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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 53 "Reaction Equilibrium for \+
Multiple Gas Phase Reactions" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "We begin by creating a list of \+
component identities and to calculate the number of components." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "components:=[A,B,C,D,X,Y,Z];
 nc:=nops(components);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+component
sG7)%\"AG%\"BG%\"CG%\"DG%\"XG%\"YG%\"ZG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#ncG\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Next, we cre
ate a list of reactions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 
"Reactions:=[R1,R2,R3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Reaction
sG7%%#R1G%#R2G%#R3G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Each react
ion may be expressed as a Maple equation:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "Reaction[R1] :=A+B=C+D;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%)ReactionG6#%#R1G/,&%\"AG\"\"\"%\"BGF+,&%\"CGF+%\"DG
F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Reaction[R2] := B+C=X
+Y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)ReactionG6#%#R2G/,&%\"BG\"
\"\"%\"CGF+,&%\"XGF+%\"YGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "Reaction[R3] :=A+X=Z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)React
ionG6#%#R3G/,&%\"AG\"\"\"%\"XGF+%\"ZG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 83 "The stoichiometric cofficients can be deduced from the se
t of reactions as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
146 "printlevel:=2:\nfor r in Reactions do \n   for i in components do
 \n     nu[i,r]:=coeff(lhs(Reaction[r])-rhs(Reaction[r]),i); \nod: od;
\nprintlevel:=1:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG%#R1G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#nuG6$%\"AG%#R1G\"\"\"" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>&%#nuG6$%\"BG%#R1G\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%#nuG6$%\"CG%#R1G!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%#nuG6$%\"DG%#R1G!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#
nuG6$%\"XG%#R1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#nuG6$%\"YG
%#R1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#nuG6$%\"ZG%#R1G\"\"!
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG%#R2G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%#nuG6$%\"AG%#R2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%#nuG6$%\"BG%#R2G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%
#nuG6$%\"CG%#R2G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#nuG6$%\"
DG%#R2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#nuG6$%\"XG%#R2G!\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#nuG6$%\"YG%#R2G!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#nuG6$%\"ZG%#R2G\"\"!" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"rG%#R3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
&%#nuG6$%\"AG%#R3G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#nuG6$%
\"BG%#R3G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#nuG6$%\"CG%#R3G
\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#nuG6$%\"DG%#R3G\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#nuG6$%\"XG%#R3G\"\"\"" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>&%#nuG6$%\"YG%#R3G\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%#nuG6$%\"ZG%#R3G!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 41 "We need mole balances for each component:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "r:='r': \nfor i in components do  \+
      \n   CMB[i] := C[i,0] = C[i] + add(nu[i,r]*e[r],r=Reactions);\n \+
  print(CMB[i]);\nod:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6$%\"A
G\"\"!,(&F%6#F'\"\"\"&%\"eG6#%#R1GF,&F.6#%#R3GF," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"CG6$%\"BG\"\"!,(&F%6#F'\"\"\"&%\"eG6#%#R1GF,&F.6#%
#R2GF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6$F%\"\"!,(&F%6#F%\"
