 61.17 67.43 68.96 75.06 66.71 71.90 70.71 70.12 69.35 69.56 70.95 62.58 74.13 74.26

3.79 2.07 5.47 3.50 3.54 0.89 3.39 2.88 0.54 2.70 2.26 3.17 3.12 4.91 1.09 0.49

3.60 2.82 0.81 1.82 1.46 4.37 2.66 2.18 7.03 2.38 3.65 1.48 1.44 1.93 5.15 5.85

standard-deviation vector (4.36, 1.37, 1.60). Rotation by —70 degrees around the third axis gives the rotation matrix 0.342 O = -0.940 0

0.940 0" 0.342

0

0

1.

and the new data matrix will be XTOZ. Rotation by 30 degrees around the first axis gives the rotation matrix 10 0 0.867 |_0 0.5

0 -0.5 0.867J

and the new data matrix will be XTOZOX. The three plots are shown in Figure 2.2: the first rotation shifts representation from the standardized CaO vs SiO 2 to an array with very large dispersion. The second rotation around this new x axis produces an unexpected linear correlation which shows that the three variables are tightly related. The original unit vectors in the standardized SiO 2 , CaO, K 2 O are also shown in order to keep track of the transformations. They are simply given by the rows of Oz in the first plot, and by the rows of OZOX in the second plot, o

64

Linear algebra

3 2 1

(a)

o o

o

o

oo°o

° VTW

0

O

Q

O

-1

0 °o O°o

-2 -3

-2

(b)

2

1

1

Loo%

0 -1

2

0

o

O

o °o o o° o

oO

o°o o o o o

o °o o

-2 -3

-2

0

2

2 <

1

8

0 -1

Qo

Standardized dataaxes l:SiO 2 2:CaO 3: K2O

-2 -3

-2

Figure 2.2 (a) Standardized SiO2, CaO and K2O data for 2.1 Ga-old granitic rocks from West-Africa (Boher et al., 1992, Table 2.1). (b) After rotation around the third coordinate axis of — 70 degrees, then (c) after rotation around the first coordinate axis of + 30 degrees.

2.2 Square matrices

65

• If A is a diagonal matrix, it corresponds to a scaling transformation. Each coordinate of the vector x is scaled by the corresponding diagonal element. Example:

Lo 1/2JL1J L1/2J • A projector Pm x m is a square matrix, normally singular, and such that PT = P (symmetric)

(2.2.22)

PP=P (idempotent)

(2.2.23)

trP=rank P

(2.2.24)

If P is a projector, l—P is a projector onto the orthogonal space since (Px)T(I- P)x = xTPTx - xTPTPx = 0

Given the m x n rectangular matrix A, the m x m projector P=A(ATA)~1AT projects each m-vector y onto the column-space of A. Py can be written

where the term in brackets is a vector of dimension m. Py is therefore a linear combination of the column-vectors of A. In addition, P is visibly symmetric and idempotent since

while the trace of the projector is

As a special case, projection onto a vector b corresponds to the matrix P, such that

PJ-^T

bb

& Find the projection y of the rock composition x given in weight percent in Table 2.2 into the diopside-olivine-silica triangular diagram of Walker et al. (1979). This diagram uses coordinates given by: plagioclase (pi) = A12O3 + Na 2 O + K 2 O diopside (di) = CaO - A12O3 + Na 2 O + K 2 O olivine (ol) = (FeO + MgO + A12O3 - CaO - Na 2 O - K2O)/2 silica (si) = SiO2 - (FeO + MgO + A12O3 + 3CaO + 1 lNa 2 O + 1 lK2O)/2

66

Linear algebra Table 2.2. Major-element composition x(%) of a rock to be projected in the diopside-olivinesilica triangular diagram of Walker et al., 1979.

M represents molar weights of oxides.

xT M

SiO 2

A12O3

FeO

MgO

CaO

Na 2 O

K2O

49 60

16 102

8 72

13 40

10 56

2 62

1 94

where oxide symbols represent molar proportions in a projection from the plagioclase apex. The solution to this problem is extremely simple by non-matrix methods, but we will take advantage of its simplicity to give the layout of an approach that can be extended by computer software to any projection. First comes a scaling step that consists in dividing weight percents by the molecular weights given in Table 2.2 to give molar proportions m. Multiplication by 1000 is convenient and results in millimoles per 100 g of rock.

— 60 0

0 1000 102

0

0

0

0

0 0

1000 72 0

0 0 0 1000 40

0 0

0 0

0 0 49

817 157

0

0

0

16 8

111

0

0

0

13

325

1000

0

0

0

0

0

0

0

0

0

0

0

0

0

0

56

0 1000 62 0

0

=

10

179

2

32

1

11

0 1000 94

This projection step first consists in recasting the molecular rock composition as linear combinations of the mineral molar coordinates pl-di-ol-si. The results represent a modified rock vector m+ related to m through the matrix Q, such that

67

2.2 Square matrices and SiO2 0 Q=

0

A12O3 1 -1

FeO

MgO

CaO

Na2O

0

0

0

1

1

0

0

1

1

1

K20

0

1/2

1/2

1/2

-1/2

-1/2

-1/2

1

-1/2

-1/2

-1/2

-3/2

-11/2

-11/2

The modified rock vector m+ =(200,65,186,16)T can now be projected onto the diopside-olivine-silica plane through the projector P, such that y=Pm+=PQm This projector P is defined by the three vectors di (diopside), ol (olivine), and si (silica) from the same space as m+. The problem is quite simple because
0

0 pi

A= 1 0

0

0 di

0

0

1 0 ol 1 si

The projector P onto the column-space of A is "0 0 0 0" P=A(AJAy1AT

=

0 10 0 0

0 10

_0 0 0 1

which, in this case, leads to y = [0,65,186,16]T. The same result is conveniently arrived at by combining the operators in the proper order and relating the weight fractions to the final projection coordinates through one single matrix product

(PQM)x =

0

0

0

0

0

0

-9.80

0

0

17.86

16.13

10.64

65

0

4.90

6.94

12.5

-8.93

-8.06

5.32

186

-4.90

-6.94

-12.5

-26.79

-88.71

-58.51

16

16.67

0

0

0

68

Linear algebra

The matrix PQM can be computed once and used for any further calculation. In order to compute the coordinates j n o r m of the rock composition in the triangular diagram, the solution y should finally be normalized to 1. The result is ^nOrm = [0.243, 0.697, 0.060]T Petrological literature is rich in various projections. The reader can try other projections in which Q and P have different form. <^. & Find the projection of the vector x (1,1) onto the vector b (1,2) and its orthogonal complement. The projector P on b is

J>

_ [1/5 2/5]

Tj "1.2/5 4/5 J

<• 4 2

The projection of x onto b is therefore 5

2/5

2/5

5JL1J |_6/5J

4/5J

and the orthogonal complement is

2/5

l

-1/5J

2.2.6 The metric tensor and oblique projections We now return to the very basic concept of vector length in vector spaces. In a Euclidian space spanned by a base of n orthogonal unit vectors eh the squared length I2 of a n-vector v is the quadratic form given by l2 = vTv = vTInv = vTfj et etTv = £ i= 1

foT»)V»

(2.2.25)

i= 1

The scalar etJv is the ith 'coordinate' vt of the vector v in the base of the orthogonal vectors eb i.e.,

In a Euclidian space, the squared length of a vector is the sum of its squared

2.2 Square matrices

69

coordinates. It is quite frequent that a vector is not or cannot be naturally expressed as coordinates in a Euclidian base of orthogonal vectors. An enlightening example is the major-element composition of a rock which can be expressed as a combination of either orthogonal oxide proportions or non-orthogonal mineral proportions. Let us call the new arbitrary unit vectors £j and write the change of coordinates as a set of linear equations n

(from the equations written above, it seems more convenient to deal with the transpose of unit vectors). In a matrix notation, we introduce the nxn matrix E of the general unit vectors Sj which is the equivalent of the matrix /„ for the vectors ei and the matrix B made of the bu components

E* = InB

(2.2.26)

Clearly B=ET, and given the special nature of the vectors et ^ = 8^

(2.2.27)

We assume that we have shifted to a 'real' new vector base, i.e., the vectors Gj are independent and B is non-singular. Now, we can go back to the v coordinates and write that, in a Euclidian space, vector length is an invariant

For convenience, we will write w = B 1v and call wf the ith coordinate of vector w in the new base of the vectors Sj. We also define the matrix G as (2.2.28)

and call it the metric tensor (a tensor is a multi-dimensional array or a generalized matrix which has special properties upon change of variables). It satisfies (length x)2 = xJGx

(2.2.29)

and is equal to the identity matrix /„ in the Euclidian vector space 9T. The concept of metric tensor becomes central whenever distances and projections are considered, particularly when least-square criterion are used, a point that will be discussed in Chapter 5. Let us ask the frequently raised question of how to find an expression in terms of old coordinates (e.g., oxide proportions) for a projection made in the non-Euclidian space. This could be the case for finding oxide abundances of a basalt composition projected in the Yoder and Tilley tetrahedron, or the oxide abundance of a metamorphic rock composition projected into an ACF diagram assuming that quartz is present.

70

Linear algebra

& A rock is made of 0.45 moles SiO 2 , 0.10 moles CaO, and 0.45 moles MgO, which we will describe with the composition vector v (0.45,0.10,0.45). If we consider this rock as being made of virtual forsterite (SiO2, 0, 2MgO), enstatite (2SiO 2, 0, 2MgO), and diopside (2SiO2, CaO, MgO), what is the projection vector v of the rock composition onto the enstatite-diopside plane? The mineralogical matrix BT is obtained by the set of equations relating the unit mineral compositions e to oxide unit fractions e e (forsterite) = l/3e SiO2 + 0eCaO + 2/3e MgO £ (enstatite) = l/2eSiOl + 0eCaO + l/2e M g O £ (diopside) = l/2e SiO2 + l/4e C a O + l/4e M g O

BT reads as 1/3

0 2/3"

J

B = 1/2

0

1/2

.1/2 1/4 1/4. and therefore 1/3

0

2/3T1/3

T

G=B B= 1/2 0 1/2 0 .1/2 1/4 I/4JL2/3

1/2 1/2"

5/9

1/2 1/3"

0 1/4 1/2 1/2 3/8 1/2 1/4 J Ll/3 3/8 3/8.

It is left to the reader to check that

l

B =

-3

3

3"

4

-6

-2

0

4

0.

and therefore the abundance vector w of virtual forsterite, enstatite, and diopside proportions is given by -3 4 0

3

3

-6 4

-

2

0.45

0.3

0.10 = 0.3

ojL0.45.

.0.4

We can check the double equality ["0.45" 0.10

0.45] 0.10 = 0.415

Lo.45.

71

2.2 Square matrices

and

p/9 0.3

1/2

1/31 0.3

0.4] 1/2 1/2

3/8

0.3 = 0.415

Ll/3

3/8J

0.4

3/8

In the virtual mineral space, the rock composition is projected onto the plane made by the vectors enstatite [0, l , 0 ] T and diopside [0,0,1] T . Although these vectors are not orthogonal in the original oxide composition space, which can be verified by constructing the dot product of columns 2 and 3 in the matrix BT, the particular choice of the projection makes the vectors orthogonal in the transformed space. According to the projector theory developed above, we project the rock composition onto the column-space of the matrix A such that 0 0" A= 1 0

Lo l. The projector P, which makes such a projection, is given by P=A(ATA)~1AT or "0 0 P= 0 .0

0"

1 0 1.

0

As expected, the projected virtual mineral abundances co of the rock are given by 0 0.3 L0.4J

We find the projected rock composition v by applying the change of variable backwards 1/3

1/2

1/2

0

0

1/4

2/3

1/2

r°

0.35

0.3 = 0.10

1/4. L0.4_

.0.25

which can be later normalized to 100% oxides. In case this calculation should be repeated over large files, writing up the oblique projector PB explicitly saves computational efforts. By forming the appropriate dot product, we check that the

72

Linear algebra

projected composition is not orthogonal to forsterite in the oxide space (oblique projection) for T0.35" [1/3

0

2/3] 0.10

0.85

L0.25J

2.2.7 Gram-Schmidt orthogonalization Given a set of independent vectors x1,x2, ..., xn, it is requested to form a new set •••> y>t» of orthogonal vectors spanning the same space. The Gram-Schmidt orthogonalization scheme sets

JI>J2>

yi=xx

(2.2.30)

The fcth step removes from xk its components along yl9 ..., j f c _

l5

e.g., for k = 2 (2.2.31)

or, in general

-Jfc-i

(2.2.32)

This procedure is important for producing orthogonal vectors and functions in numerical analysis. It is not unique since it can be started from any xk. <& Construct orthogonal vectors from xx = [1,2,0] T and x2 = [1,1,2] T . The first step is

Given the products i=5

and

the second step is 1 2/5 3 2 = -1/5 1 y2= 1 - - I = J 2 . _2_ .0.

T

It is easily verified that y1 and y2 are orthogonal.

2.3 Eigencomponents

73

2.3 Eigencomponents 2.3.1 General • Most square matrices Anxn can be factored using a diagonal matrix An and a regular matrix Unxn in the following forms A = UAU 1oAU=UA<>A=U ^AU

(2.3.1)

The ith diagonal element A, of the matrix A is called the ith eigenvalue of A. The columnvectors (**!, ..., «„) of £/ are usually taken (although not necessarily) of unit length and are called the eigenvectors of the matrix A. • If the inverse A ~l of matrix A exists, then it can be decomposed as A-1 = UA-1U~l

(2.3.2)

A matrix and its inverse therefore have the same eigenvectors but reciprocal eigenvalues. Each pair of eigenvector and eigenvalue (eigencomponents) can be related through a linear system of equations such that Aut = XiUi

(2.3.3)

Example:

-\ -3 The relationship between u2 and X2 is

[-: -a-H-3 The order of eigencomponents is arbitrary, although they are commonly ranked in the order of increasing or decreasing eigenvalues. Some properties result from those of the determinant, e.g., det A = det U x det A x det U

1

n

= det U x det A/det U= ]~] K

(2-3.4)

which shows that, for a matrix to be singular, at least one of its eigenvalues must be zero. From the properties of the trace, we get

t r / * = t r £ / A { r 1 = t r A ( T 1 £ / = t r A = £ Xt The trace of a matrix is equal to the sum of its eigenvalues.

(2.3.5)

74

Linear algebra 2.3.2 Computation of eigencomponents AUi = kt u( o det(A - kti) = 0

(2.3.6)

The second equation is called the characteristic equation of the matrix A. Once lt is known, the vector ut is computed as the unit vector solution of the linear system. It is left to the reader to show that the eigenvalues of a 2 x 2 matrix A can be found as a solution of the equation 0

(2.3.7)

Example: the eigenvalues and eigenvectors of the matrix A given by

'-[.; -a are obtained by solving the characteristic equation

{LL - l2~X - 12-Aj1 = 0 or (2-/l)2-l=0 and has the solutions kx = 3 and X2 = 1. The x and y coordinates of the eigenvector associated with kl satisfy

These equations, which give an eigendirection and not the coordinates of an eigenvector, are equivalent. An eigenvector, usually with unit length, must be chosen along that eigendirection, so that

One solution is x= 1/^/2

and

y=-l/y/2

For the same eigenvalue, the opposite vector in the same eigendirection is the solution — x, — y, also acceptable. The eigenvector associated with k2 = 1 is

2.3 Eigencomponents

75

and finally

V -i / v /2 1/V2JL0 Efficient computer programs rarely use this method. The numerical algorithms used to search for eigenvalues are countless and difficult to implement without a dramatic loss of accuracy. Use of the best canned routines saves time and produces better results. 2.3.3 Eigencomponents of symmetric matrices If A is a symmetric matrix, then the matrix U is orthogonal. This can be shown by considering two eigencomponent pairs i and j

Multiplying the first equality by the transpose of Uj gives u/AUi = (Aujfui = Xju/ui = Ajif/ifi

(2.3.8)

which is possible only for i=j or orthogonal, unit-length eigenvectors. The eigenvectors u{ make an orthogonal base of unit vectors in the space 9?" A = UAU

l

= UAUJoAU=UAoA=UTAU

(2.3.9)

If A is a symmetric matrix, it can be shown that its eigenvalues are real. Product-matrices such as ATA or AAT are special cases of symmetric matrices and will be shown in the next section to have non-negative eigenvalues. • A symmetric matrix A, can usually be factored using the common-dimension expansion of the matrix product (Section 2.1.3). This is known as the singular value decomposition (SVD) of the matrix A. Let Xt and ut be a pair of associated eigenvalues and eigenvectors. Then equation (2.3.9) can be rewritten, using equation (2.1.21) A = UAUT= £ XiUiu?

(2.3.10)

A symmetric matrix therefore can be expanded as a weighted sum of the outer product matrices i#, utJ, the weights being the eigenvalues kt (Figure 2.3). This will prove to be a useful concept in principal component analysis (Chapter 4). For large matrices (n > 30), the modulus of eigenvalues may span tens of orders of magnitude. In contrast, the eigenvector components are bounded since eigenvectors are of unit length. A large matrix is therefore dominated by the eigencomponents corresponding to large eigenvalues, whereas the eigencomponents corresponding to the smallest eigenvalues are seen mostly as noise.

76

Linear algebra T

=

A,

Figure 2.3 Singular value decomposition of a matrix A into the weighted sum of the outer product of its eigenvectors.

• A projector is another case of a symmetric matrix. Since it is idempotent, its eigenvalues must be either 1 or 0. Indeed, idempotence relative to eigenvectors ut implies

Moreover P(Put) = Pui = Xiui

which requires that any eigenvalue and its square are equal. Such a condition is only true for 0 and 1. The square-root A112 of a symmetric matrix A is defined as Al/2 = A1/2UT

(2.3.11)

which enables the matrix A to be decomposed as A=(A1/2)TA1/2

The elements of Ai/2 may be complex numbers. Al/2 is the product of a simple rotation and a scaling, carried out in that order. Example: r

2 -1~| 1 / 2 = | V 3

L-l

00T1/V2

-1/J2ljj3j2

2j Lo yrl

I/V2J Ll /v /2

-V372]

l/v/2~J

& Describe the geometric transformation applied by multiplying the vector x = [1,2]T by the matrix A such that

-[

!

"

Multiplication of vector JC by matrix /4 produces the vector y and therefore y = Ax=UAUTx

2.3 Eigencomponents

11

x2

[-3 Ax = UAUTx

(iii) AUTx

-2

-1

Figure 2.4 Decomposition of the product of a vector by a 2 x 2 symmetric matrix A = UAUT into a sequence of (i) rotation (ii) scaling (iii) rotation. The last and the first rotations are opposite of each other.

Using the results from the previous exercise, the first transformation UTx is a rotation that can be written as

L1/V2

1/V2I2J L 3/V2J

(Figure 2.4). The second transformation scales the resulting vector UTx

3/^/2

The third transformation counter-rotates the vector \UTx back to the initial frame

-1/V2 I/V2JL 3/yiJ L3 Example: Let us check the singular value decomposition for the matrix must prove the following equality A = Ajlljfl!

=

2

L-l

M.We 2j

78

Linear algebra

With the help of the results from previous exercises, we get

1/V2 J

1/2 - l / 2 j L—1/2 1/2J

p/2 1/21 I" 2 -11 L1/2 1/2J L - i 2J

2.4 Quadratic forms and associated quadrics 2.4.1 Quadrics associated with symmetric •

matrices

T

Given An^n a symmetric matrix, the quadratic form S = x Ax can be rewritten as S = xTU\ UTx = (UTx)TA( UTx) = (A1/2 UTx)T\ljl

UTx

or, using SVD

S=t Wurfx = t U»iTx)Wx i = 1

(2.4.1)

i= 1

and finally

S=I^ 2 =I Z i 2 i=l

(2.4.2)

i=l

where ^• = 11^

(2.4.3)

represents the zth component of x in the frame of the vectors i#, and

z^V'V*

(2-4.4)

its ith component in the frame of the vectors h112^. • In space W, S = constant is the equation of a hyper-quadric whose principal axes are colinear with the eigenvectors ut and have a half-length of A,~1/2. The simplest case occurs when all At are positive, which happens in particular when A is a product of real matrices such as BTB or BBT. Then the hyper-quadric is a hyper-ellipsoid and, from the above equations, 5 is positive whatever the vector x. The matrix A is said to be positive definite. Similarly, the equation xJA-lx=l

(2.4.5)

represents the equation of an hyper-quadric whose principal axes are colinear with the eigenvectors ut and have a half-length of V / 2 . This is easily shown by using the eigencomponent decomposition of the matrix inverse A'1. If A is definite positive, the associated quadric will be a hyper-ellipsoid with half-length principal axes equal to kt ~1/2.

2.4 Quadratic forms and associated quadrics

79

1

0.5

-0.5 -

-1

=i

i

-1

-0.5

y- \j 0.5

Figure 2.5 The ellipse 2x2 — 2xy + 2y2 = 1. ux and u2 are the eigenvectors associated with the quadratic form, 3 and 1 the corresponding eigenvalues.

Given the ellipse (Figure 2.5) defined by the equation

find the matrix associated with this quadratic form. Find the appropriate change of variable to obtain the equation of a circle. Clearly

[-; -3 since

Using the factorization of the symmetric matrix A as UAUT, we get, upon a change of coordinates

80

Linear algebra

which is equivalent to

\x]=u\*U bJ

LyJ l-

In the previous two equations, the old coordinates x and y are set as linear combinations of the eigenvectors. The following equation is therefore arrived at s=3x i 2 +y 2 =i which is the equation of an ellipse with its principal directions colinear with the eigenvectors and with half-axes equal to l/y/3 and 1/^/L Changing this equation into the equation of a circle is achieved through

lA/ilx or \X

-

LyJ "'"

LyJ L-1/V2 1/V2JL 0

The change of variables

x" y=- —- +

y

therefore leads to the expression

which is easily simplified into x i2 +y i2 = i i.e., the equation of a circle with unit radius and centered at the origin, o & Table 2.3 lists some Na and Cl data (in mmoles per liter) on rainwater from the Amazon basin (Stallard and Edmond, 1981). Draw the ellipse associated with the

2.4 Quadratic forms and associated quadrics

81

Table 2.3. Na and Cl concentration (\imoll~x) of rainwater samples from the Amazon area (Stallard and Edmond, 1981). Na Cl

18.6 25.9

14.2 14.0

19.5 18.0

21.8 24.4

20.6 22.7

16.2 13.8

9.8 10.5

9.9 8.4

13.2 14.0

7.4 5.7

9.2 1.6 11.5 1.9

quadratic form |_c -x N a ,c

-x c l ja

_

-i

LQ-XciJ

where xNa and xcl are the mean value of the sample Na and Cl concentrations, respectively, and S is the covariance matrix. The purpose of this exercise is to learn how to draw a probability ellipse from the mean values and covariance matrix, a topic to be further developed in Chapter 4. Na and Cl are incorporated into clouds during evaporation of seawater and are therefore strongly correlated. Let us call x the vector of Na and Cl concentrations, x the vector of sample means and S the symmetric, positive-definite covariance matrix, i.e., the 2 x 2 matrix with variances on the diagonal and the covariance between Na and Cl concentrations as off-diagonal terms. The equation of the ellipse to be drawn can be written (x-x)TS-\x-x)=l The covariance matrix is factored using the diagonal matrix A and the eigenvector matrix U as UAUT. Since S is symmetric and positive-definite, the eigenvalues are positive and the eigenvectors orthogonal. The inverse S~ * of S can be expanded as UA ~x UT and the transformation z = A~i/2UT{x-x) transforms the ellipse equation into the unit circle equation zTz=l Conversely, given a vector z on the unit circle, we go back to the original coordinates through

The mean value of Na and Cl concentrations are 13.50 and 14.23 and the covariance matrix S is =p7.56

42.791

L42.79

55.09J

82

Linear algebra

The eigenvalues can be found by solving equation (2.3.7) X2 - A(37.56 + 55.09) + (37.56 x 55.09 - 42.79)2 = X2 - 92.65/1 + 238.2 = 0 giving /Ij =2.643 and X2 = 90.01. The coordinates x and y of the first eigenvector are found by solving one of the eigenvector equations

or y= -34.92/42.79*= -0.816* Combining one of these equations with the condition for a unitary vector x2 + / = l or x 2 (l+0.816 2 )=l we get x = 0.775 and y= — 0.632. Proceeding identically with the second pair of eigenvalue and eigenvector, we obtain the factorization

s-u\uj-l

0>775 a 6 3 2 2 6 4 3

L -0.632

T-

0.775 J|_0

° T

a775

90.01 JL-0.632

°-632T 0.775 J

We now calculate the coordinates of an arbitrary number of points zi (i= 1,2,...) on the unit circle, most easily by incrementing an arbitrary angle (pt from 0 to 2n and taking zi = (cos cph sin cp^T. For instance, for q> = n/6, z = [y/3/291/2]T and i3.50~| |~ 0.775 0.632X^2.643 0 T N /3/2"|_ri7.59 l4.23j + L-0.632 0.775JL 0 y^OOTJL l / 2 j " L l 7 . Enough points have been calculated in this way to draw the ellipse shown in Figure 2.6. o

2.4.2 Gerschgorin's circles theorem This theorem has important implications in the box model theory. It states that every eigenvalue of AnXn, possibly complex, lies in the complex plane inside at least one of the circles centered at the diagonal entry au and with a radius equal to the sum Z \atj\(i / ; ) of all the off-diagonal elements of the ith row. In order to prove this theorem, let X and u [ul9 w2, • • •, wJ T be a pair of eigenvalue

2.4 Quadratic forms and associated quadrics

83

30 o

25

o o

^»—1

20

yS

i u

15 -

O

-

o so

10 -

-

/ o

5-

_

o

0

0

10

15

20

25

Na (junol I"1) Figure 2.6 The ellipse (x — x?S~x(x-x)=\ and Edmond (1981) listed in Table 2.3.

for the Amazon rain Na and Cl data of Stallard

and eigenvector of the matrix A, which therefore satisfy Au — Xu

Let ut be the component of u with the largest absolute value. Then, by the rule of matrix product (2.4.6)

We rewrite the last equality as != X auuj or (2.4.7)

Taking the modulus of each side and applying the rule for sums of modulus (Schwarz

84

Linear algebra

inequality) gives

-a«l= Z Since u( is the component of u with the largest absolute value, we get

iA-aui< i K\

(2.4.8)

which means that, in the complex plane, the eigenvalue X lies within the circle centered at au and with radius r, such that (2.4.9) j*i

The same argument applies to AT and may be used to calculate Gershgorin's circles with respect to rows instead of columns. <& Find the Gershgorin's circles for the matrix A, such that • - 1

A=

0

r

3

I

1 1

l.

0

Imaginary part

Real part

Figure 2.7 The Gershgorin's circle theorem. The three eigenvalues of the matrix A are located within the Gershgorin's circles.

2.5 Systems of linear differential equations

85

Using commercial software, we find the eigenvalues kx = -1.45,^2 = 1.00, ^ 3 = 3.45

We can check that

|A2-3|
and the corresponding circles are drawn in the complex plane of Figure 2.7. <=

2.5 Systems of linear differential equations 2.5.1 First-order linear homogeneous equations

Let us first investigate the properties of equations that involve only first-order derivatives, i.e., equations such that (2.5.1)

at

where f(t) and g(t) are functions of t only. g(t) is known as the input or forcing function (see Chapter 7). If g(t) = 0, the equation is said to be homogeneous. Given the following system to solve in the n time-dependent unknowns xb with the atj being constant coefficients: dxj

= allxl+al2x2+

... + alHxH

dx2 -— = a2lxl+a22x2+ dt

... +a2nxn

at

dxn — =fl w ix 1 +an2x2 -f ... +annxn dt

Let us lump the n xt (i= 1,2, ..., n) together into the n-column vector x and the atj coefficients into the matrix AnXn which is not assumed to be symmetric. We now note the system of equations in the matrix form (2.5.2)

86

Linear algebra

The matrix A is first diagonalized as UAU'1

and, premultiplying by IT"1, we get

I A^ l ,^ A l T ' , dr dr

(2.5.3)

Defining the n-vector y as U~ xx, the matrix equation can be rewritten dv

-f = Ay dt

(2.5.4)

which amounts to the n equations

dr where /,- are the diagonal entries (eigenvalues) of A. This system has a unique set of solutions (i = 1,2, ..., n) ^ = y,°e^

(2.5.6)

which we recombine into the matrix form x=Uy=UeAtU-1x0

(2.5.7)

where y0 = [ j ^ 0 , . . . , y n °] T is the vector j , and x0 the vector JC, both taken at t = 0. e A ' is a diagonal matrix with entries eA'f. Reverting to the JC values, the solution is JC =^v=i7e

Af

f/- 1 jc 0

(2.5.8)

The matrix Ue^U'1 is usually denoted eAt. In the common-dimension expansion form (Section 2.1.3), this matrix product reads x= £ e;'f(/th column of (7)(/th row of U~l)x0= ^ e^.-JCo i= 1

(2.5.9)

i- 1

where the matrix 21, is defined as ^lf- = (ith column of U)(ith row of f/~1)= Ue^U'1

(2.5.10)

The relationship between the 21/s and the matrix exponential tAt is eAr= f ^"^i

(2.5.11)

i= 1

The final step can be taken by defining the n x n matrix Q by its ith column

9X£JC0.

2.5 Systems of linear differential equations

87

Then the solution becomes

(2.5.12)

The solution vector x therefore can be viewed as the weighted sum of n 'components' eAr with a corresponding weight matrix Q = Z5IiJt0 (i = 1,..., n). This form is extremely useful to predict the rate of growth or decay for each individual component of the solution. &

Solve the system

dt

dt

• = xl+x2

with the initial conditions x1 = 1 and x2 = — 1 at t = 0. Defining the column vector x as (xux2)T, we can write djc_p

01

d7~|_l

U

Ax

The matrix A can be factored as

-C 3-

p> OT-72 01 .0 l l - l lj

UAU l = L 2

The solution is

or

- 'ill The alternative formulation

Linear algebra

or, in its more conventional form

x2 = e 2 r -2e r shows that the two components (1,1) and (0, — 2) of x increase at different rates, o ^ Given a number of nuclides at t = 0, calculate the distribution of nuclides in the U-Th decay series at any time t. This problem has important applications to the various methods of dating collectively known as radioactive disequilibrium and to the evolution of hazard in nuclear wastes containing mixtures of radioactive nuclides with different periods. The natural nuclides 2 3 8 U, 2 3 5 U, and 2 3 2 Th decay to different lead isotopes along a series of radioactive isotopes of different element (Pa, Ra, Rn, ...). For one specific decay series, let us call Nt the number of atoms of the ith present at time t in the system, which we assumed to be closed to any exchange with the surrounding medium, and kt its decay constant [in (time unit)" 1 , not to be confused with eigenvalues]. Let n be the number of nuclides in the series, i.e., i= 1 for 2 3 8 U, 2 3 5 U, and 2 3 2 Th and i — n for the stable lead isotopes 206,207, and 208. The amount of each nuclide is increased by the decay of its parent isotope and decreased by its decay into the daughter isotope, hence

at

with the Xt canceling for i = 1 and i = n. Although the solution to the system of equations for several nuclides has been known for quite a while (Bateman, 1910), matrix formulation has become the most flexible approach. It is common practice to deal not with number of atoms but with activities [N J = AjiVj (number of decay events per time unit). Therefore, multiplying both sides

dt

or, in a matrix form -Xi

dN

~dt~

o

x2 -x2

...

...

0

0

•••

0

0

••

0

0

0

0

-L. /L

N=AN 0 -X

where TV is the /t-column vector of nuclide activities [ A ^ ] , [N2\

•••, [ N J . The

2.5 Systems of linear differential equations

eigenvalues of the matrix A can be found by solving 0

0

0

0

0 =0

det 0

0

0

0

...

0

-k^i-fi

-k

L

for the eigenvalue \i. Expanding this determinant along the last column shows that the only product which does not vanish is that along the diagonal. Hence

and the system eigenvalues tx1,...,fin are simply the negative of decay constants Al9A2>-~9 K- Th e solution therefore can be written as a linear combination of the negative exponentials

where a/ are constant terms and the sum is limited to j = i because the number of atoms of a given nuclide does not depend on its descendants. For all three radioactive series, the decay of the first nuclide is the rate-limiting step so that Xx «kj, except for the terminal lead isotopes (Faure, 1986). After some typical time t(*, such that — «t t * « —, for all j ^ i, yV 1 all exponential terms become negligible relative to the term in e~A|'. This situation is known as secular equilibrium and requires

Taking the derivatives and comparing with the decay equation, we obtain dN —— ~ —A^cCi e

at

— A, (a,_! e

— a£ e

j

which can be rearranged into i

~ai

This condition is only satisfied when

which establishes the well-known result that, at secular equilibrium, activities are equal.

90

Linear algebra

Let us take the simple example of the two radioactive isotopes us call ^238u a n ( * ^234u their decay constants, respectively d238 U dr d 234 U dr

.

J

238

U and

234

U. Let

238U

LF — A 2 34U

U

or, in activities d[ 238 U] dr ^234U([U][U])

dr In matrix form, this system of equations becomes 0

V[ 2 3 8 U]

A

The matrix admits — /I238u a n ^ ~^234u f° r eigenvalues. Let V be the eigenvector matrix of the matrix on the right-hand side (for obvious reasons, we do not want any confusion with U chemical symbol). The coordinates of the eigenvector associated with — A238U are determined from the equation

o which, combined with the normalization condition

yields 1

_

where — V (^234U~

2.5 Systems of linear differential equations

91

The coordinates of the eigenvector associated with — 2 2 3 4 U satisfy

or

which requires ^ i 2 = 0 and v22 = \. The eigenvector matrix F i s therefore

v= It can be verified that the eigenvectors are not orthogonal. Using the standard procedure, this matrix is easily inverted into

The solution may now be expressed as the sum of the individual components

238

U and

234

U

[ 238 U] [ 234 U]

» - A234U*

which is recombined as = [238U0]e~A238ur a

- A234U*

Although this way of deriving a classical result seems rather awkward and timeconsuming compared to more direct elimination methods, it has the advantage that it can be extended to any combination of isotopes in a decay series. <>

92

^

Linear algebra

Apply the previous procedure to the decay scheme

using the following decay constants: " 99 aa " 1 /L234U = 2.79 xlO" 6 a~ ] ^230Th = 9.21 x 1 0 " 6 a " 1 , 10" 238U = 0.155 125 x 4 A226Ra = 4.27 x 10" a ' \ and A210Pb = 3.23 x 10" 2 a " \ Assume [ 238 U 0 ] = 1, [ 234 U 0 ] = 5, [ 23O Th o ] = 0.2, [ 2 2 6 Ra 0 ] = 0.02, and [ 2 1 0 Pb 0 ] = 2.0. The choice of activities at t = 0 amounts to normalising activities to [ 2 3 8 U]. Once the matrix A

0

238U

*234U

0

0

0

0

0

0 0

0

0

0

0 0

0

is built on a computer from the numerical values given above, the following eigenvector matrix V is calculated "0.44719

[ts inverse V

0

0

0

0"

0.447 21 0.37194

0

0

0

0.447 22

0.533 57

0.568 90

0

0

0.447 22

0.53708

0.58144

0.70239

0

0.447 22

0.53713

0.58161

0.71180

1

is "

2.236 -2.689

l

V~ =

0

0

0

0

2.689

0

0

0

1.758

0

0

1.424

0

-1.013

1

-2.522

0.769 -2.064x10"

4 10

3.301 x 10"

0.0316

-1.455

-3.821 x l O "

6

0.0134

The eigenvalues are known to be the negative of the decay constants. Finally, the matrix Q is computed as 1 0

0

0

0

1 4.0000

0

0

0

0

0

1 5.7382

-6.5383

1 5.7760

-6.6824

-0.073606

0

5.7765

-6.6844

-0.074592

1.9824.

.1

2.5 Systems of linear differential equations

93

-a <

lO"1

103

106

109

Time (years) Figure 2.8 Evolution of the activity of the five nuclides 2 3 8 U, 2 3 4 U, 2 3 0 Th, 2 2 6 Ra, 2 1 0 Pb for the initial activity conditions [ 2 3 8 U O ] = 1, [ 2 3 4 U 0 ] = 5, [ 2 3 O Th] o = 0.2, [ 2 2 6 Ra] 0 = 0.02, [ 2 1 0 Pb 0 ] = 2.0.

The evolution of activities is shown in Figure 2.8. We can see that about one million years (1 Ma) is required to reach full secular equilibrium. <=> & Kinetic theory of oxygen isotope exchange: a granite is made of two mineral phases, quartz (i=l) and feldspar (* = 2), plus interstitial hydrous fluid (w). Initially, 8 18 O 1 =9, 8 18 O 2 = 8 (common late magmatic values) and 5 18 O w =—5 (meteoric water). The system is assumed to be closed and the mass fractions fx= 0.3,/ 2 = 0.6, and / w = 0.1 of oxygen held in mineral 1, mineral 2, and water relative to the total in the rock do not change during the isotopic exchange process. Equilibrium oxygen isotope fractionation between a mineral (i=l,2) and the surrounding hydrous fluid depends on a simple function af (T) of the temperature

(18o/16ox-

( 1 8 O/ 1 6 O) W ~

Calculate the evolution of 518O for each phase assuming that temperature is such that cc1 = 1.010, a2 = 1.005 and first-order exchange kinetics apply with time constants kt and k2(kjk2 = 0.2). We are going to present a method slightly modified from that of Criss et al. (1987) in order to account for some forms of oxygen isotope disequilibrium among minerals in ultrabasic, metamorphic or hydrothermal assemblages. When a mineral assemblage is invaded by a fluid which is not in equilibrium with it, isotopic exchange takes place in such a way that the whole system tends to a new state of equilibrium. Criss et al. (1987) suggest that approach to equilibrium takes place through first-order

94

Linear algebra

kinetics, i.e.,

_ or

where the temperature-dependence of at is made implicit. Dividing both sides of each equation by ( 18 O/ 16 O) SMOW , we write df (18O/16O)t 18 drL( O/ 16 O) SMOW or, upon multiplication by 1000

dt which can be expressed as a deviation from the initial conditions (superscript 0)

dt -M518O.- 0-a,.5 18O w0)-1000/^(1 -a,) Finally, we assume no precipitation and no dissolution of solid phases. Closed system and mass conservation imply

Dividing all by ( 18 O/ 16 O) SMOW , subtracting/i + / 2 + / w , and then multiplying by 1000 results in

or 618Ow-518Ow°=--(618O1-518O10)-^(518O2-618O20) Jw

Jw

Relabelling the variables (1 = 1,2) in such a way that

2.5 Systems of linear differential equations

and i° +1000(1

-aJ-a

results in a system of non-homogeneous linear differential equations

dt

Let us define the matrix A as

/w

A=

We can write

dr

Introducing the change of variables

the system of equations becomes dz dt

which can be solved as a homogeneous system. The solution is

This equation can be reformulated as

The eigenvalues of A are found from

95

96

Linear algebra

with

and ,

ai/i /w

/w

Let us express the time in units of l/kl9 which amounts to assuming k1 = \9 k2 = 5, and work with a non-dimensional time T = k1t. The matrix A is calculated as " / ~~ V

1.010 xO.3\ 0.1 ) 1.005x0.3 01

1.010x0.6 01 / ~ V

1.005x0.6 01

_f" -4.030 L-15.075

-6.0601 -35.150J

and the vector b as b

_|~-1(9 + 1000(1-1.010)- 1.010(-5)}l_r -4.0501 ~L-5{8 +1000(1-1.005)- 1.005(-5)}_r[_-4O125j

Therefore \ 2.001J Solving the characteristic equation, we find that the eigenvalues of A are X1 = —1.3289 and X2= —37.851 with the eigenvector matrix Uand its inverse given by _[" 0.9134 01764"! ~L-0.4071 O9843J

_1__Tl.0139 ~|_0.4193

-018171 0.9408J

The matrix exponential is calculated from the common-dimension expansion as

or, inserting numerical values, as ^ t = e -i.328J~ a 9 1 3 4 ][i. O 139 L-0.4071J

-0.1817]+ e- 3 7 - 8 5 1 { 0 1 7 6 4 ][0.4193 L0.9843J

0.9408]

Remembering that x0 is the deviation of 8 18 O from the values at t = 0, i.e., x0 = [0,0] T , we get the final solution as

0.975e—3.+

L

2.5 Systems of linear differential equations

97

This result can be made explicit as

The new equilibrium state can be calculated by letting f-»oo

eldsp =

5 18 O feldsp°-2.0 = 6.0

5 18 O W can be calculated using the closure equation 618Ow = 5 1 8 O w 0 - — ( + 2 ) - — ( - 2 ) = - 5 - 6 + 1 2 = + 1 The reader will check that 5 18 O rock ( + 7) does not change in the process,

2.5.2 Linear equations of order higher than one A simple example will show how higher-degree linear equations reduce to a system of first-order equations.

Solve

dr2

dt

with the conditions that x(0)= 1 and dx(0)/dt = 0. Let us make the change of variables xx = x(t) and x2 = dxjdt, with xx = 1 and x2 = 0 at t = 0 .The second-order differential equation can be transformed into the following system of two first-order equations

dr dx^ dt Defining the matrix A as

••[-:

98

Linear algebra

and the vector x as (xl9x2)T, the last equation can be recast in matrix form as — = Ax dt which can be solved as a system of equations of order one. Finding the eigenvalues of matrix A amounts to solving the characteristic equation of our differential equation

X = 1 and X = 3 are the solutions of this equation. It can be verified that the eigenvector matrix U and its inverse are given by

^^2

1 / ^

^

fi 3/yioJ Proceeding as before we obtain the solution 1 1

r i i

7 i

10 /2

/2JLO

3 10J

u im

which can be rearranged as rx.1

UJ

3/2-3/2JLe3'

It can be checked that -e r — 2 2

is the solution which satisfies both the differential equation and the initial conditions, o 2.5.3 Stability of solutions to linear systems of differential equations All the previous examples happened to provide matrices with real and negative eigenvalues but this is by no means required to be a general situation. The reader is referred to advanced textbooks on eigenvalue theory (e.g., Wilkinson, 1965) for the demonstration that the complex eigenvalues of real general (non-symmetric) matrices form pairs of conjugate complex numbers. Let us consider one of these eigenvalues

2.6 Linear function spaces

99

k which we write in its complex form

where a and b are the real and imaginary parts of A, respectively. The corresponding time-dependent exponential that appears as a component of the solution to the system of differential equation is

which is the product of a real exponential term by a periodic complex term of modulus unity and frequency 2n/b. zh grows exponentially for a > 0 and the solution is unstable for large values of time. When a is negative (which will be shown in Chapter 7 to be the case for linearly coupled geochemical reservoirs), eAr decays towards zero after some oscillations if b is not zero. For a = 0, the system oscillates endlessly. This state of sustained oscillation is known as a limit cycle and separates stable (a < 0) from unstable (a>0) conditions. Many textbooks in applied mathematics offer an excellent discussion of the stability theory for differential equations (e.g., Strang, 1986; Logan, 1987; Zwillinger, 1989).

2.6 Linear function spaces 2.6.1 General

The idea of a vector space is usefully extended to an infinite number of dimensions for continuous functions. Given a function /(e.g., / = sin x) and a definition domain (e.g., 0 to 2TT), the coordinates of/ = sin x will be the infinite number of values of the function over the definition domain. This definition is consistent with that of Euclidian spaces if a metric is defined. In about the same way as the squared norm of the n-vector x(xi,x2, ..., xn) is

the squared-modulus of the infinitely dimensional vector / = sin x over the [0, 2TC] interval divided in segments of length Ax = 2n/n is 2 71/Ax

| / | 2 = lim Ax->0

£ sin2(/Ax)Ax

(2.6.1)

i=l

i.e., \f\2=\

sin 2 xdx Jo

(2.6.2)

100

Linear algebra

In general, the squared-modulus of the vector function f(x) over the domain 3) is

2 I/I2 = =

==[

/2Md*

(2-6-3)

(note the alternative formulation), whereas, the dot, or inner product, of the vector functions f(x) and g(x) is

fT

»=«»>=l

f(x)g(x)dx

(2.6.4)

Two functions f(x) and #(x) are orthogonal over Q> if for #0

and

<^O

(2.6.5)

then = 0

(2.6.6)

2.6.2 Fourier series A widely used example of orthogonal functions is the set of sines and cosines. For example, given any real number a, and the function sin nx for integer values of n, @ is equal to [a,a + 27i]. We can check that srr, f =

• 2 , f sin2nxdx=

JJoe

lcos2nxj [x dx= -

Ja J

2

L 4« J a

L2

(2.6.7)

where the subscript value for the quantity in brackets is subtracted from the superscript value. Also C2n . . f27r cos(n-m)x-cos(n + m)x j J \h9/— sin nx sin mx dx = dx Jo Jo 2 /r

x

and

r T r T o

2L

n-m

Jo

2L

n-fm

Jo

Similar orthogonality relationships can be shown for cosines. In addition, sines and cosines are mutually orthogonal, i.e., for any m and n <sin 2nnx, cos 2nmx} = 0

(2.6.9)

2.6 Linear function spaces

101

Just as a vector is projected as components on orthogonal axes, a given function defined on a given domain can be projected onto an orthogonal set of functions. The Fourier series decomposition of a function/(x) defined over the interval [ — X,X] is a convenient example /(*)= f;Ksin27rn^ + &Bcos27rn^)

(2.6.10)

A/

A

As in the case of regular vectors, the components an and bn can be found by forming the inner products upon multiplication of the /(x) expansion by the adequate sine or cosine /

x\ / x x /, sin 2nn —) = an\ s*n ^nn —> s^n ^nn ~ X/ \ X X

We note that, with the appropriate variable change, equation (2.6.11) reads

)-x

sin^ nn — ax = — X n}.n

sin nn — dl n— ) = — n = X X V X) n

with an identical result when the integrand is a cosine. We therefore obtain an and bn from ( /, sin 2nn — ) \

( /, cos Inn — '

Y

an = ±

YI

\

?L9b =± X

n

^L

(2.6.12)

X

From these last expressions, it is apparent that a0 is null whereas b0 represents the mean value of the function over the interval. We will now work out two examples, the results of which will be used in Chapter 8. & Find the Fourier series expansion of the boxcar function (Figure 2.9) defined over the interval [ - X , Z ] by f(x)=l

0<x<X

/(x)=-l

-X<x<0

f(x) = 0

x=0

If this pattern is repeated over all intervals [(2n-l)X, (2n+l)X] for integer values of n from — oo to + oo, the result is a periodic function of period 2X. an is calculated as If0 an = —\ Xj-x

x 1 C+x x (~l)sinnn — dx-\— (l)sinnn — dx X XJ0 X

Linear algebra

102

1.0

0.0

i

-1.0

-X

-2X

2X

x Figure 2.9 The periodic boxcar function with period 2X. or x~ -x cos nn — X nn/X an =

X +x cos nn — X nn/X 0

+

0

X

= — ( 1 — cos nn) nn

Only the odd terms are non-zero and equal to 4(nn) i. The bn coefficients such that b

If0 x 1 C+x x n = —\ (-l)cosmr — dx + — (l)cosmr — dx X J -x X X Jo X

can be rewritten x

x

{

L ! f" A bn = — cos nn — dx H—

x)0

x

r cos nn — dx X

x)0

Changing x to — x in the first integral and recognizing that cos is an even function shows that bn is zero for all n. The Fourier series expansion of the boxcar function is therefore /(*)=

x 4 an sin nn — = A nr

1

x sin(2w+l)7c —

103

2.6 Linear function spaces

1.0

xlX

0.8 0.6 0.4

\ Wlr

-1 //

\\ \1

0.2 0 0

X

Figure 2.10 Partial sums of Fourier components of the boxcar function up to p = 9. Convergence to the boxcar functions is achieved rapidly although 'ears' appear next to discontinuities (Gibbs effect).

Addition of thefirstcomponents up to p = 9 is shown in Figure 2.10. Reconstruction of the boxcar function is rapid although 'ears' persist next to the edge, a feature characteristic of discontinuities and known as the 'Gibbs effect'. <= # ^ Find the Fourier series expansion of the ramp function (Figure 2.11) defined over the interval [ — X,X] by f(x) = x f(x) = 0

-X<x<X x=±X

Again, reproducing this pattern over intervals [(2^-1)^, (2n+ \)X~\ for integer values of n from — oo to + oo results in a periodic function with period 2X. an is calculated as flfI = — x sin nn — dx I-x X

Integrating by parts gives

x sin nn — dx= — X

x cos nn — X nn/X -x

-r-

x\ cos nn — \ X\ dx=nn/X

J-x\

and therefore 2X2(-l)n/nn X

nn

-+0 nn

104

Linear algebra

1.0

0.0 / -1.0

-2X

2X

Figure 2.11 The periodic ramp function with period 2X.

It will be left as an exercise for the reader to show that the cosine terms bn, including the mean value b0, are zero. The Fourier series expansion of the ramp function is therefore ;(x)=

2X » (-1)" . x > sinmr — n

n=i

n

X

2.6.3 Legendre polynomials The powers of the variable x (1, x, x 2 ,..., x n ,...) are not orthogonal functions over a unique interval. However, particular sets of polynomials present the orthogonality property. A simple and useful example is that of Legendre polynomials. Let us choose over the range [—1, +1] the first two polynomials P0 = l and Px=x Po and Px are orthogonal since c= 2

2.6 Linear function spaces

105

and

while

In the space defined by the non-orthogonal 'vectors' 1, x, x 2 ,... let us find the Legendre polynomials of higher degree by Gram-Schmidt orthogonalization. A polynomial P 3 is found by removing the components of x 2 along P o and P x as in equation (2.2.32)



or x 3 dx 2

P, = x - ^ ^

1-i^i

x 2

x dx Finally 2_

0

x

2

3

Alternatively, a recursion formula can be used (e.g., Scheid, 1968) (n+l)P n + 1 =(2n+l)P n -nP n _ 1 which, from the same seed P o = 1 and Px=x

produces the sequence

P 2 = i(3x 2 -1)

= !(35x 4 -3Ox 2 + 3) 8 = i(63x 5 -7Ox 3

(2.6.13)

106

Linear algebra

1.0

Legendre Polynomials

n-\-l

0.5

-0.5

-1.0 -1.0

0.5

-0.5

1.0

Figure 2.12 The first Legendre polynomials.

Because of a different normalization , the coefficients of the parentheses are not identical for the Gram-Schmidt orthogonalization and for the recursion formula. The first Legendre polynomials are depicted in Figure 2.12. It will be checked that a polynomial of order n has exactly n zeroes in the range [— 1, + 1 ] .

2.6.4 Associated Legendre polynomials Legendre polynomials are one specific variety of a more extended class of orthogonal polynomials called associated Legendre polynomials. An associated Legendre polynomial Pim(x) is defined relative to an ordinary Legendre polynomial Pt(x) through

dxn

(2.6.14)

Alternative definitions lack the (— l) m term. Pt(x) is therefore a concise expression for Pl°(x). Note that, because of the derivative term in equation (2.6.14), if m > /, P*m(x) = 0. Advanced calculus would show orthogonality properties with =0 for /I ^/2

2.6 Linear function spaces

107

Table 2.4. The first associated Legendre polynomials up to l = m = 2. /

m

0 1

0 0 1 0 1 2

2

Pim(x) 1 X

-0-X2) 1 ' 2

l/2(3x2-l) -3(l-x 2 ) 1/2 x 3(1 -x2)

and

2/+1 (l-m)\ The numerical generation of associated Legendre polynomials is discussed by Press et al. (1986). These authors use the following recurrence on / x)

(2.6.15)

and the two starting values Pwm(x) = ( - l)m(2m-1)!!(1 ~x2)m/2

(2.6.16)

where the double factorial n\\ denotes the product of all the odd integers ^ n , and P£+1(x) = x(2rn + l)Pmm(x)

(2.6.17)

Examples of ordinary Legendre polynomials with m = 0 are P 0 °(x)=l P 1 °(X) = X X 1 X P 0 ° ( X ) = ^

(2-0)P2°(x) = xx 3 x P! 0 (x)- 1 x P0°(x) = 3 x 2 - 1 The first polynomials P,m(x) are given in Table 2.4. The number of associated Legendre polynomials up to the order / is (/+1)(/ + 2)/2. 2.6.5 Spherical harmonics

Some global problems deal with the variations of some geochemical parameters on the surface of the Earth. Problems of that sort have recently appeared when, for example, the world-wide distribution of the 206 Pb/ 204 Pb and other isotopic ratios

108

Linear algebra

in oceanic basalts permitted Dupre and Allegre (1983) to identify large-scale anomalous regions in the southern hemisphere for which Hart (1984) coined the name of 'Dupal anomaly'. The need to account for these variations with series of functions orthogonal on a sphere meets that encountered by geophysicists when trying to extract the most significant part of the variations in the gravity equipotential (geoid). These problems can be dealt with adequately using spherical harmonics. Spherical harmonics closely resemble normal Fourier harmonics except that they are functions of both the latitude and the longitude instead of the linear abscissa on a standard axis. Bi-dimensional Fourier analysis on a plane exists but is inadequate since the most desirable property of the requested expansion is the orthogonality of its components upon integration over the surface of the Earth, assumed to be spherical for most practical purposes. The standard coordinates on the surface of a sphere of radius r, i.e., the spherical coordinates, are the longitude and the co-latitude 9 (co-latitude is n/2 minus the latitude). Orthogonality over the surface of a sphere of two distinct functions /((/>, 6) and #(, 6) takes the usual form but in two dimensions

J the Earth surface

f(,e)ds(,e)=o

(2.6.18)

/ 2 (4>, 6) dS{(j), 6) = const

(2.6.19)

and

r

I

the Earth surface

where the functional dependence of the surface element dS with <\> and 6 is to be determined. From Figure 2.13, we see that the arc length elements d/^ = r sin 6 d(/> and dle = rd6 satisfy the length conditions since (1) integrating rsin0d at constant 0 for (j) varying from 0 to 2n gives the circumference 2nr sin 6 of the small circle, and (2) integrating rdO at constant (f> for 6 varying from — n to n gives the circumference 2nr of the large circle. Likewise t h e surface element dS = dle d/^ satisfies t h e surface c o n d i t i o n

m

'2n

"I

C~TC2n

r sin Odd) \rd(b = r2

o

C

2n

J

f

1

1 d
Ji Uo

J

= r2 \ d(f)\ d(cos 9) = r2 x 2TT X 2 = 4rcr2 (2.6.20) Jo J -l which is the surface S of the sphere. Further calculations will be carried out with reference to a sphere of unit radius, which can always be arrived at by proper

2.6 Linear function spaces

Figure 2.13 The system spherical coordinates: is the longitude and n/2-6

109

is the latitude.

normalization. We get S=

dS=

d(/> d ( - c o s 0 ) =

whole sphere

which shows that the surface element is dS = — dcf) d(cos 6). As in the case of Fourier components, we will refrain from using complex variables. Instead, we will handle spherical harmonics as two separate sets of orthogonal functions C™ (0,0) and S,W(<M) such that

Cr(,0)=

(2.6.21)

4TT

(2.6.22)

™^ 9) =

4n

Example: —^ -si 47r(2+l)! i.e.,

/

i 16n

Linear algebra

110

Longitude

Latitude Figure 2.14 The spherical harmonic S32((p,6).

Figure 2.14 shows the spherical harmonic S32 (,9). The normalization property of associated Legendre polynomials stated above, guarantees that these functions are orthogonal over the surface of a sphere

[2

(2.6.23)

Jo A bounded function /((/>, 6) can be expanded as an infinite series of C™ (0,6) and S (0,6) over the surface of the sphere

1=0

m

=o

(2.6.24)

where the oclm and f$lm coefficients are to be found from an integral expression similar to that used for Fourier coefficients. For a given / there is a total of (/ + 1)(' + 2) — (/ + 1) = (/+ I) 2 oilm and Plm coefficients (since all fil0 are zero), e.g., 49 coefficients are needed to expand a function in spherical harmonics to the order 6. Likewise, a 00 is the average value of the function over the sphere. A typical application of spherical harmonics will be given in Chapter 5.

3 Useful numerical analysis

3.1 Functions of a single variable 3.1.1 Derivatives The derivative of the function /(x) with respect to the dependent variable x is the scalar number defined as d/W /(x + Ax)-/(x) kim ffx (x) = (x) = kim x dx Ax-o Ax fx'(x0) represents the slope of the tangent to the curve y = f(x) at x = x0. An extremum, i.e., either a minimum or a maximum, corresponds to a null derivative. The quantity

is the derivative of order n of/(x) with respect to the variable x and is obtained by applying n-times the derivation formula. • The log derivative of a function /(x) is the derivative of ln/(x), i.e., dln/(x) dx

=

fx'(x) f{x)

1

The logarithmic differential of the function/(x) is defined as d/(x)//(x). If/(x) can be written as a ratio of two functions g(x)/h(x\ then ^

^

fix)

g(x)

^

(3.1.2)

h(x)

Example:

1+x

x2

)

1+x

V^

1+x/ dx

<& Optimum spike addition (Webster, 1960). We now deal with a specific example

that can be extended to any isotopic pair. Let us assume that 150 neodymium spike is added to 100 mg of a sample with ca. lOppm Nd in order to measure sample 111

112

Useful numerical analysis

neodymium concentrations. Isotopic proportions of 148 and 150 isotopes are 5.73 and 5.62 percent in natural Nd (molar weight 144.24), and 0.40 and 97.25 percent, respectively, in the Oak Ridge 1 5 0 Nd spike. Calculate the optimum amount of a spike containing 12.5 nmol 1 5 0 Nd per gram of solution to be added to this sample which minimizes the error on the calculated sample concentration. Assume that sample and spike Nd isotope compositions are perfectly known. Let us consider the two isotopes 1 4 8 Nd and 1 5 0 Nd and measure the isotope composition of the spiked mixture. Using the subscripts sa, sp, and m for sample, spike, and mixture, it was shown in Section 1.3 how to calculate the spiking ratio r 8

N,sp

Nd

8

150 Nd

Nd\ 50 Nd/ m

/ 1 4 8 Nd\ / 148 Nd V150Nd

where Afsa'5°Nd and iVspl5°Nd refer to the numbers of 1 5 0 Nd atoms of sample and spike, respectively, present in the mixture. Using x to represent the 1 4 8 Nd/ 1 5 0 Nd ratio, we get the more compact form

r

_

and try to find the value of x m which makes the relative error on r a minimum. Assuming that the isotopic compositions xsa and x sp are perfectly known for the spike and sample, the relative error dr/r on the spiking ratio r is dr = dlnr

_

~ dx ™

dx

™

_

*m(s,p-xj

d x m _ dxm

The relative error on the measurement dx m /x m is amplified by the factor y to give the relative error on the spiking ratio dr/r. The amplification term y goes to infinity for the extreme cases x m = x sa (no spike) and x m = x sp (no sample). Given that (xsp — xsa) is constant, y is minimum for

dx m (x s p -xj(x m -x s a )

-=0

or ~ X m ( - X m + X sa + X sp - X m )

=0

which is equivalent to (3.1.3)

3.1 Functions of a single variable

0

0.05 148

0.1

113

0.15

0.2

Nd/150Nd ratio in the sample-spike mixture, xm

Figure 3.1 Optimization of spike addition for the isotope dilution technique: y is the error amplification factor.

and hence 1-

-^ (3.1.4)

^sp

Relative error on isotope dilution measurements is minimum whenever the mixture isotopic ratio is the square root of spike and sample isotopic ratios. In the present case, xsa = 5.73/5.62 =1.0196, and x sp = 0.40/97.25 = 0.004 113. We can plot the amplification factor modulus |y| as a function of the isotopic ratio in the mixture (Figure 3.1). Minimum amplification is obtained for

hence the assay contains approximately O.lOOgxlOxKT 6 = 6.93nmolNd 144.24 gmol" 1

114

Useful numerical analysis

or 0.0562 x 6.93 nmol = 0.3895 nmol of natural

r

150

Nd. Addition of

0.06351

from the spike, amounting to 6.133nmol150Nd 12.5 nmol 150Nd per gram of solution of spike solution, minimizes error propagation on concentration. <>

3.1.2 Equation of the tangent to a curve Tangents to curves and surfaces play a key role for a certain number of petrologic or geochemical systems which undergo infinitesimal changes. Finding the compositional changes associated with the segregation of a mineral phase and deciding whether a system is stable relative to small perturbations are problems that commonly need the tangent equations to be found. Given the curve with equation y =/(x), the limit of the chord intersecting the curve in X and x is the tangent to the curve at x = xO. The slope s of this tangent is given by r s= lim

f(x)-f(x0) x — x0

df(x) dx

while the tangent equation must satisfy

and therefore (x-x0)

(3.1.5)

<& Cumulate control lines vs liquid lines of descent. In a study of the 1931-1986 basaltic eruptions from the Reunion Island (Indian Ocean), Albarede and Tamagnan (1988) found the Ni and Cr concentrations (in ppm) listed in Table 3.1 and shown in Figure 3.2. Samples with Ni>100ppm were found to contain large amounts of cumulus olivine. The last five samples of Table 3.1 are picrites. The smooth trend observed for all the rocks is suggestive of a complementary fractionationaccumulation relationship out of a single magma batch. Therefore, it is asked whether the trend observed for Ni and Cr in basalts with less than 100 ppm Ni may be ascribed to the removal of the olivine found in the cumulates. When minerals fractionate from a crystallizing magma, mass balance requires that, in a linear plot of Cr vs Ni, the points representing the composition of the parental

3.1 Functions of a single variable

115

Table 3.1. Ni and Cr concentrations in ppm in Piton de la Fournaise lavas, Reunion Island, 1931-1986 (Albarede and Tamagnan, 1988).

Ni

Cr

Ni

Cr

Ni

Cr

Ni

Cr

55 56 61 62 65 67 70 71 72 73

58 76 66 82 150 128 106 123 124 160

74 74 74 75 75 77 77 78 79 79

191 181 145 224 177 157 203 184 181 172

79 80 81 83 88 88 96 100 112 118

230 166 189 223 236 248 272 363 380 442

144 146 157 203 462 802 870 890 975

494 448 523 555 903 1547 1700 1635 1740

1000

100

Picrites =

;

Basalts

1000

100

Ni Figure 3.2 Plot in log-log scales of the Ni and Cr concentrations (ppm) in post-1930 basalts (open symbols) and picrites (solid symbols) from the Piton de la Fournaise volcano, Reunion Island, Indian Ocean (Albarede and Tamagnan, 1988).

magma, the residual magma and the average cumulate define a straight line (see Chapter 1 and Figure 3.3). In order to define the composition of the solid in equilibrium with the magma at a given point of its fractionation history, let the residual magma composition approach that of the parental magma. We conclude that the instantaneous cumulate must lie on the tangent to the locus of liquids (the so-called liquid line of descent) at the point representing the magma (Figure 3.3). The tangent line is also the locus of all combinations between the magma and the instantaneous cumulates, i.e., the locus of the cumulative rocks belonging to a magmatic stage with a unique differentiation extent: this is the cumulate control line (by reference to the widely used olivine control line). The model developed below is from Albarede (1976).

Useful numerical analysis

116

Parent magma

(Linear scales)

u

Instantaneous cumulate M

Bulk cumulate

Residual liquid

Ni Figure 3.3 Schematic residual magma-cumulate relationships for instantaneous and average magmatic products in a linear plot of Ni and Cr concentrations. For mass balance to be obeyed, the instantaneous cumulate must lie on the tangent to the liquid line of descent at the point representing the instantaneous liquid.

In the present case, the question is whether Reunion olivine-rich rocks belong to a cumulate control line of the basaltic rocks. Through a simple log-log regression, we find that the basalt trend ( = the liquid line of descent) can be approximated by a power law, a form justified in Sections 1.5 and 9.3. liq

=«(Ciitl

)

(3.1.6)

The equation of the tangent to this curve in the (Ni, Cr) plane at CNi = C liq Ni and c ''liq ~ IS (3.1.7)

or -

C

li

(3.1.8)

3.1 Functions of a single variable

117

From the last two equations, we get CCr = C l i q C r + ab(CUqNi)b-

l

(CNi - C liq Ni )

(3.1.9)

which can be rewritten C Cr -C Hq Cr = a 6 ^ ! ! r ( C N i - C l i q N i )

(3.1.10)

Qiq

Finally, comparing with equation (3.1.6), we get s~
(~* Cr

_

Z^S-=b-^-

^Ni

r

Ni

f* r

Cr

(3.1.11)

Ni

W1-11/

The slope of the linear array defined by the olivine-rich lavas (olivine control line) with more than lOOppm Ni is found by linear regression to be (CCr -C liq Cr )/(C Ni -C liq Ni )=1.6 whence we conclude that the liquid which could be involved in the olivine-rich lavas has a ratio Cr/Ni= 1.6/b= 1.6/2.8^0.6, which is well out of the range of the Cr/Ni ratios in basalts (1 to 3). Therefore, olivine-rich lavas are not cumulative rocks genetically related with the basaltic sequence. The large value of b is probably related to the presence of spinel on the liquidus. This does not exclude, however, that they may be cumulates from an earlier differentiation stage with a smaller b value , i.e., before spinel saturation.
b

sol •

At first sight, the answer would be AG = AG0 = Gliq(XUqb)-Gsol(Xsolb\ but we will show that this is wrong. We assume that dn = dn a + dn b moles are transferred from the liquid to the solid phase. Let us assign the symbol fi to chemical potentials, e.g., juliqa for species a in the liquid. Then dn AG = /<sola dnsola + /isolb dnsolb + ^liqa dnliqa + /zliqb dnliqb We have assumed that dn = dnsola + dnsolb and therefore

(3.1.12)

118

Useful numerical analysis

hence dn AG = (^sola - A*liqa)cKola + (//solb - /*liqb)dnsolb

(3.1.13)

The molar fraction of b in the newly formed solid is defined as

y b_ ^ sol

with a similar definition for the liquid fractions. The change AG in Gibbs energy upon transfer of dn moles is b q

-AO^soi b

(3.1-14)

which, for reasons which will appear later, is rewritten as AG = V b = XXsol V

This equation can be recast into AG = Xsol V sol b + (1 - * S O , W - [*nq Vuq b + (1 - * n q b K q a ] b

-X s o l b )( M l i q b -^ l i q a )

(3.1.15)

The terms on the right-hand side of the first line represent the difference between the molar free enthalpies of the solid and liquid solutions at composition Xsolb. In addition, a standard result for binary systems (e.g., Swalin, 1962) states for G and fi the following relationship

Applying equation (3.1.16) to the liquid results in AG = Gsol(JTsolb) - G liq (X liq b ) -

(3.1.17)

Let us label with an asterisk the value of the free enthalpy taken at Xsolb along the tangent to the curve of the liquid free enthalpy at X liq b (Figure 3.4) so that

liq l A sol ) — ^ l i c ^ l i q

dGUt

dX

(3.1.18)

3.1 Functions of a single variable

119

A

A

/

/

/

\

\

V

/

]~~

Gsol

\

•

AG i

AG 0 '

/ <

• — _ _ _ _ _

'liq

sol

Figure 3.4 Change of Gibbs free enthalpy at the onset of crystallization of a solid solution with composition Xsolb from a liquid solution with composition Xliqb. The change corresponds to AG, not to AG0.

which gives the simpler expression = G s o l (X s o l b )-G l i q *(X s o l b )

(3.1.19)

The change in Gibbs free enthalpy is therefore the difference measured at Xsolb between the G value of the solid and that taken along the tangent to the liquid curve (Figure 3.4). At equilibrium, Gliq and Gsol have a common tangent and therefore AG = 0. <=• Such a calculation can be extended to other molar quantities such as the volume change (AV) or the entropy change (AS). Unlike AG, the quantities AV and AS do not vanish at equilibrium. Although they derived the same tangent rule through a significantly more complicated theory, Walker et al. (1988) showed that in the context of olivine flotation in melts at high pressure, the slope of pressure-temperature (P, T) equilibrium univariant curves given by the Clapeyron rule (e.g., Denbigh, 1968) cannot be used to retrieve the relevant information on the actual change in density upon melting and crystallization in the mantle.

120

Useful numerical analysis 3.1.3 Leibniz's rule for the derivative of a definite integral

• Given /(x, t) a function of time and of another variable x, and given the time-dependent limits a(f) and /?(f), Leibniz's rule states that

11""/™*,. r ^ Jm

drj a a(r) (r)

^

-

J

dt

M dt

*

,3,.20, dt

Example: d Ckt

—

Ckt

sin(2f x) dx =

dtJo

2x(cos 2tx) dx + (sin 2kt2) xk- (sin 0) x 0

Jo

3.1.4 Taylor series • The Taylor expansion of the function f(x) about the value x 0 is

2!

X0

or in compact form

where the exclamation mark stands for the factorial expansion. The Taylor-McLaurin expansion of the function/(x) about x o = 0 is a particular case of equation (3.1.21) 4- ...

It is particularly useful for approximating some functions in the vicinity of x = 0: (a) the exponential function — e° + — e ° + . . . = l + x + — + — + . . . 2! 3! 2! 3!

(3.1.22)

(b) the natural logarithm function x2

1

1+0 *

-.x-xl 2 (c) the power series

x +

'2!(l+0)

2

.l+...

4

1

3!(l+0)3 x

l+ 3

x3

1

(3.1.23)

3.1 Functions of a single variable

121

with the particular case for a = — 1 x3+ ...

(3.1.24)

Isotopic fractionation provides illustrative examples of first-order expansions of unknown functions. In general, the mass spectrometric measurement r/ of the ratio between two isotopes of mass mt and m, of the same element, differs from the natural value RJ. Only a very small fraction of the original sample produces ions and different processes taking place in different parts of the mass spectrometer act differently on the sensitivity of each isotope. We assume that instrumental isotopic fractionation is mass-dependent.

Equilibrium fractionation. A simple fractionation law, called the linear law (e.g., Hofmann, 1971), relates the measured and natural isotopic ratios through a function /(Am/) of the mass difference Am/ = m7 — m, between the isotopes defining the ratios )

(3.1.25)

As Russel et al. (1978), we write the Taylor expansion of/(Am/) about Am/ = 0 and get /(Am/) = /(0) + Am//'(0) 4- higher-order terms and drop the terms which involve derivatives of order higher than one. As one isotope cannot be fractionated relative to itself (Am/ = 0),/(0) = 1 and the linear fractionation law reads r/

= K/(l+Am/(5)

(3.1.26)

where 8 = /'(0) is a constant coefficient called the mass discrimination or mass bias per mass unit. Let us take the example of the 148 Nd/ 150 Nd ratio

(1 2S)

~

natural

In a ratio-ratio plot (Figure 3.5), typically r/ 2 vs r/ 1 , the ratios combine as r/2-iV2_R/2Am/2_^ n r/ 1 -/*/ 1 K/'Am/ 1

/2Am/

2

Am/1

which shows that they are linearly related.

Mass discrimination with distillation effects. Let us assume that the isotope composition of an element is being measured by thermal ionization. This method consists in ionizing the sample atoms by evaporation on a metal filament. Statistical thermodynamics (e.g., Denbigh, 1968) tells us that, while vapor pressure is a function

Useful numerical analysis

122

O

i 42 o C/5

Domain of the linear law

Isotopic ratio 1 Figure 3.5 The domain of the linear law for mass-dependent discrimination between two isotopic ratios.

of the molecular weight of the isotope, the fraction evaporated and ionized is a complex function of ionization potential, temperature, work-function of the filament, etc. At a given time, we assume that the ionization parameters are fixed. Therefore, the ionization probability of an isotope i and, equivalently, the proportion of atoms on the filament that per unit time eventually ionizes is a function of mt only. Let g(mt) be that proportion. If we call nt the number of ionized atoms of isotopes i, we can write dw.

(3.1.27)

For two isotopes i and j , we combine expressions (3.1.27) as duj

drii

dlnn,— dlnrc,

d In rJ(i)

n{dt

dt

dt

Expanding the function g in a Taylor series to the first order with respect to m, we get the approximation

3.1 Functions of a single variable

123

or dlnr,'(f) — ss - (m,- - mi)gm (m,)

Upon integration, this equation becomes r/(0 = rIj(0)exp[- Am, V f a M =r/(0)[exp (-gJimdt)^

(3.1.28)

where the pre-exponential term represents the isotopic ratio of the first fraction evaporated from the filament. This ratio is at equilibrium with the ratio of the sample initially on the filament. From the earlier discussion, we can express r,J(0) as

Two limiting cases arise depending on the intensity of the distillation effect. If distillation is not important, we can expand the distillation exponential term to the first degree as rt\t)« /V(0)(l + Am/ <5)[1 - A w ^ W f l * M<>){ 1 + Am^-^'(m ( )t]} where the second-order terms are neglected. [8 — gJim^t] is known as the time-dependent mass fractionation per mass unit. This is the widely used timedependent linear law of mass fractionation suitable for large samples. For small samples, however, mass fractionation is important and the cumulative effects are dominant so 8 can be neglected resulting in

This relationship is known as the power law of mass fractionation (Wasserburg et al, 1981; Hart and Zindler, 1989). Writing oc(t) for -^ w / (mI )r for the mass fractionation per mass difference unit, the 148 Nd/ 150 Nd ratio (Am/= —2) would change with oc(t) according to a power law if

3.1.5 Roots of implicit equations and extrema of functions: the Newton method Some equations such as/(x) = 0 cannot be explicitly solved for x. If multiple solutions are not expected in a narrow range, Newton's method is often simple to implement and has faster convergence than the natural method of interval splitting. The method is recursive and uses the first-order expansion of/(x) in the vicinity of thefcthguess / [ x ( * + 1J] % /[x (fc) ] + [x(fc + 1 } - x (fc)]/'[x(fc) ]

124

Useful numerical analysis

Table 3.2. Calculation of yj2 as a solution to the equation x2 — 2 = 0 by the Newton method. Compare with the true value of 1.41421. Step/c 0 1 2 3 4

7.00000 1.36111 0.137 80 0.00222 0.00000

3.00000 1.833 33 1.46212 1.41500 1.41421

Expressing our goal that/[x (fc

+ 1)

6.00000 3.66667 2.92424 2.83000 2.82843

1.16667 0.37121 0.047 12 0.00078 0.00000

] = 0 results in

The Newton method can also be used to find the extremum (maximum or minimum) of a function/(x), i.e., the value for which the first derivative f\x) is zero. The iterative search for the extremum is implemented by a formula derived from equation (3.1.29)

i

i

i

The extremum X is a minimum if the second derivative/"(X) is positive, a maximum if f'\X) is negative. <& Find the square-root of 2. This amounts to solving

Pretending that we ignore the result, we calculate/'(x) = 2x and take 3 as the initial guess x(0). Hence,/[x ( O ) ] = 3 2 - 2 = 7,/'[x ( O ) ] = 2 x 3 = 6 and x(1) = 3-(7/6)= 1.8333 Table 3.2 lists the results for four iterations. After four iterations, the estimate reproduces the true value of yjl to the fifth decimal place, o Find the roots of the equation

1+x2

3.1 Functions of a single variable

125

which appears in connection with diffusion from a sphere into a finite volume (see Chapter 8). x (in radians) is the intersection of the periodic function tan x with the single value function x(l-hx 2 )" 1 . Since tanx goes to infinity for x = (2n+l)n/2 (n = 0, ± 1 , ± 2 , ...), there is one of these intersections per interval [_(2n — l)n/29 (2n +1)TT/2]. Consequently, we take 2nn/2 = nn as the initial guess for each interval. In addition, let us replace g(x) by f(x) such that /(x) = ( l + * 2 ) t a n x - x which is easier to handle and has the same roots since (1 +x 2 ) is strictly positive. The derivative f'(x) is +x 2 Xl+tan 2 x)-l Let us try for n = + 5, i.e., with 5TT = 15.708 as the initial guess x(0) . The first step gives /'[x ( 0 ) ] = (1 + 25TT2)0 - 5TT = -15.708 / ' [ x ( 0 ) ] = 2 x 5TT x 0 + (1 + 25TT2)(1 + 0 2 ) - 1 = 246.739 x{1) = 5n-(- 15.708/246.739)= 15.772 and the second step / [ x ( 1 ) ] = ( l + 15.7722)tan 15.772-15.772 = 0.1492 /'[x ( 1 ) ] = 2 x 15.772 x tan 15.772 + (1 +15.7722)(1 + tan 2 15.772)-1 =251.770 x ( 2 ) = 15.772-(0.1492/251.770)= 15.771

The improvement is not significant, so we stop here, o & A primary isochron 2 0 7 Pb/ 2 0 4 Pb vs 2 0 6 Pb/ 2 0 4 Pb on a series of rock samples gives a slope of 0.256. Calculate the age T of the isochron. Standard textbooks on chronology (e.g., Faure, 1986) give the slope of a 207p b/ 204p b

v s

206p b/ 204p b

i s

35L 1 f(T) = — — e'~ *238ur - 0.256 = 0 T 137.88 e '-l

Taking the derivative relative to T, we get 1

*

/-''35UT/ /-")38L^

1\

"

/'"'38U7'/ /" ) 35U^

1\

Using / 2 3 8 U = 0.155125Ga" 1 and / 2 3 5 U = 0.98485 0 a " 1 and an initial guess T 0 = 5Ga, Table 3.3 lists the results of the first five iterations, o

126

Useful numerical analysis

Table 3.3. Iterative calculation of the age of an isochron with a slope of 0.256 in the vs 206Pb/204Pb diagram by the Newton method. Step/c 0 1 2 3 4 5

jn

5.00000 4.01055 3.41007 3.234 33 3.222 31 3.222 26

207

Pbl204Pb

/[7^]//TT<*>] 0.58927 0.17202 0.032 60 0.001 97 0.00001 0.00000

0.595 55 0.28647 0.18549 0.163 59 0.16219 0.16219

0.989 45 0.60048 0.175 74 0.012 02 0.000 05 0.000 00

# I n a Concordia diagram 2 0 6 Pb/ 2 3 8 U vs 2 0 7 Pb/ 2 3 5 U, a series of zircons give a good alignment with a slope a = 0.043 633 and an intercept /? = 0.094613. Calculate the ages at which this line intersects the Concordia. Equation of the Concordia is (Wetherill, 1956; Faure, 1986)

whereas the straight-line equation reads y-ax-j5 = 0 These equations can be combined as /(T) = e;238uT-1 -a(e ; 2 3 5 u r - 1)-0 = which has for derivative

Using two different initial guesses (5 and OGa) in order to approach the two intersections from different directions, Table 3.4 gives the results of the first iterations towards each intersection. The zircon alignment intersects the Concordia at 1.00 and 2.00 Ga.<^ & A basaltic liquid with an FeO content C 0 FeO of 10 percent and an MgO content C0MgO of 12 percent crystallizes olivine (ol). Calculate the FeO and MgO contents CliqFeO and CliqMgO after 15 percent crystallization. Irvine (1977) has shown (see Section 1.5) that

CfaFeO and CfoMgO being the contents of FeO in fayalite and MgO in forsterite, the

3.1 Functions of a single variable

127

Table 3.4. Iterative calculation of the two age intercepts with the Concordia curve for a zircon linear array having a slope a = 0.256 and an intercept /? = 0.094 613. Step/c First intercept: 1 2 3 4 5 6 7 8 9

/'[T<*>]

/[r<*>]//'[r<*>]

5.0000 4.1243 3.3571 2.7447 2.3202 2.0875 2.0091 2.0001 2.0000

-4.8824 -1.6891 -0.5581 -0.1715 -0.0465 -0.0095 -0.0009 0.0000 0.0000

-5.5755 -2.2017 -0.9113 -0.4039 -0.1999 -0.1213 -0.0990 -0.0965 -0.0965

0.8757 0.7672 0.6124 0.4245 0.2326 0.0784 0.0090 0.0001 0.0000

Second intercept: 1 0.0000 2 0.8436 0.9883 3 4 0.9999 5 1.0000

-0.0946 -0.0113 -0.0008 0.0000 0.0000

0.1122 0.0782 0.0671 0.0661 0.0661

-0.8436 -0.1447 -0.0116 -0.0001 0.0000

solution is (Chapter 1 and Albarede, 1992) Q

FeO

Q MgO

(3.1.32) where KD is the ratio (FeO/MgO) ol /(FeO/MgO) liq and the parameter z is defined as C1/^

MgO//^1 MgO

Z — r Ujiq

/U 0

c\ i 'l'W

yD.l.DD)

Taking the derivative of/(z) with respect to z gives

f'(z)=-KD

r

F

FeO

(3.1.34)

Using a table of molar weights, we get C fa FeO = 2 x 71.85/203.78 = 70.52 percent CfoMgO = 2 x 40.31/140.71 =57.30 percent A natural starting value for z if F. The calculation listed in Table 3.5 was stopped once the relative deviation in z was less than 0.001. The final z value of 0.4306 is converted through equation (3.1.33) into CliqM*0 = 0.4306 x 12/0.85 = 6.08 percent

128

Useful numerical analysis Table 3.5. Iterative calculation of the z value, equation (3.133), for thef(z)=0 equation (3.1.32). Step/c

f(z)

f'(z)

z

0 1 2 3

-0.1121 0.0054 0.0000

-0.2556 -0.2867 -0.2842

0.8500 0.4115 0.4305 0.4306

and through equation (3.1.31) into C liq FeO = 6.08 x (10/12) x (0.4306)° 2 9 ~ * =9.21 percent.

<& Upon heating for a time t = 1 hr at 800°C, a plagioclase crystal with a radius a = 1 mm has lost 40 percent of the radiogenic argon it initially contained. Calculate the argon diffusion coefficient 2 at this temperature. Assume that the plagioclase crystal may be considered as a sphere with isotropic diffusion properties and that, initially, radiogenic argon was homogeneously distributed. Let us define the dimensionless number

From Section 8.6, the fraction F(T) of argon left in the sphere at x can be written F(T) = -^-t\exp(-n2n2T) n2n=ln2

(3.1.35)

Hence, the implicit equation to be solved for T is /(T) = F(T)-0.6 = 0

The derivative of /(T) is /'(T)=

- 6 £ exp(-n27r2T)

(3.1.36)

The first seven iterations produced from an arbitrary trial value T = 0.0001 are listed in Table 3.6. The final result is exact to better than four decimal places. This result allows the diffusion coefficient to be calculated as

3.1 Functions of a single variable

129

Table 3.6. Iterative calculation of the % value corresponding to a lost fraction of radiogenic argon of 40 percent. Spherical geometry is assumed, equation (3.1.35).

Stepfc

F(T)

/'M

/W//'(T)

T

1 2 3 4 5 6 7

0.9664 0.9664 0.8444 0.6932 0.6145 0.6004 0.6000

-166.26 -166.26 -32.261 -14.028 -10.168 -9.6357 -9.6216

-0.0022 -0.0076 -0.0066 -0.0014 -0.0000 -0.0000

0.0001 0.0023 0.0099 0.0165 0.0179 0.0180 0.0180

Find numerically the mininimum over [0, 1] of the binary entropy function /(x) = x l n x + ( l -

The obvious result x = 0.5 can be arrived at in a number of ways. The first and second derivatives are /'(x) = ln[x/(l-x)]

and r ( x ) = x"

With the trial value x(0) = 0.1, we obtain /'[x ( 0 ) ]=-2.1972,/"[x ( 0 ) ] = 11.111, and x(1) = 0.2978. The result 0.5000 with four correct decimal places is obtained with x(3). o

3.1.6 Ordinary differential equations: the Euler method Quite commonly, differential equations appear in the form

(3.1.37)

and cannot be solved explicitly. We have to resort to one of the many numerical methods of which the simplest versions are given here. The Euler method has little practical value, but forms the basis for most of the more elaborate methods. It consists in a first-order expansion of the derivative. The approximation at step tin+1) is (3.1.38)

130

Useful numerical analysis

Table 3.7. Solution of the differential equation dy/dt = -2ty by the Euler method with a time step of 0.1 (left) and 0.01 (right).

t

y

-2ty

True value

t

v

-2ty

True value

0.00 0.10 0.20

l 1.0000 0.9800

0 -0.2000 -0.3920

1 0.9900 0.9608

0 0.01 0.02

1 1.0000 0.9998

0 -0.0200 -0.0400

1 0.9999 0.9996

0.90 1.00

0.4655 0.3817

-0.8379 -0.7634

0.4449 0.3679

0.19 0.20

0.9663 0.9627

-0.3672 -0.3851

0.9645 0.9608

& Solve the equation that results from introducing the Boltzmann variable into the diffusion equation (Chapter 8) y'(t)=-2ty given the initial value y(0)= 1. This equation has an exact solution y = exp(-t2). For a rather coarse time step of 0.1, we obtain .1-0.0)(-2x0xl)=l .2) = )/(0.1) + (0.2-0.1X-2x0.1 x l) = 0.98 Table 3.7 compares the accuracy for two different time step sizes. Obviously, final accuracy depends on the time step chosen and considerable computational effort would be required for a good approximation, <=>

3.1.7 Ordinary differential equations: the Runge-Kutta method Although Press et al (1986) compare the use of the Runge-Kutta method to 'ploughing the fields' and that of high-order predictor-corrector schemes to 'racing on the fast lane with a sports car', we are still dealing with a reliable method easy to implement and quite successful. The Runge-Kutta method is sketched in Figure 3.6 and uses successive approximations of the function on the [t{n) — r(w+1}] interval. Let us define

3.1 Functions of a single variable

131

t Figure 3.6 Numerical solution of ordinary differential equations: sketch of the four steps of the Runge-Kutta method to the order four giving the n + 1 th estimate y ( n + 1 ) from the nth estimate y{n).

and the intermediate evaluations

Then

f-fc4)

(3.1-39)

o This scheme has tight connections with the Simpson's rule for numerical integration.

&

Let us solve the same equation as above y'(t)=-2ty

with initial value y(0)= 1 and a constant time step tin+1) — t(n) = OA.

Useful numerical analysis

132

Table 3.8. Solution of the differential equation dy/dt = —2ty by the Runge-Kutta method to the order four with a time step of 0.1.

Step n

y> *i

k2

K exp(-r2)

0

2

1

0 1 0.0000 -0.0100 -0.0100 -0.0198 0.9900 0.9900

0.1 0.9900 -0.0198 -0.0294 -0.0293 -0.0384 0.9608 0.9608

0.2 0.9608 -0.0384 -0.0471 -0.0469 -0.0548 0.9139 0.9139

4

3 0.3 0.9139 -0.0548 -0.0621 -0.0618 -0.0682 0.8521 0.8521

0.4 0.8521 -0.0682 -0.0736 -0.0734 -0.0779 0.7788 0.7788

5 0.5 0.7788 -0.0779 -0.0814 -0.0812 -0.0837 0.6977 0.6977

Let us show how the method is implemented on the first time step y(0)=l

r r /c3 = 0.1x

/

o.i\ /

o\i

/

oi\ /

-0.01M

-2x(0+ —jx( 1+

j

=-0.00995

/c4 = O.lx [ - 2 x(0 + 0.1)x (1-0.00995)] = - 0 . 0 1 9 8 0 1

hence (0-2x0.01-2x0.00995-0.019 801) -=0.99005

y(0.1)=l-h-

A few more steps with results trimmed to the fourth digit are computed in Table 3.8. The Runge-Kutta provides a robust and reasonably precise answer to most differential equations, o 3.1.8 Interpolation with spline functions Interpolating discrete data is an old concern of physics but, in the case of numerous data points, the conventional collocation and osculating polynomials are of too high degree to be really useful (Scheid, 1968). Interpolation of n + 1 data points yOiyl9..., yn tabulated at x0 xx ' xn for intermediate values of the dependent variables x can be done by a number of methods but one of the simplest and most elegant is the constructions of cubic splines. On each interval, the 'data' function will be approximated by a cubic polynomial such that, at their common point, the polynomials of neighboring intervals have identical values, slopes and curvatures (Ahlberg et ai, 1967). Let us consider the interval (xt_ x - x£). A third degree polynomial

3.1 Functions of a single variable

133

has a linearly changing second derivative y", hence y"(x) = yyi-l" + (\-y)yi'\

for ^ ^ x ^ x , -

(3.1.40)

where y is a factor depending on x. Solving at the extremities x t _! and x, of the interval, we get X( X

/(x)=

~

ft-r + fl

^—)yi",

for x.-.^x^x,-

(3.1.41)

while over the interval xt and xi+1 the expression becomes +

X Xi

~

yi+l",

for x ^ x ^ x I + 1

(3.1.42)

These expressions can be integrated relative to (x, — x) 1 l(x,-x) 2 y(x)=--

yi.l 2x I -x,_ 1

I" 1 (x,-x)2l n - x,-x-y, +cx for x^ L

x-x,--

2x,-x I _J l

I_U/' + _

2 X, +

t

—y.

+ l»

+ C2 for x,

2 Xf + j — X,-

— Xf J

where c t and c2 are two constants of integration. A second integration gives for Xt-

6x l-xI _1 2 — ly/' + —yi+i" + c2(x-Xi) + c4 for x,y(x) = -\ 3(x-x,) 6L x« + i-XiJ 6x l + 1 - x f

Writing that the first cubic goes through the points (Xi-^y^^ yi-i = -(xi-xi-l)2yi-1" o

+ -(xi-xi-1)2yi"-cl(xi-xi-.l) 3 y i

hence

r)

+ c3

3

+yi~yi-1

U

and (x,-,^,-), we get

(3.1.43)

where yi'i~) is the left-derivative at x = xt. Likewise, writing that the second cubic goes through the points (xf,yf) and (xi+1,yi + 1), we get

3

o

134

Useful numerical analysis

hence (3.1.44)

where y/ ( + ) is the right-derivative at x = xf. The first left- and right-derivatives must be equal for the common point x = xh hence

X: —Xf_!

which is recombined into

For n data points, (n— 1) equations, such as the one above, can be written with (n+ 1) unknowns y/' (i = 0,..., n). Two additional equations are needed, which most often are end conditions at i = 0 and i — n. The two conditions specifying the slopes

(3.1.45)

(3.1.46) xM-xM_1

make it possible to complement the system of equations and solve it for the (n +1) unknowns y" (i = 0,..., n). In a matrix form, this is written as AM=D

(3.1.47)

where the current element atj of the (n +1) x (n + 1) tri-diagonal matrix A are given by

0
(3.1.48)

otherwise

M i s the (n+1)-vector of the unknowns y{\ while D is the (n+1)-vector with current

3.1 Functions of a single variable

135

Table 3.9. Chondrite-normalized Ce/Yb ratio of some recent lavas of the Piton de la Fournaise, Reunion Island (Albarede and Tamagnan, 1988).

i Year (Ce/Yb)N

0

1

2

3

4

5

6

7

1948 20.9

1953 21.2

1956 22.0

1966 20.8

1972 21.7

1975 22.4

1981 21.3

1985 18.9

element

(3.1.49)

n

= 6\ y n ' {

>

Once the linear system is solved for the unknowns y", the value interpolated at any arbitrary x can be calculated from either formula for x ^ ^ x ^ x , (3.1.50)

— 6

l

-(x — xt)3

for x I ^ x ^ x l + 1

(3.1.51)

xi+l-Xi

where the left- and right-derivatives y{( } and y{{ + ) are given by equations (3.1.43) and (3.1.44), respectively. It is often preferred to choose the values of the derivatives instead of the curvatures as unknowns. In particular, the right-derivatives y{{ + ) at the data points (breakpoints) appear in the standard pp-form (piecewise polynomial) used in many software packages (de Boor, 1978). This transformation is a simple task using the relationship between derivatives and curvatures, preferably in a matrix form. & The Piton de la Fournaise volcano (Indian Ocean) erupts basalts with chemical compositions that change with time. The rare-earth elements have been measured on eight dated historic lavas (Table 3.9 and Figure 3.7, Albarede and Tamagnan, 1988), and chondrite-normalized (Ce/Yb)N ratios over the time interval 1948-1985 are given in Table 3.9. Calculate an annual interpolation of these results. Let us build first the matrix A of coefficients atj and the right-hand side vector D with the - admittedly questionable - assumption that the derivatives at the end-points are zero. Matrix A is calculated from equations (3.1.48), e.g., aoo = 2 x (1953-1948)= 10,

136

Useful numerical analysis

1940

1950

1960

1970

1980

1990

Year Figure 3.7 Spline interpolation of the chondrite normalized Ce/Yb ratio in recent lava flows of the Piton de la Fournaise volcano (Albarede and Tamagnan, 1988). The end derivatives are supposed to be zero.

a01 = 1953-1948, and so on. Vector D is calculated from equations (3.1.49), e.g., dn =

u

l

/ 21.2-20.9 \ 1 9 5 3 - 1948

,1953 - 1 9 4 8 /

-0

=0.36

r 1956-1953; V

so we arrive at

A=

10

5

0

0

0

0

0

0

0.36

5

16

3

0

0

0

0

0

1.24

0

3 26

10

0

0

0

0

2.32

0

0

10 32

6

0

0 0

0

0

0

6

18

3

0 0

0

0

0

0

3

18

6 0

0

0

0

0

0

6

20 4

0

0

0

0

0

0

4

8

and D =

1.62 I 0.50 2.50 2.50 3.60

Solving the matrix equation AM=D,we get the vector M of unknowns y" (i = 0,..., 7), which enables the left- and right-derivatives to be calculated through equations (3.1.43) and (3.1.44) (Table 3.10). Let us give an example by calculating the (Ce/Yb)N value interpolated for the year 1960 with thefirstinterpolation formula. Clearly i = 3 (1966), so x3 - x = 1966-1960 = 6, then x3-x2 = 1966-1956 =10. Moreover, y3 = 20.8, ^ ( " } = -0.0423 and ^ = 0.0919.

3.2 Functions of several variables

137

Table 3.10. Second derivatives, first left- and rightderivatives calculated for each data point listed in Table 3.9 through equations (3.1.43) to (3.1.46).

Finally, y3"-y2"-

i

y"

0 1 2 3 4 5 6 7

-0.0185 0.1090 -0.1371 0.0919 0.0085 -0.0683 -0.2161 0.5581

yti+)

0 0.2262 0.1840 -0.0423 0.2589 0.1693 -0.6839 0

0 0.2262 0.1840 -0.0423 0.2589 0.1693 -0.6839 0

=0.0919-(-0.1371) = 0.2290, hence \x3*)

=y3-

+>V(*3*)\ 2 6 x3 — x2

(

3

)

which, upon replacement with the actual values, yields y(t =1960) = 20.8-(-0.0423) x 6 + -(0.0919) x 6 2 - - ( — ) x 6 3 = 21.88 2 6 \ 10 / The calculation can be made for an arbitrary number of points provided their abscissa lie inside the range of x values. Figure 3.7 shows the characteristic features of spline interpolation, a very smooth aspect although with some 'overshooting' problems, i.e., extrema located between the data points. Alternative interpolation schemes are discussed by Wiggins (1976). o 3.2 Functions of several variables

3.2.1 Introduction • Let / ( x 1 , x 2 , . . . , x j be an n-multivariate scalar function of the dependent variables x 1 ,x 2 ,...,x n . An equivalent notation is/(jc), where JC is the vector made of the n variables Xj, x 2 ,..., xn. The partial derivative of the function/(jc) with respect to the dependent variable x, is defined as

df(x)

/ ( x 1 , x 2 , . . . , x l + Ax,,...,x n )/(x 1 ,x 2 ,...,x l ,...,x M ) = lim

(J.z.l)

A Higher-order partial derivatives can be defined in a similar way. Using equation (3.2.1), we can show the important equality

d2f(x) dx dy

djm dy dx

(3 2 2)

138

Useful numerical analysis

• Let u{t) be an n-vector with components (ul9 u2,..., un) depending on a single scalar variable t. The derivative of the vector u with respect to the scalar t is the n-vector defined as du

— = lim dt Ar^o

u(t + At)-u(t)

(3.2.3)

At

It makes a vector with n components dujdt (i= 1,2,..., n). Example: with respect to t is the vector = The derivative of u(t) = \ |_sin t J dr |_cos t J The gradient vector grad/(jc) of the scalar function/(JC) is an n-vector defined as df(x)dx1 df(x)/dx2

grad/(jc) =

(3.2.4)

df(x)/dxn The nabla notation V/(JC) is commonly used. Example: the gradient vector grad /(JC) of the function defined as

grad/(jc) = .COS X 3 _

As a particular case, the gradient of a scalar quantity obtained as the dot product of the constant column vector u(u1,...,ut) and the column vector x (xu...,xn) is the vector u itself for

grad wx = Vi#T x =

duJx/dx2

u2

(3.2.5)

JuTx/dxn_ Example: the gradient vector of the scalar 2xt 4-3x2 is the column vector [2, 3]T . The variation df(x) of an n-multivariate scalar function/(x) along a displacement direction x can be written as

<*/(*)= I ^ - d x dx

(3.2.6)

where do: is the n-vector with elements dx^dx^...,dx M . The variation d/(x) therefore has the meaning of the scalar product between the gradient and the displacement vector. Particular displacement vectors dx generate contour lines with constant/(JC) values such as df(x) = 0. The dot product vanishes and the gradient is therefore a vector perpendicular

3.2 Functions of several variables

139

to the contour lines of the function. The gradient vector points uphill. If d/(jt) is negative, the function f(x) decreases along the direction x and the opposite is true if it is positive. Example: At point (1,2, n/2), the direction [-1,3,2] T parallel to the infinitesimal change 3) =

djc(-d/c,3d/c,2d/c)

with d/c>0 induces a change of the function

defined previously, such that d/(jt) = -dkx (22) + 3 dk x (2 x 1 x 2) + 2 dk x [cos(;r/2)]

The direction [— 1, 3, 2] T corresponds to an increasing value of/(jc). • The divergence of a vector v with components v^v2 ..., vn is the scalar number noted either divr or, with the nabla notation V», defined as ivi> = V-i>= V —*dX

(3.2.7)

The Taylor's expansion of a bivariate function /(x, y) in the neighborhood of the point (xo Jo) is obtained by expanding /(x, y) with respect to x at constant y then with respect to y at constant x (or the other way around) and can be*written as

f +highe,orderterms

Defining AJC as the column vector with elements (x —x0) and (y — yo\ an alternative matrix formulation is /(x, y) = /(x 0 , y0) + AJCT grad / + - AjtT//Ajt + higher-order terms

(3.2.8)

where the 2 x 2 symmetric curvature matrix or Hessian H is defined as d2f(x,y) H=

dx2

d2f(x,y) dx dy

d2f{x,y) dx dy d2f(x,y)

(3.2.9)

dy2

An extremum is a minimum if any fluctuation of the coordinates about this point causes the function to increase. It is a maximum if any fluctuation causes the function to decrease. In any other case, we are dealing with a saddle point. We write the fluctuation A/of/(x,y)

140

Useful numerical analysis

about the extremum with coordinates (x*, y*) as A/ % f(x, v) - /(x*, y*) = - AxTHAx In general, / / can be diagonalized as

H=U\U ~l where A is the diagonal matrix of real eigenvalues and U an orthogonal and therefore invertible matrix. The fluctuation A/now writes

Af*-A Introducing the new vector z such that z=U~ 1Ax, we obtain Afx-zTAz

= -trAzzT = -(A1zl2 + A2z22)

where the properties of the trace of a matrix have been used (Section 2.2.4). If all eigenvalues are positive, the extremum (x*,/*) corresponds to a minimum. If they are negative, the extremum corresponds to a maximum. If they are of mixed sign, we are dealing with a saddle point. Figure 3.8 depicts the different cases. These concepts are easily extended to more than two variables. &

For x = (w, v)9 calculate the Hessian of the function f(x) defined as f(x) = — sin u cos v

and map curvature changes. This function is plotted on Figure 3.9 and shows a regular pattern of maxima and minima. Its Hessian matrix is Tsin u cos v cos u sin v~\ |_cos u sin v sin u cos vj The relation tr H= sin2 u cos2 v>0 shows that the eigenvalues of H have the same sign. Curvature changes sign whenever the determinant of H vanishes, i.e., for det H— sin2 u cos2 v — cos2 u sin2 v = 0 or det H= (sin u cos v — cos u sin v) x (sin u cos v + cos u sin v) — 0

3.2 Functions of several variables

141

Figure 3.8 Curvature of a function f (x,y). Top: the Hessian H has two negative eigenvalues. Middle: two positive eigenvalues. Bottom: mixed-sign eigenvalues.

and, finally det H= sin(w — v) x sin(w + v) = 0 which holds for u — v = n or u + v = n. o

142

Useful numerical analysis

Figure 3.9 Plot and curvature of the function/(w,v) = — sinwcos i; for —

3.2.2 System of implicit non-linear equations: the Newton-Raphson

method

The Newton method can be extended to several variables in order to find the zeroes of n functions/ in n variables xi9 which we lump as the vector x = [ x 1 , . . . , x J T , i.e., to solve the system of equations fi(xl,...,xn) = fi(x) = 0

(3.2.10)

for i = l , . . . , n . We start with an initial guess jc(0) = [x 1 ( 0 ) ,...,x n ( 0 ) ] T of x, expand the function to the first order, and make the result equal to zero. We get / T v (!) v t 1 )!— / T v (°> v- (°>~l _l_ «ro«l / T v (0) v (0)1 • A *•(!) — n JilXi ,..,xn j — jiix1 ,...,xn JH-graajfLXi ,...,xM j A X — U

where Ax(1) is the vector of the increments xiil) — xiiO)(i=l9...9ri). in expanded form the n equations of this type, we obtain

ML dxl

dxn

I

k

Lumping together

(3.2.11) Y

(1)

Y

(0)

dxn

Let us call/[jc ( 0 ) ] the n-column vectors of the n values of/ calculated at x{0\ and (0) D[JC ( 0 ) ] the matrix of the derivatives. We assume that Z)[x ] is non-singular, i.e., that its determinant (the Jacobian of the ^^/transformation) is different from zero. The increment Ax(1) is calculated as (3.2.12)

3.2 Functions of several variables

143

and the iteration repeated as far as needed. This formula extends equation (3.1.29) to multiple dimensions. Although several examples implementing the Newton-Raphson method for the computation of chemical equilibrium are developed in Chapter 6, we will now present some simple applications that illustrate its basic principles. & Let us assume that, at high temperature and ambient pressure, the binary system albite-anorthite (ab-an) is ideal. The temperature Tf and enthalpy AH{ of melting of each component is Tfab=1373K Atffab = 64.3kJmor 1 Tfan = 1830 K A//fan = 133.0 kJ mol" J Assuming that AHf are constant, calculate the composition of the solid and liquid coexisting at 7 = 1600 K. The variable X referring to mole fractions, equilibrium is achieved when the chemical potentials ji(ln X) for each element are equal in each phase

an

2

= //solan(0) + 0tT In Xsolan

where M is the gas constant and /i(0) the Gibbs energy of the standard state (pure phase or end-member) at the same temperature and pressure. Closure requires the conditions AY

liq

ab

_i_ AY a n — 1l ' Hq ~~

sol

ab i y an i i~A sol ~~ x

y

A

Equilibrium conditions can be recast into two equations in the two independent variables X liq an and X sol an hq

an liq

sol

-lnX sol an ] = 0

(3.2.14)

We recognize in the first two terms on the right-hand side of each equation the Gibbs energy of melting AGf of each end-member which, assuming that AH{ is constant, can be expressed as

and therefore AGfab = 64.3 x (1-1373/1390)=-10.6kJmoP 1 AGfan= 133 x(l-1600/1830)=+ 16.7kJmor 1

Useful numerical analysis

144

Table 3.11. Iterative calculation of the solution to equation (3.1.23) through the Newton-Raphson method. The vector x is the set of the two variables Xliqan and Xso*n. D(n)

n

-Ax-/)"1/

fin)

0 1 2 3 4 5 6

x( w + 1 ) -x ( n ) + Ax 0.6 0.3

-18075 25 937 2329.8 5041.5 561.32 -2738.5 -7.7393 -238.17 0.0369 -2.7995 -1.88xlO" 6 -3.76 xlO" 4

- 3 3 257 22171 -19197 43 328 - 1 5 585 90857 -16142 75 625 -16215 74079 -16216 74061

Let us define x = derivatives D(x) is

19004 -44343 50857 -18015 36148 -21049 35 874 -21143 36055 -21081 36057 -21080

0.29297 -0.43843 0.16061 0.10644 -0.02949 0.00281 -0.003 67 -0.00187 -0.00004 -0.00002 5.8 xlO" 9 2.7 x H T 9

0.30703 0.73843 0.14642 0.63199 0.17591 0.629 17 0.17958 0.63104 0.17962 0.63106 0.17962 0.63106

9.99 x 108 3.08 x 107 7.81 x 106 5.68 x 104 7.84 x 10° 1.41 xlO" 7

*0, A M l a n ] T a n d / ( J C ) = [ / 1 ( J : ) , / 2 ( x ) ] T . The matrix of partial

SIT (3.2.15)

D(x) = Sf2 Y

an

^Miq

Let us choose the initial guess for JC(0) = (0.6, 0.3). Successive steps produce the results shown in Table 3.11. Figure 3.10 shows the Gibbs free enthalpy of the liquid and solid mixtures together with the final result JC(6) = (0.179 62,0.631 06). The last column in Table 3.11 lists the squared-modulus s = / T / o f the vector/as a convenient measure of convergence. <j= Not all calculations converge so nicely, especially when the derivatives of higher order are large and variable. The choice of variables as well as their value at the starting point may turn to be critical in achieving reasonable convergence.

3.2.3 Extrema: the steepest-descent method A large category of problems consists in finding the extremum of a function with respect to several variables. Finding the maximum of a function/(JC) is equivalent to finding the minimum of —f(x), so the discussion will be restricted to the search for minima. Let us assume that/(jc) is a function in the n variables xl9 x2, >,xn collected

145

3.2 Functions of several variables

I o 0.2

0.4

0.6

0.8

1

Figure 3.10 Computation of equilibrium concentrations for coexisting solid and liquid solutions.

Figure 3.11 The steepest-descent method: the search in one direction is discontinued when no further decrease is possible, i.e., when the search direction is parallel to the local contour line. The next step starts in a perpendicular direction, i.e., in the direction opposite to the local gradient.

into the vector x. Since the grad f(x) vector points towards the maximum increase of/(jc) (Figure 3.11), minimizing/(JC) may be iteratively achieved using (3.2.20)

where a is a constant. The linear search for the optimum value <xm of a is carried out by either bisection or by more efficient method such as Davidon's cubic interpolation (e.g., Walsh, 1975; Fletcher, 1987). A measure of how fast/(jt) decreases from x{k) to xik+1) is the scalar gik + 1\ such that (k + 1)

=

1) _

(3.2.21)

Useful numerical analysis

146

Table 3.12. Search for the minimum ofthe function f(x) = exp(0c12 steepest descent.

by the method of the

The scalar a, equation (3.2.20), is estimated by crude linear search, g is the convergence criterion given by equation (3.2.21). Values in italic refer to the minimum along the search direction.

a 0 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0 0.1 0.2 0.3 0.4 0 0.15 0.30 0.45 0.60

1.0 0.9197 0.8393 0.7590 0.6786 0.5983 0.5179 0.4376 0.3573 0.4376 0.3829 0.3283 0.2736 0.2189 0.1642 0.1096 0.0549 0.1096 0.0872 0.0649 0.0425 0.0202 0.0425 0.0297 0.0169 0.0042 -0.0086

x2

/

-1.0 -0.8393 -0.6786 -0.5179 -0.3573 -0.1966 -0.0359 0.1248 0.2855 0.1248 0.0936 0.0624 0.0312 0.0001 -0.0311 -0.0623 -0.0935 -0.0623 -0.0369 -0.0115 0.0140 0.0394 0.0140 0.0123 0.0130 0.0137 0.0144

20.0855 9.5322 5.0811 3.0422 2.0459 1.5453 1.3111 1.2494 1.3373 1.2494 1.1784 1.1225 1.0798 1.0491 1.0293 1.0200 1.0207 1.0200 1.0104 1.0045 1.0022 1.0035 1.0022 1.0012 1.0006 1.0004 1.0005

df/dx, 40.171

df/dx2

g

-80.342

8069

1.0935

0.6236

-6.18

1.0935

0.6236

-6.18

0.2235

-0.2542

0.0859

0.2235

-0.2542

0.0859

0.0852

0.0559

0.0048

0.0852

0.0559

0.0048

0.0083

0.0549

0.0038

At the minimum along the kth search direction, the old and the new directions should be orthogonal, i.e., (3.2.22) grad/[jc(k+ ^-grad/iy 0 ] =0

Calculate by the steepest-descent method the minimum of the function

using the starting point ( + 1 , - 1 ) . The solution is obviously (0, 0). The first four stages of linear search are given in Table 3.12. <^

3.2 Functions of several variables

147

The steepest-descent method does converge towards the expected solution but convergence is slow in the vicinity of the minimum. In order to scale variations, we can use a second-order method. The most straightforward method consists in applying the Newton-Raphson scheme to the gradient vector of the function/to be minimized. Since the gradient is zero at the minimum we can use the updating scheme grad f[x(k

+1

>] = g r a d / [ j c ( f c ) ] + / / < k ) [ > ( * + l)- jc ( k ) ] = 0

or *<*+1> = x <*>-[#<*>]-i g r a d / [ x ( k ) ]

(3.2.23)

where H{k) is the curvature matrix (Hessian) o f / a t step k. Equation (3.2.23) is the extension of equation (3.1.30) to multiple dimensions. As will be seen in Chapter 5, this method is extremely useful for refinement near the minimum but may otherwise run away towards any point where the gradient vector vanishes, such as saddle points or even maxima. This inconvenience may be overridden by using the mixed scheme due to Marquardt (1963) and which is written grad /[jc(fc)] + [#<*> + a/J[jc(fc + 1 } - *<*>] = 0 where a is a parameter and /„ the (n x n) identity matrix. This scheme allows the user to shift from the most reliable gradient method far from the minimum for large values of a to a fast-converging Newton-Raphson method for small values of a near the minimum (Fletcher, 1987). Other methods, which also have the property of accelerated convergence near the minimum, take advantage of how the gradient varies locally, or, in other words, build up a local second-order approximation to the function/ The most commonly applied methods are those of conjugated gradients due to Fletcher and Reeves and the variable metric methods, e.g., the Davidon-Fletcher-Powell (DFP) and Broyden-FletcherGoldfarb-Shanno (BFGS) algorithms. These methods require more substantial theoretical developments, which may be found in the books by Walsh (1975), Press et al. (1986), Fletcher (1987) and are found in most major software packages such as MatLab. 3.2.4 Constrained minimization Constraints are relations of equality or inequality, which must be exactly obeyed by the unknowns of a model. A familiar example is the mineral abundances in a rock or the end-member proportions in a mixture, which must sum up to unity whatever the errors on the data. • When the objective function to be minimized is linear, the problem is relatively simple (Figure 3.12) and is the substance of linear programming. Let us take a simple example in n = 2 dimensions and assume that we are looking for the minimum of a linear function/(JC), where JC is the vector [x 1 ,x 2 ] T , with the equality constraint bTx = q (b and q being known constants) and the inequality constraints xt ^ 0 , x2 ^0. In geochemistry or thermodynamics, the equality constraints are typically conservation equations, while phase or end-member

148

Useful numerical analysis

x\\\\\\v<x\\\\\\\

Figure 3.12 Minimum of the function/(x) submitted to the equality constraint bTx = q, and to the inequality constraints x ^ O , x 2 ^0. The feasible set is the segment of the constraint line located in the positive quadrant. The minimum occupies an edge of the feasible set. proportions are usually required to be non-negative. Constant values of/(jc) define straight lines in the plane (xl5 x2). The inequality constraints split the space in two subspaces. The feasible set of solutions is the convex subspace that comprises all the values of the vector x which satisfy both the equality and inequality constraints. A corner is a vector of n values (a point), an edge is a segment associated with a linear relationship between the n variables. In the case of Figure 3.12, we see that the feasible set is a segment of a straight line. It could be an empty set if the equality constraint was entirely contained in the forbidden subspace of the inequality constraints. The vector corresponding to the minimum, if it exists, may be shown to occupy a corner of the feasible set (e.g., Strang, 1976). The idea of the simplex method is to select any corner of the feasible set and to proceed from corner to corner along the edges in a direction that minimizes the function f(x) until no further reduction can be achieved. A linear programming solution devised by Wright and Doherty (1970) is widely used in the literature for the problem of finding modal abundances, while an example of free energy minimization will be discussed in Chapter 6. When constraints are equality only, the method of Lagrange multipliers is of broad applicability. Let us consider, as in Figure 3.13, a function/(JC) to be minimized on which we impose the constraint that g(x) = 0. From the previous discussion on the gradient properties, we know that — grad / is a vector perpendicular to the curves of constant /(JC) pointing towards decreasing values, while grad g is orthogonal to the locus of x vectors such as g(x) = 0. The orthogonal projection of —grad/on the constraint g(x) = 0 represents a direction of decreasing/(JC). A minimum will be reached once —grad/and grad g are

3.2 Functions of several variables

149

Figure 3.13 Constrained minimization: the minimum of a function f(x) submitted to the constraint g(x) = 0 occurs at M on the constraint subspace, here on the curve g(x) = 0 where V/(JT) + XVg(x) = 0. P is the unconstrained minimum of f(x). This principle is the base for the method of Lagrange multipliers.

collinear, i.e., grad / + X grad g = 0

(3.2.24)

where X is a constant called a Lagrange multiplier. If more than one constraint should be obeyed, as many Lagrange multipliers as there are constraints should be used. In practice, this procedure amounts to increasing the number of independent variables in the system by as many new variables as there are constraints. Equation (3.2.24) is indeed equivalent to finding the minimum of (3.2.25) The derivatives with respect to x produce the set of equations represented by grad 5 = grad / + X grad g = 0

(3.2.26)

150

Useful numerical analysis

while the derivative with respect to k gives 9(x) = 0

(3.2.27)

which is precisely the constraint which we want to be verified. &

Find the minimum of y2

submitted to the constraint

This problem amounts to minimizing

where / is a Lagrange multiplier, relative to x, y, and L The derivatives relative to each variable cancel at minimum dS dS dS — = 2x + ^ = 0 , — = 2 y + A = 0, a n d — = dx dy dX Adding the first two equations and subtracting the third twice results in /.= - 1 , x= 1/2, and y= 1/2

& Distribution of energy states. According to quantum theory, the energy states £ o>£i>£2>--- t h a t atoms in a gas, a liquid or a crystal can reach are distinct and have an equal probability of being taken by an atom. Standard textbooks (e.g., Swalin, 1962) show that the entropy 5 of a population of N atoms, nt being in the energy state ei9 is S = — k Y n.In — t

N

where k is the Boltzmann constant. Find the values of nt for maximum entropy 5 for constant total energy E. The first constraint is a fixed total atoms number £ > = iV or X d w . = ° i

(3.2.28)

i

and the second is a fixed total energy Y4eini = E or YJsidni = ° i

i

(3.2.29)

3.2 Functions of several variables

151

Let us minimize the function S1" given by

S+ = 5 + xfc eini - E \ + fih nt where a and ft are two Lagrange multipliers. We first observe that

and therefore

Each derivative relative to n 0 ,M 1? n 2 ,... must cancel, so for any state i A: In — - a e , - ^ = N

which we rearrange as

Summing up over all the energy states and applying the constraint (3.2.28), we get

and therefore n

N

Yea£'/fc

Introducing this expression into dS and equating to (Lejdnj)/T would show that a is equal to — T " 1 and produce the familiar Boltzmann distribution.o When the minimum of a function f(x) is sought not algebraically, as in these examples, but numerically, different techniques exist, which are described in a specific and abundant literature. Once more, a simple approach makes use of the minimization properties of the gradient direction. The so-called gradient projection method consists in using the projection of the gradient direction onto the constraint subspace as the search direction (Figure 3.14). It works best with linear constraints and an example will be given in Chapter 6.

Useful numerical analysis

152

\ \

-grad/

\ \ \ \

£(*) = const

\ /

\ \ \ x2

\ \

/x\ \

/decreases

gradg

\ \ \

\%

\

\

\ \

\ Figure 3.14 Search direction for the minimum of a function/^!, x2) submitted to the constraint g(x1,x2) = 0. The optimum direction is the projection of the opposite of the gradient onto the constraint subspace.

3.2.5 The Runge-Kutta method for a system of differential equations The single-variable method can be extended to a system comprising any number of differential equations. The Runge-Kutta method is commonly found in software packages. For simplicity, it will be described for a system of only two equations — = *r =f(t,x,y) dt

(3.2.30)

dy —=yt'=g(t>x>y) dt

where / and g are two known functions of t, x, and y. Let us define

Given the intermediate evaluations kA = h(n)f[t{n\ x(n), /">], /c 2

t<»» +

"

x<»>

+ 2 i , y»» + !i I

», x
],

;3 = l4 =

r

L(«)

3,

y(n) + / 3 ]

3.2 Functions of several variables

153

the values of x and y at time t{n+1) are

<#^ Solve the following equation, which appears in connection with some diffusion problems, in C(t) from t = 0 to t = 0.5 " + 2fC'-0.5C =

(3.2.31)

with the conditions C= 1 and C' = 0 at t = 0. Let us use a time step h of 0.1 and define x = C, y = C. Equation (3.2.31) becomes a system of first-order equations

' = 0.5x-2ty

(3.2.32)

with the conditions x=l and y = 0 at t = 0. With reference to equation (3.2.30), the functions / and g are defined as (3.2.33)

Let us work out the first step in detail using t(0) = 0, x ( 0 ) = 1, >>(0) = /cx =0.1 x0 = 0 / 1 = 0 . 1 x ( 0 . 5 x l - 2 x 0 x 0 ) = 0.05 /c2 = 0.1 x (0 + — J = 0.0025

/2 = 0.1x o.5n + -J-2K)+ — /c3 = 0.1

- ^ ) ] = 0.04975

= 0.002488

kt = 0.1 x (0 4- 0.049 814) = 0.004 981 /4 = 0.1x [0.5(14-0.002488)-2(0 + 0.1)x(0 + 0.049814)] =0.049 128

Useful numerical analysis

154

Table 3.13. The first five iterations (n = l, ..., 5) of the Runge-Kutta solution to equation (3.2.31) for a time-step of 0.1.

n

0

1

2

3

4

5

t(n)

0 1 0 0 0.0500 0.0025 0.0498 0.0025 0.0498 0.0050 0.0491 1.0025 0.0497

0.1 1.0025 0.0497 0.0050 0.0491 0.0074 0.0480 0.0074 0.0481 0.0098 0.0466 1.0099 0.0977

0.2 1.0099 0.0977 0.0098 0.0466 0.0121 0.0447 0.0120 0.0448 0.0142 0.0425 1.0219 0.1424

0.3 1.0219 0.1424 0.0142 0.0426 0.0164 0.0400 0.0162 0.0401 0.0183 0.0373 1.0382 0.1824

0.4 1.0382 0.1824 0.0182 0.0373 0.0201 0.0343 0.0200 0.0345 0.0217 0.0312 1.0582 0.2167

0.5 1.0582 0.2167 0.0217 0.0312 0.0232 0.0279 0.0231 0.0281 0.0245 0.0247 1.0813 0.2447

x{n) y(n)

k1

h

k2

h

u

x(n+l)

which gives the x and y values at the next step u_

u

0 + 2x0.0025 + 2x0.002488 + 0.004981

=0+

Table 3.13 gives the results up to £ = 0.5. <= & Convective dispersal of a conservative tracer in a velocity field. This calculation is of major interest for many problems such as the geochemical evolution of the mantle, the mixing time of the ocean, the understanding of magma mixing processes, ... Justification of the equations used is presented in Chapter 8. Solve for t = 0.02, 0.04, 0.06, ... the system of differential equations dx . (n — = — n sin nx sin dt \2

&y n fn — = — cos nx cosl dt 2 \2

ny y

(3.2.34)

7iy v

given x= y = 0.l at t = 0. It is left to the reader to build Table 3.14. 3.2.6 Interpolation with spline functions When a bi-dimensional table of data with unevenly distributed coordinates is available (e.g., data points on a map), the need for computing interpolated values is frequently

3.3 Partial differential equations: the finite differences method

155

Table 3.14. The first five iterations of the Runge-Kutta solution to equation (3.2.34) for a time step of 0.02.

n t(n)

xin) y(n)

f G

h

k2

h K U(»+D X

y(n+D

0 0 0.1 0.1 -0.9233 0.4616 -0.0185 0.0092 -0.0167 0.0097 -0.0169 0.0097 -0.0153 0.0103 0.0832 0.1097

1 0.02 0.0832 0.1097 -0.7638 0.5129 -0.0153 0.0103 -0.0138 0.0108 -0.0139 0.0108 -0.0126 0.0113 0.0693 0.1205

2 0.04 0.0693 0.1205 -0.6303 0.5670 -0.0126 0.0113 -0.0114 0.0119 -0.0115 0.0119 -0.0104 0.0125 0.0578 0.1324

3 0.06 0.0578 0.1324 -0.5191 0.6244 -0.0104 0.0125 -0.0094 0.0131 -0.0095 0.0131 -0.0085 0.0137 0.0484 0.1455

4 0.08 0.0484 0.1455 -0.4268 0.6854 -0.0085 0.0137 -0.0077 0.0143 -0.0078 0.0144 -0.0070 0.0150 0.0406 0.1599

5 0.10 0.0406 0.1599 -0.3506 0.7501 -0.0070 0.0150 -0.0063 0.0157 -0.0064 0.0157 -0.0057 0.0164 0.0343 0.1756

encountered. A most common application is the drawing of contour maps (such as isopleths). Several procedures exist but the success of cubic splines makes this technique easily available from software packages. Two-dimensional interpolation is carried out as a successive construction of one-dimensional splines along rows followed by one-dimensional splines along columns (e.g., Press et a/., 1986). Other useful spline variants in multiple dimensions are described by Sandwell (1987). 3.3 Partial differential equations: the finite differences method

Partial differential equations (PDE) involve relationships between partial derivatives of a function. One of the most common PDE problems to be solved in geochemistry appears to be the diffusion equation, which, in the jargon of PDE specialists, is said to be parabolic, because of the values of each derivative's coefficients, in contrast with elliptic equations (e.g., Laplace equation in a square) or hyperbolic equation (e.g., the wave equation). Many methods exist which can solve these problems, some of which appeared in the last 20 years, e.g., the collocation and least-square methods (Finlayson, 1972), the variational finite-element method (Zienkiewicz, 1977) and the multigrid method (Hackbusch, 1985). However, the easiest route is by far the finite difference method (e.g., Mitchell, 1969). Although computational effort with finite differences is small, the results, surprisingly enough, are often quite satisfactory. In addition, although being rather slow, the finite differences are easy to learn and easy to implement. The increasing power of computers makes it possible to consider this method as a competitor to the most efficient and sophisticated methods, sometimes difficult to master for a non-specialist, always time-consuming to implement. Only for natural objects with irregular geometry, such as plutons, could alternatives to the finite differences result in a net gain in the time and effort spent in solving the problem.

156

Useful numerical analysis 3.3.1 One-dimensional diffusion problems: general

Let us consider the diffusion of a homogeneously distributed substance out of a slab with unit thickness in the direction x and infinite size in the other two dimensions. C(x, t) being the concentration at time t at a distance x from the origin, the diffusion equation is £OM)*C<M) dt dx2 where 3f is the diffusion coefficient of the substance. The following initial condition holds C(x,0) = Co(x) while at the boundaries, the conditions are

The first step is to discretize the (x, t) domain using a uniform mesh x = iAx, i = 0,l,...,n=l/Ax t=jAt, 7 = 0,1,...,00 and call Ctj the concentration at the mesh point (node) x = i Ax, t =j At (here i and 7 have do not have their usual meaning of element and phase). Let us define the central difference operator 8X as

From this definition, the second-order difference is

or 5 x 2 C/=C l _ 1 ^-2C/ + Cl + 1^

(3.3.2)

In the finite difference schemes, the derivatives are replaced by difference operators, e.g., the first derivative dc(xt)

8cy

^x

Ax

c ; - c ;

3

Ax

and d2C(xj) ^ Sx(dxCA6xCi AxV Ax /

CI-/2C/CI^ Ax 2

Ax2

3.3 Partial differential equations: the finite differences method

157

The time derivative is evaluated at mid-point between t and t + At, i.e., dc{x,t) dt

6 r c^ + 1 / 2 At

c/+1-cy At

(3.3.5)

and the second derivative with respect to x is replaced by a linear combination of the second-order finite differences at t and t + At ,,,6,

Axz

dx

where 9 is a parameter defined over (0,1]. Rewriting the diffusion equation into a partial difference equation leads to

At

|_

Ax2

(3.3.7)

Ax2 J

Two well-known schemes can be derived from this equation. For 0 = 0, the second-order difference is evaluated at t = i Ax and the scheme is called explicit

Ax2

At

The 'molecule', that defines the nodes involved in the calculation for a pair of i, j values, is shown in Figure 3.15. Defining the parameter r as

(3-3-8)

r~ Ax2 and recombining J 1"

(3.3.9)

or equivalently, in terms of the central difference operator

For r < 1/2, the explicit scheme is shown to be stable, e.g., does not make errors grow exponentially with time (e.g., Mitchell, 1969). Alternatively, for 6 = 1/2, i.e., evaluation of the second-order difference as an average of the values at t and t + At, leads to the robust Crank-Nicholson scheme ^

^

+ CI. + ^ + 1 )

(3.3.10)

The 'molecule' of the Crank-Nicholson scheme is also shown in Figure 3.15.

158

Useful numerical analysis

Equivalently, in terms of the central difference operator 8X, we can write

Multiplication by 2 and recombination leads to the linear system of equations (3.3.11)

Given the boundary conditions of zero concentration for i = 0 and i = n9 the equations can be recast into matrix form 1 -r

0

0

0

0

0

2(1+r)

-r

0

0

0

j+1

0

0

-r

2(1 + r)

-r

0

0

0

-r

2(1 + r)

-r

0

0

0

0

-r

2(1 +r)

-r

0

0

0

0

0

0 - - .

0

0

0

0

0

0

0

1

0

0

r

2(1-r)

r

0

0

r

2(1 - r )

r

0

1

C J

!

0 0

0

0

0

0

r

2(1 - r ) ..-Or

..

0

0

r

0

-r)

r

0

1

Let us lump the concentrations at the nodes f = 0, 1, 2,..., n— 1, n into a single vector c7 and defining the (n+ l)x(n-|-1) matrices An and ^ n by their current element atj and bip respectively, such that

.= - r

bij = r

for | i - ; | = l

j= 0

6l7 = 0

otherwise

The matrix form of the Crank-Nicholson implicit scheme becomes AncJ+l=BncJ

3.3 Partial differential equations: the finite differences method

159

Implicit CrankNicholson scheme

Explicit

i+l

i+l

i-l

i-l

Time Figure 3.15 Finite difference 'molecules' for explicit (left) and implicit Crank-Nicholson (right) schemes.

or (3.3.12) which shows how the node concentrations at time t + At are calculated from those at time t. Although the Crank-Nicholson scheme is unconditionally stable, use of small r values such as r< 1/2, improves accuracy (Mitchell, 1969). &

Solve with both the explicit and implicit methods the equation

ct

with the initial condition C(x, 0) = 1 and the boundary conditions C(0, t) = C(/, t) = 0 for / = l c m and ® = 0.005cm 2 s" 1 . For the purpose of illustration, a very coarse mesh size will be used with Ax = 0.25, i.e., w = 5, and At = 2.5 s. Therefore

r = — r2 = Ax

0.005 x 2.5 z— = 0.2 0.252

Useful numerical analysis

160

Figure 3.16 Results of the explicit difference scheme applied to the following diffusion problem: (a) initial concentration equal to unity (b) end concentrations at x = 0 and x = 1 cm are zero (c) Qi = 0.005 cm2 s" 1 . Length increment is Ax = 0.25, i the number of length increments. Time increment is At = 2.5 s, j the number of time increments.

We first implement the explicit scheme. The initial conditions require C,° = 1 = 1 for i = 1,2,3,4 (initial condition) Q>° = 0

(boundary condition)

C 5°

(boundary condition)

Points i = 5, 4, and 3 may be obtained by symmetry from points 0, 1, and 2. Hence at t = At = 2.5s l

= Co°

(boundary condition)

1

C 1 = 0 . 2 x C 0 0 + 0.06xC 1 0 + 0.2xC 2 ° = =0.2 x ( V + O^ x C2° + 0.2 x C 3 °= 1

The same calculation can be can be carried on as far as needed. The results up to r = 4Af=10s are depicted in Figure 3.16. Let us now turn to the implicit Crank-Nicholson method and form the matrix An as 1

0

0.2

2(1+0.2)

0 A =

-0.2

0 -0.2 2(1+0.2) -0.2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-0.2 2(1+0.2) -0.2 0

-0.2 2(1+0.2) 0

0 -0.2 1

3.3 Partial differential equations: the finite differences method 1

0

0

-0.2

2.4 -0.2

0

-0.2

0

0

-0.2

0

0

0

0

0

0

0

0

0

0

0

2.4 -0.2

2.4 -0.2 -0.2

0

2.4 -0.2

Likewise 1

0

0 «„=

0

2(1-0.2)

+ 0.2

+ 0.2

+ 0.2

0

0

0

0

0

0

0

0 0

0

0

0

0

0

0

+ 0.2

2(1-0.2)

+ 0.2

0 0

+ 0.2

2(1-0,2) + 0.2

0

2(1-0.2) + 0.2

0

1

0

"1

0

0

0

0 "

0.2

1.6 0.2 0

0

0

0

0.2

1.6 0.2 0

0

0

0

0.2

1.6 0.2 0

0

0

0

0.2

1.6 0.2

0

0

0

0

0

1

therefore "1.0000 0

0

0

0

0.1678

0.6784

0.1409

0.0118

0.0010

0.0001

0

"

0.0141

0.1409

0.6902

0.1418

0.0118

0.0012

0.0012

0.0118

0.1418

0.6902

0.1409

0.0141

0.0001

0.0010

0.0118

0.1409

0.6784

0.1678

0

0

0

0

0

1.0000

•

•

We can now calculate the approximation cx for i= 1, 2, 3, 4 at t = At as fl.OOOO 0

0

0

0

0.1678

0.6784

0.1409

0.0118

0.0010

0.0001

0

" "0" 1

0.8321

0.0141

0.1409

0.6902

0.1418

0.0118

0.0012

1

0.9847

0.0012

0.0118

0.1418

0.6902

0.1409

0.0141

1

0.9847

0.0001

0.0010

0.0118

0.6784

0.1678

0

1

0.8321

0

0

0

0

1.0000

0m

•o

_0

161

Useful numerical analysis

162

o

Figure 3.17 Same as Figure 3.16 but for the implicit Crank-Nicholson scheme. then iterate over successive time steps. The results up to t = 4 At = 10 s are depicted in Figure 3.17. o

3.3.2 More boundary conditions

For boundary conditions which require a prescribed value of the flux instead of concentration, we introduce what is usually called fictitious points. Let us assume that at x = 0, the condition is dC(xj)

(3.3.13)

ex We introduce the approximation valid for any value of t, hence of j dC(x, t) ex

l

<~ - 1

(3.3.14)

where i = — 1 refers to the point symmetrical to the mesh point i = 1 relative to the interface. Then we write that the finite difference approximation holds for the mesh point / = 0 at the interface, e.g., for an implicit scheme -rC^j+1 +2(1 Combining the two equations, the fictitious point concentrations can be eliminated

or (3.3.15)

3.3 Partial differential equations: the finite differences method

163

As will be shown below, writing this equation as a matrix equation is rather straightforward. Similar techniques can be used for the so-called 'radiation' boundary conditions which involve linear combinations of concentration and concentration gradient and appear in connection with elemental fractionation between adjacent phases. & Solve with the Crank-Nicholson scheme the diffusion problem described in the worked example in Section 3.3.1 d2C(x,t) <3 dx2

dC(x,t) ct

with C(x, 0)= 1, C(/, r) = 0 for /= 1 cm, and 2 = 0.005 cm2 s~ \ but with no flux at x = 0. Let us keep the same mesh size Ax = 0.25, i.e., n = 5, and Ar = 2.5s. Let us form the matrices

A =

2(1 + r )

—r

0

0

0

0

—r

2(1 + r)

—r

0

0

0

0

—r

2(1 + r)

- r

0

0

0

0

—r

2(1 l-r)

—r

0

0

0

0

2(1 + r)

—r

0

0

0

0

1

" 2.4

-0.4

-0.2

r 0

0

0

0

0 "

2.4

-0.2

0

0

0

0

-0.2

2.4

-0.2

0

0

0

0

-0.2

2.4

-0.2

0

0

0

0

-0.2

2.4

0

0

0

0

-0.2 1

0

and 2(1 - r )

2r

0

0

0

0

r

2(1-r)

r

0

0

0

0

r

2(1-r)

r

0

0

0

0

r

2(1 -r)

r

0

0

0

0

r

2(1-r)

r

0

0

0

0

0

1

164

Useful numerical analysis 1.6

0.4

0

0

0

0 "

0.2

1.6

0.2

0

0

0

0

0.2

1.6

0.2

0

0

0

0

0.2

1.6

0.2

0

0

0

0

0.2

1.6

0.2

L0

0

0

0

0

1

and the vector of flux conditions at x = 0 -4rg0Ax 0 =0

V= 0 0

The matrix equation reads (3.3.16)

We obtain "0.6903 0.2837

0.0238

0.0020

0.0002 0.0000"

0.1419

0.7022

0.1429

0.0120

0.0010

0.0001

0.0119

0.1429

0.6904

0.1419

0.0118

0.0012

0.0010

0.0120

0.1419

0.6902

0.1409

0.0141

0.0001

0.0010

0.0118

0.1409

0.6784

0.1678

0

0

0

0

0

1

We can now calculate the approximation cx at t = At fO.6903

0.2837

0.0238

0.0020

0.0002 0.0000" "1"

0.1419

0.7022

0.1429

0.0120

0.0010

0.0001

1

•i.oooo0.9999

0.0119

0.1429

0.6904

0.1419

0.0118

0.0012

1

0.9988

0.0010

0.0120

0.1419

0.6902

0.1409

0.0141

1

0.9859

0.0001

0.0010

0.0118

0.1409

0.6784

0.1678

1

0.8322

0

0

0

0

0

1

The results up to t = 4 At = 10 s are depicted in Figure 3.18.

_0_

0

3.3 Partial differential equations: the finite differences method

165

1 0.8 0.6 0.4 0.2 0

1

0

2

3

4

5

i Figure 3.18 Same as Figure 3.17 but with no flux at x = 0.

3.3.3 A word about advection Solving the purely advective equation or even introducing an advection term into the diffusion equation is a source of numerical difficulties. The simplest advection equation of a medium moving at velocity v in one dimension can be written dC(xj)

dC(xj) = —v-

(3.3.17)

dx

ct

Most simple finite difference schemes tend to be either unstable, inaccurate or both. Strang (1986) recommends using Friedrichs scheme

C,- +1

— C,--!

At

(3.3.18)

With r = vAt/Ax, the finite difference equation becomes (3.3.19)

and the stability condition is — l ^ r ^ - h l . For the cases where advection appears as an additional term in the diffusion equation, the reader is referred to the discussion in Chapter 9 of the book by Fletcher (1991). 3.3.4 Two space coordinates: The ADI method Given a rectangle R {a ^ xl ^ b, c ^ x2 ^ d) and the two-dimensional diffusion equation x 1 > x 2 > r) = dt

p 2 C(x l t x 2 > r) L

d*i

2

d2C(xux2,t)~

(3.3.20)

166

Useful numerical analysis

with the initial condition C(xu x2, 0) = C o(x 1 , x2) and the boundary conditions C(a,x2,t) = Ca(x29t) C{b9x2,t) = Cb(x29t)

The grid points are ilAx (Jl=0, 1, ..., wl), i2Ax (i2 = 0, 1, ..., nl\ andjAr. Explicit different schemes show poor stability properties (Mitchell, 1969). In terms of the central difference operator, it may be shown that an accurate implicit equation is 1--^ which is the extension of the Crank-Nicholson implicit formula to two dimensions. The widely used Peaceman-Rachford method deals with the Laplacian operator in two steps. Approximation is carried out on one space derivative with an explicit scheme and an implicit scheme for the other derivative (hence the name of Alternating Direction Implicit or ADI method), which results in the two successive formulas

I,i2

H/2

Developing, the two steps can be written

(3.3.21)

+ ui2i+V2

(3.3.22)

The 'molecule' of the difference scheme over the time steps is given in Figure 3.19. The half-step may not have in general the significance of an intermediate time step and is more a matter of computational convenience. For sake of demonstration, let us assume that concentration on the edge is constant with time c C

J—c 0i2 — e 0 i 2

with similar equations for the other three edges. Let us first compute the condition along the vertical (y) edges for the first half-step. For il = 1 and 1 < f 1 < n\ — 1, equation

3.3 Partial differential equations: the finite differences method

167

a

il

Figure 3.19 Finite difference 'molecule' for the alternating-directions implicit method (ADI) of solving the two-dimensional diffusion equation.

(3.3.21) becomes Ili o

(3.3.23)

iUn2

(3.3.24)

A similar equation holds for Q = nl—\ and 1 < i \
and for the horizontal (x) edge. For il = 1 and 1 < il < nl — 1, equation (3.3.23) becomes 2(l+r)CUi2j+1'2-rC2,2j+^2 and for il = nl,

= rCui2_^ + 2(\-r)Cui2^rCUi2

j +l

+ rC042

(3.3.25)

I
(3.3.26) We now turn to the conditions at the corners. As an example, let us compute equation (3.3.21) of the ADI scheme for il = i2= 1 (3.3.27)

168

Useful numerical analysis

and for i\=n\ — 1 and i2 = n2—l.

(3.3.28)

Let us form the system of equations for I
0 -r

..• 0

0 0

7 + 1/2 2

7+1/2

0

0

-r

-r2(l+r)

-r

0

•.

0

-r

2(l+r)_

2

C

1 , «"2 — 1

C

0,i2

r

0

2(1 - r ) + r C«l-2,i2-l

j

L

0

r

For f2 = 1, the system becomes "2(1 +r)

-r

0

•••

0

-r

2(1+r)

-r

0

0

0 0

0 •••

-r 0

2(1 +r) -r -r 2(1 +r)_

c ^2,0

nl - 2 , 1 .^nl-1,1

C

nl-2,

U

« l -1,2 _

while for i2 = n2-l "2(1+r)

-r

0

•••

0

-r

2(1+r)

-r

0

0

0 0

0 ...

-r 0

2(1+r) -r -r 2(l+r)_

c2

l

7+1/2

3.3 Partial differential equations: the finite differences method CUn2-2J

169

C\,n2-\

J

C2,n2-2

C2,«2-V r

L2(l-r). mCnl,n2

- 1 + Oil - I,n2_

All these equations can be stacked as •2(1+r)

-r

0

-r

2(1+ r)

-r

0

-r

2(1 + r)

-r

0

-r

(1+r).

"2(1 -r)

r

0

0

r

2(1-r)

r

0

0

0

r

0

...Or e

o

^2,0

2(1 -r)

r 2(1 -r\

r

r

...

o

Cnl,n2-2

c2

Cnl,n2-l+Cnl-l,

where C J is the (n\ — 1) x (n2— 1) matrix at time step 7 of concentration values at all the inner grid points (not including the boundaries). In a compact matrix form, the equation reads (3.3.29) In this equation, Anl _ x is an (nl — 1) x (nl — 1) tridiagonal matrix with 2(1 + r) on the main diagonal and — r on the other two diagonals. Bn2-1 is an (n2— I)x(n2— 1) tridiagonal matrix with 2(1—r) on the main diagonal and I r o n the other two diagonals. H is an (nl — 1) x {nl— 1) matrix built as indicated. For the second half-step described by equation (3.3.22), we can write (J'2 = 2, 3, ..., n2-2, Ci,i2J C2,i2j

—r 2(1 +r) —r

1

Useful numerical analysis

170 2(1 - r )

r

0

r

-r)

r

0

0

0

0

r

2(1 -r)

r

0

r

2(1 - r ) _

0

0

. J+l/2

C

J+l/2

-2,i2

cnl - I i 2

J+l/2 i+1/2

and expressions similar to those of the first half-step for concentrations on the edges. In a matrix form, the system of equations can be written (3.3.30)

In this equation, An2 _ x and Bnl _ x are defined as before, while His an (nl — 1) x (nl — 1) matrix. Rewriting the expression (3.3.29) for the first half-step as

and inserting into equation (3.3.30), the matrix equation becomes

and finally ,-'

(3.3.31)

We now get a fairly good feeling that handling boundary conditions in two dimensions is significantly more difficult than those of one-dimensional problems. Flux conditions can be applied to the boundaries using the method offictitiouspoints. Solve the two-dimensional diffusion equation C(x,y,t)] d2C(x

dC(x,y,t) dt

dy2 J

in the rectangle defined by a^x^b = a + L and condition C(x, y, 0) = 0 and the boundary conditions

with the initial

C(a,y,t) = O •(b,y,t) =

^

d-c

b-a

for a diffusion coefficient Q) = 0.005 cm2 s i. Let us mesh the rectangle assuming Ax = 0.2, which is equivalent to nl=ny = 4, n2 = nx = 5, and At = 0.1, i.e., r = 0.0125 (note that x and y are switched relative to the

3.3 Partial differential equations: the finite differences method

171

indices n\ and til). We now build the time-independent matrices 2.0250

-0.0125

-0.0125

2.0250

-0.0125

-0.0125

2.0250

-0.0125

-0.0125

2.0250

0 0

0

0

0 0

and "1.9750 0.0125 0

0

0.0125

1.9750 0.0125 0

0

0.0125

1.9750 0.0125

0

0

0.0125 1.9750

which are combined into 0.9754 0.0122 0.0001 0.0000" 0.0122 0.9755 0.0122 0.0001 0.0001

0.0122 0.9755 0.0122

0.0000 0.0001

0.0122 0.9754

Likewise 2.0250

-0.0125

-0.0125

2.0250

0

-0.0125

0 -0.0125

and

1.9750

0.0125

0

0.0125

1.9750

0.0125

0.0125

1.9750.

L0

2.0250J

so 0.9754

0.0122

0.0001

0.0122

0.9755 0.0122

LO.0001 0.0122 0.9754

H is built from the boundary conditions as 0.2000 0.4000 0.6000 1.55'

H= 0

0

0

0.50

.0

0

0

0.25.

hence 0.0025

0.0049

0.0075

0.0190"

0.0000

0.0000

0.0001

0.0062

.0.0000

0.0000

0.0000 0.0031.

Useful numerical analysis

172

The concentration matrices will be shown as Cj fringed on each edge by the boundary values written in italic, i.e., as (n y +l)x(n J C +l) = 6 x 5 matrices, and a symbols with overbar Cj will be used. Given the initial concentration matrix '0 0.2 0.4 0.6 0.8 1.0 ' 0 0

0

0

0

0.75

C° = 0 0

0

0

0

0.50

0 0

0

0

0

0.25

0 0

0

0

0

0

we can calculate concentrations at t = At '0 0.2

0.4

0.6

0.8 1.00'

0

0.0025 0.0049 0.0075 0.0190 0.75

0

0.0000 0.0000 0.0001 0.0062 0.50

0

0.0000 0.0000 0.0000 0.0031

0.25

0

0

0

0

0

0

0.4

0.6

0.8 1.001

t = 2At V0 0.2 0

0.0049 0.0098 0.0149 0.0372 0.75

2

C = 0 0.0000 0.0001 0.0003 0.0124 0.50 0

0.0000 0.0000 0.0001 0.0061

10 0 and so forth. <=•

0

0

0

0.25 0 A

4 Probability and statistics

4.1 A single random variable 4.1.1 General

In most natural situations, physical and chemical parameters are not defined by a unique deterministic value. Due to our limited comprehension of the natural processes and imperfect analytical procedures (notwithstanding the interaction of the measurement itself with the process investigated), measurements of concentrations, isotopic ratios and other geochemical parameters must be considered as samples taken from an infinite 'reservoir' or population of attainable values. Defining random variables in a rigorous way would require a rather lengthy development of probability spaces and the measure theory which is beyond the scope of this book. For that purpose, the reader is referred to any of the many excellent standard textbooks on probability and statistics (e.g., Hamilton, 1964; Hoel et al, 1971; Lloyd, 1980; Papoulis, 1984; Dudewicz and Mishra, 1988). For most practical purposes, the statistical analysis of geochemical parameters will be restricted to thefieldof continuous random variables. • A random variable X is defined by (a) a continuous domain Q of definition in % e.g., ] - oo, 4- oo[, [0, + oo[, or [ - 1 , + 1]. (b) its probability distribution function (sometimes called the cumulative distribution function or, for short, distribution function), i.e., a continuous non-decreasing function F{x) defined for each value x in Q, such as F(x) = #{X^x}

(4.1.1)

where & stands for 'probability of. Strict notation conventions should be observed: an upper-case letter should refer to a random variable, while the same letter in lower case refers to the values taken by this variable. Writing the same relation at x + cbc

and subtracting, we get

The probability density function f(x) (also the density function or frequency function) is the 173

Figure 4.1 Relationship between the probability density function/(x) of the continuous random variable X and the cumulative distribution function F(x). The shaded area under the curve f(x) up to x0 is equal to the value of f(x) at x0.

4.1 A single random variable

175

derivative of the distribution function F(x). These two functions relate through dd

/ (() )d d # { ^ ^

+

}

(4.1.2)

dx f(x) therefore has the significance of a probability per unit of X in the neighborhood of the value x. Because F(x) is non-decreasing, its derivative/(x) is non-negative. Conversely, if Q is the domain ] — oo, +oo[, the cumulative distribution function F(x) relates to the probability density function/(x) through =\

f(u)du

(4.1.3)

J—

F(x) represents the surface under the curve of the function/(u) up to u = x (Figure 4.1). Since the total probability over the domain is 1

1 •

(4.1.4)

If p is an integer between 0 and 100, the pth percentile is the value xp of the random variable X which limits to the right p/lOOths of the surface S under the probability curve (Figure 4.2). 4.1.2 Expectation and moments

Let us consider a continuous random variable X defined over a given domain Q of ^ , such as [0, ..., +oo[ or ] - o o , ..., +oo[ and let/(x) be its probability density function over Q. Let g(X) be a function of the random variable X. •

The expectation $(g) of g(X) is <%)= I g(x)f(x)dx

(4.1.5)

and is usually denoted with greek letters. Expectation is therefore a linear operator. The expectation has the meaning of a centroid with the local' probability /(x)dx as a weight-function. • Using parentheses to emphasize power functions, the nth moment is the expectation $\_(X)n~\ of X". The mean \i is the first moment H=

x/(x)dx

(4.1.6)

JQ

• The nth central moment is g[_(X-n)n] or £{[X-g{X)~]n}. central moment, i.e.,

The variance a2 is the second

\rckr(Y\ rr 2 jPf F Y /?( YY\2\ /P(Y2\ JCP( Y\/P( Y\ y a\\J\ ) — Ox — 0 I L ^ — ^ V^ )J ( =z & V-A ) — <5 ^A )0 \J\ )

(A \ H\ yr.l.. I)

• The standard deviation a is the square-root of the variance and has the same unit as the random variable. A random variable is standardized (or reduced) if its variance is unity and centered if its mean is zero.

Probability and statistics

176

100 100

Figure 4.2 Definition of the percentile p: the curve is the probability density function /(x) of the continuous random variable X. xp is the pth percentile when the surface up to xp represents p percent of the total surface S under the curve. ^

Calculate the mean and variance of the uniform density f(x)=l/2a

for

The mean is the first moment, i.e., c+a i rx 2 i + f l li = £'(X)=\ x —dx= — =0 J_fl 2a L4«J-fl while the variance is the second central moment i fx 3 ~l +a a3
2

&

2

2

—a3

a2 =— 6a 3

<^

Find the variance of the Cauchy distribution

n

1+x2

Because f(x) is symmetric about the mean, its mean must be zero. Its variance should be Tll-hX 2

but this integral diverges when x->oo. The Cauchy distribution has no finite variance, o • The moment generating function M(t) = $(etX) is particularly useful to compute moments. Expanding the exponential, we get = S[ l + tX+l-X2+

...]=\+t£(X)+t-g(X2)+

...

4.1 A single random variable

111

Alternatively, M(t) may be expanded in Taylor series in the neighborhood of t = 0 M(t) = M(0) + tM'(0) + - M'(0) + ...

(4.1.8)

where the number of primes indicates the order of the derivative. Comparing the last two equations shows that the moment of order n is equal to the nth derivative of M(i) for t = 0 (4.1.9) Jt = O

& Calculate the moment generating function of the general normal distribution f(x) given by

From the definition of the moment generating function, we write

Multiplying and dividing the right-hand side by the exponential of fit + a2t2/2, we get / M(t) = exp( fit H

\

(T2t2\C

+ co

I

1

(x-u)2-^rT2t(Y-n\^rT4t2l

f exp

^ / J - ao ^/luo

L

which we rearrange as 2<72

J

The integrand is almost the normal density. Introducing the variable u= -

M(t) can be rewritten as

<

A*f +

a2t2\ 2 /

x

1

f +0

x /2^J-oo

On the right-hand side of the multiplication sign, we recognize the integral of the reduced normal density function which sums up to one, and therefore e^t+£ r

(4.1.10)

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Probability and statistics

The first and second derivatives are

M"(t) = G2M{t) + (/i + (72t)M'(0 = (T2 Af (0 + (// + a2t)2M{t) The values of M(t) and its first two derivatives at f = 0 are M(0)=l, M/(O) = iu, and M"(0) = //2 + (72 The mean of the normal density function is therefore &(X) = M'(0) = fi

(4.1.11)

and its variance g(X2)-g(X)g(X)

= M'\Q>)-[M\0)]2 = ii2 + G2-v2 = o2

(4.1.12)

two well-known and quite useful results. 4.1.3 A compendium of some common probability density functions • A random variable is said to be distributed as the uniform distribution if f(x) is given by /(x)=-i-, b—a

a<x
(4.1.13)

and/(x) = 0 otherwise Because of its importance in natural phenomena, e.g., radioactive decay and population dynamics, let us introduce the exponential distribution through an illustration, n atoms are assumed to decay over the time interval [0 — 0], each atom having the same probability of decaying at any time in this interval. In other words, the time at which an atom decays is uniformly distributed over [0 — 0]. Let us call N(t) the number of decay events between 0 and t. The probability that a single atom has not decayed at time t, is 1 —1/6 (Figure 4.3). The probability that none of the n atoms has decayed at time t is (4.1.14) The probability that the first decay takes place before t is therefore the complement of this expression to one since

The cumulative distribution function of the time T to the first decay is F(t) = ^{T^t}=

&{N(t) # 0} = 1 - 0>{N(t) = 0}

and therefore F(t)=l-(l--'

4.1 A single random variable

o

t

179

e

Time Figure 4.3 Distribution of events on the time axis in a Poisson process over the time interval [0 — 0]: the probability of occurrence for one event in a time interval depends on its duration only (see text). The corresponding density of probability function f(t) is obtained as the derivative of F(t) f(t) = — = -

1--

dt e\

(4.1.15)

e)

Let us assume that probability of decay per unit time is constant, i.e.,

We rewrite equation (4.1.15) as

and 'enlarge' our view by increasing the number of decays n->oo, which because of constant decay rate implies 0->oo. For finite time t, At«n and, expanding the logarithm to the first order (see Section 3.1), we get the density of the exponential distribution defined over [0, + oo[

XCXP x f nn 1 ^ 1 L n\

(4.1.16)

The exponential distribution has mean /x = X and variance a2 = X2. i The most widely used distribution is the normal distribution (also Gaussian) with the familiar bell shape defined over ] — oo, + oo[. Its most general form is (4.1.17) with mean \i and variance a2. The moment generating function is given by equation (4.1.10). It is said to be centered when fi = 0 and reduced when a2 = l. • A random variable X is distributed as a log-normal distribution if In X is distributed as a normal distribution. If U is a random normal variable, a log-normal distribution is the distribution of a variable X such as (4.1.18) or X = fiaRu

(4.1.19)

crR can be viewed as a 'relative' standard deviation: for U= ± 1, X is multiplied, respectively divided, by aR.

180

Probability and statistics

Figure 4.4 Gamma probability density functions for scale parameter p = 1 and different shape parameters a = 1, 2, and 5.

The Cauchy distribution, also bell-shaped, is defined over ] — oo, + oo[ and reads (4.1.20) Its mean is zero but we have demonstrated that it does not have a finite variance. Several distributions belong to the family of gamma distributions defined over ]0, + oo[

fix)-

1

(4.1.21)

where a and p are parameters with a and p > 0. a is often referred to as the shape parameter and p as the scale parameter. This function is plotted in Figure 4.4 for /?= 1 and a = 1, 2, and 5. F(a) is the gamma Eulerian function

,-JV..

x*-le~xdx

F(a)=

(4.1.22)

Jo Jo

and is also well known in its form of the factorial function for integer values of a (4.1.23) Additional properties of this function are the recursion formula (4.1.24)

and the special value (4.1.25)

4.1 A single random variable

181

The mean of the gamma distribution is a/? and its variance a/?2. The moment generating function is (4.1.26) The parameters a and /? can be computed from // and a2 using a = (/V
p = <J2/n

(4.1.27)

Setting a = 1 and /?= I/A, we recognize the exponential distribution of equation (4.1.16). • For a = v/2 and /? = 2, the gamma distribution becomes the chi-squared (x2) distribution A

(

4

.

1

.

2

8

)

with mean v and variance 2v. For reasons that will appear later, v is known as the number of degrees offreedom. The pth percentile of the chi-squared distribution with v degrees of freedom is denoted xP,v2• A special case of the gamma distribution is obtained by replacing a by n+1, where n is an integer, and x by /to (this can be easily achieved using the change of variable technique discussed below). The resulting distribution f(x) is /(x) = i x n e - x \

(4.1.29)

and is known for integer values of x as the Poisson distribution. A random variable X is distributed as the beta distribution over the range [0,1] if its density function is given by

l^"\-\l-xr-i

(4.1.30)

with the two parameters ax and a 2 > 0 . This function is plotted in Figure 4.5 for different combinations of ax and a2. Its mean and variance are ^

(

4

.

1

.

3

1

)

The parameters ax and a2 can be computed from \i and a2 using «i = ^ ( l - M ) - ^

and

a2 = ^ ^ - / i - ( l - M )

(4.1.32)

No simple form of the moment generating function exists. In the special case where a i = a 2 = 1» t n e b e t a distribution reduces to the uniform distribution over [0, 1]. • Finally, we will frequently refer to Snedecor's F-distribution. A random variable defined over ]0, +oo[ is distributed with the F-distribution with Vi and v2 degrees of freedom

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Probability and statistics

0

0.2

0.4

0.6

0.8

1

X Figure 4.5 Beta probability density functions for different parameter pairs ax and a2.

when its density function is given by

2

;

/Vl

i/2 (vi+v2)/2

/ ( * ) = •

(4.1.33)

The pth percentile of the F distribution with vx and v2 degrees of freedom is denoted F p V) vv The order of vt and v2 is critical since the F-distribution is not symmetric in these variables. Its mean is

v,-2

v2>2

(4.1.34)

and its variance , 2v22(v1+v2-2) 'v,(v 2 -2) 2 (v2-4)'

v2>4

(4.1.35)

• Making the change of variable x = t2 in the F density for Vj = l and v2 = n, we get the Student's t-distribution defined over ]-oo, + oo[

f(t) =

1

V2

(4.1.36)

nn

Its mean is zero and its variance n/(n — 2). The pth percentile of the t distribution with v degrees of freedom is noted tPtV. The Student's f-distribution converges rapidly towards the normal distribution: in practice, when v > 30, the two distributions become indistinguishable.

4.1 A single random variable

183

Useful relations among percentiles are ti-.P/2,,v2 = fi-p,v

(4-1-37)

and (4.1.38)

4.1.4 Some relationships between fundamental distributions Although the formal proof of some relationships will be postponed until the necessary background is exposed, it is probably necessary at this point to justify the rather lengthy compendium of distributions of Section 4.1.3. The exponential distribution with parameter k is the distribution of waiting times ('distance' in time) between events which take place at a mean rate of k. It is also the distribution of distances between features which have a uniform probability of occurrence (Poisson process), such as the simplest model of faults on a map. The gamma distribution with parameter n and k'1, where n is an integer is the distribution of the waiting time between the first and the nth successive events in a Poisson process. Alternatively, the distribution/(t), such as f(t) = -(h)nQ-Xt n\

(4.1.39)

represents the probability that n events have occurred over the time t. The associated gamma and Poisson distributions are well-suited to describe non-negative quantities that result from the addition of a finite number of units. Detrital input to a sedimentary layer from a particular drainage basin through river transport can be treated as a Poisson process by assuming that small batches of sediments are carried to the sea at random times. Each single layer represents a time interval over which the total sedimentary mass added from that drainage basin may be treated as a gamma variable. The same simple approach can be used to model the distribution of contributions from a magma source to a given batch of magma. Deloule et ai (1991) have shown that the energy spectrum of hydrogen atoms in amphiboles measured by ion probe fits a gamma distribution consistent with the atom being ejected from the sample after absorption of about seven electrons by a neighboring iron atom. Given two gamma variables X and Y with parameters ax, /? and ay, /?, respectively, the proportion X/(X + Y) is distributed as a beta distribution with parameters ax and ocY. This relationship between the exponential, gamma and beta distributions is useful in handling mass balance problems. Using once again the sedimentary example, if only two basins contribute material to a detrital layer with quantities being described by Poisson processes of identical rate k but different numbers of batches nx and nY, the proportion of each component is distributed as a beta distribution with parameters nx and nY. Although this model may appear a little simplistic, it is still appealing enough to use the beta distribution for mass fractions in mixtures.

184

Probability and statistics

The physical and conceptual importance of the normal distribution rests on one unique property: the sum of n random variables distributed with almost any arbitrary distribution tends to be distributed as a normal variable when n->oo (the Central Limit Theorem). Most processes that result from the addition of numerous elementary processes therefore can be adequately parameterized with normal random variables. On any sort of axis that extends from — oo to + oo , or when density on the negative side is negligible, most physical or chemical random variables can be represented to a good approximation by a normal density function. The normal distribution can be viewed a position distribution. The ratio of two normal random variables with zero mean is distributed as a Cauchy variable. Isotopic ratios such as 2 0 6 Pb/ 2 0 4 Pb and 2 0 7 Pb/ 2 0 4 Pb therefore should not be described as normal variables since ratios of ratios (e.g., 2 0 7 Pb/ 2 0 6 Pb) should be distributed with a consistent distribution. A consistent distribution for isotopic ratios is the log-normal distribution. The square of the distance between two points with position distributed normally is distributed as the x2 distribution with one degree of freedom. The sum of n such distances is distributed as the x2 distribution with n degrees of freedom. The ratio of squared distances, which is used for instance to test the ratio of variances or the ratio of a squared distance to a variance, is distributed as an F-distribution. 4.1.5 Estimators The set of all possible outcomes of a measurement considered as a random variable is usually called the population. The parameters of the density function associated with a particular population, e.g., mean or variance, are not physically accessible since their determination would require an infinite number of measurements. A measurement, or more commonly a set of measurements ('points' or 'observations'), produces a finite set of outcomes called a sample. Any convenient number describing in a compact form some property of the sample is called a statistic, e.g., the sample mean

(4.1.40)

where m is the number of observations Xj in the sample, or the sample variance

s2 = —

(4.1.41) m-1

The (m— 1) factor will be dealt with below. An alternative expression for s2 is obtained by developing the squared parentheses as

(4.1.42)

-(xf[ m —1

m—1 (.

m

)

4.1 A single random variable

185

where the last term is often read as the mean square minus the squared mean. Although this expression is usually the easiest to implement on a computer, it may lead to devastating roundoff errors. If the value of a sample statistic #is used to estimate a parameter 6 of the population, this statistic is called an estimator and its value for the sample the estimate. Sample mean x and variance s 2 are the usual estimators of the population mean ji and variance a2. Estimators are most useful and reliable when they are convergent, i.e., |£_0|_»O when m->oo

(4.1.43)

i(6) = 0

(4.1.44)

and unbiased, i.e.,

Sample mean x and variance s2 are convergent and unbiased estimators (e.g., Hamilton, 1964), which implies that the so-called empirical variance 62 given by m

mm

m

has a bias s2/m. o2 is nevertheless convergent since the bias tends to zero when m-> oo. If the estimator S is not biased, we can write its variance a@2 as G62 = &{[6-&(G)-\2}

= S\_(6-0)2-]

(4.1.46)

The distribution of a sample statistic is a sampling distribution. A particularly important result concerns the variance of the sample mean given by a-2 = S\_{x - //)2] = — m

(4.1.47)

A proof of this statement can be found in Hamilton (1964). The square-root of a^2 is usually referred to as the standard error of mean. 4.1.6 Change of variable Let cp be a monotonous function and note q> ~ x its inverse function which is assumed to be single-valued (i.e., a value of the independent variable is associated with one value of the dependent variable). If the random variable X has the density function fx(x\ then Y = (p(X) has the density function fY(y) given by

£

(4-1.48)

dy A more rigorous statement can be found in standatd textbooks such as Hoel et al. (1971). In order to demonstrate this relationship, let us call Fx and FY the distribution

186

Probability and statistics

functions corresponding to fx and / y , respectively. Let us assume first that