\"\"&%\"eG6#%#R1G!\"\"&F-6#%#R2GF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/&%\"CG6$%\"DG\"\"!,&&F%6#F'\"\"\"&%\"eG6#%#R1G!\"\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/&%\"CG6$%\"XG\"\"!,(&F%6#F'\"\"\"&%\"eG6#%#R2G!\"\"
&F.6#%#R3GF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6$%\"YG\"\"!,&&
F%6#F'\"\"\"&%\"eG6#%#R2G!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%
\"CG6$%\"ZG\"\"!,&&F%6#F'\"\"\"&%\"eG6#%#R3G!\"\"" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 13 "where we use " }{TEXT 256 1 "e" }{TEXT -1 35 " to \+
denote the extent of reaction. " }{TEXT -1 69 "The initial concentrati
on is denoted with a second subscript of zero." }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "The reaction
 equilibrium constants may be expressed as" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 107 "i:='i': for r in Reactions do\n  Kdef[r]:=K[r] = m
ul ((C[i])^(-nu[i,r]),i=components);\n  print(Kdef[r]);\nod:" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/&%\"KG6#%#R1G**&%\"CG6#%\"AG!\"\"&F*6#%\"BG
F-&F*6#F*\"\"\"&F*6#%\"DGF3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"KG
6#%#R2G**&%\"CG6#%\"BG!\"\"&F*6#F*F-&F*6#%\"XG\"\"\"&F*6#%\"YGF3" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"KG6#%#R3G*(&%\"CG6#%\"AG!\"\"&F*6
#%\"XGF-&F*6#%\"ZG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "We cr
eate a set of equations to be solved" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "Eqns := \{seq(CMB[i],i=components),seq(Kdef[r],r=Reac
tions)\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%EqnsG<,/&%\"CG6$%\"AG
\"\"!,(&F(6#F*\"\"\"&%\"eG6#%#R1GF/&F16#%#R3GF//&F(6$%\"BGF+,(&F(6#F:F
/F0F/&F16#%#R2GF//&F(6$F(F+,(&F(6#F(F/F0!\"\"F>F//&F(6$%\"DGF+,&&F(6#F
KF/F0FG/&F(6$%\"XGF+,(&F(6#FRF/F>FGF4F//&F(6$%\"YGF+,&&F(6#FYF/F>FG/&F
(6$%\"ZGF+,&&F(6#FjnF/F4FG/&%\"KGF2**F-FGF<FGFEF/FMF//&F`oF?**F<FGFEFG
FTF/FenF//&F`oF5*(F-FGFTFGF\\oF/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
31 "and a set of unknown variables:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "Vars := indets(Eqns,name);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%%VarsG<6&%\"CG6$%\"AG\"\"!&F'6#F)&%\"eG6#%#R1G&F.6#%#
R3G&F'6$%\"BGF*&F'6#F6&F'6$F'F*&F'6#F'&F.6#%#R2G&F'6$%\"DGF*&F'6#FB&F'
6$%\"XGF*&F'6#FG&F'6$%\"YGF*&F'6#FL&F'6$%\"ZGF*&F'6#FQ&%\"KGF/&FUF>&FU
F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "The number of degrees of fr
eedom follows directly" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "D
egFree := nops(Vars)-nops(Eqns);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
(DegFreeG\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Case 1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Specs[1] := C[A,0]=1.5;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&SpecsG6#\"\"\"/&%\"CG6$%\"AG\"\"!$
\"#:!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Specs[2] := C[
B,0]=1.5; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&SpecsG6#\"\"#/&%\"CG
6$%\"BG\"\"!$\"#:!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "S
pecs[3] := seq(C[i,0]=0,i=[C,D,X,Y,Z]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%&SpecsG6#\"\"$6'/&%\"CG6$F+\"\"!F-/&F+6$%\"DGF-F-/&F+6$%\"XGF
-F-/&F+6$%\"YGF-F-/&F+6$%\"ZGF-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
91 "At the conditions of interest the reaction equilibrium coefficient
 has the following values" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
42 "Kvals := \{K[R1] = 1, K[R2]=2.63, K[R3]=5\};" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&KvalsG<%/&%\"KG6#%#R1G\"\"\"/&F(6#%#R2G$\"$j#!\"#/&F
(6#%#R3G\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "We now augment t
he set of equations with the set of specifications" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 53 "AllEqns:=Eqns union \{seq(Specs[k],k=1..3),
op(Kvals)\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(AllEqnsG<6/&%\"CG6$
%\"BG\"\"!$\"#:!\"\"/&F(6$%\"AGF+,(&F(6#F2\"\"\"&%\"eG6#%#R1GF6&F86#%#
R3GF6/&F(6$%\"XGF+F+/&F(6$%\"DGF+F+/&%\"KGF9F6/&FH6#%#R2G$\"$j#!\"#/&F
HF<\"\"&/&F(6$%\"ZGF+F+/&F(6$F(F+F+/F',(&F(6#F*F6F7F6&F8FKF6/FX,(&F(6#
F(F6F7F.FhnF6/&F(6$%\"YGF+F+/FC,&&F(6#FEF6F7F./F?,(&F(6#FAF6FhnF.F;F6/
F^o,&&F(6#F`oF6FhnF./FT,&&F(6#FVF6F;F./FG**F4F.FfnF.F[oF6FcoF6/FJ**Ffn
F.F[oF.FgoF6F[pF6/FQ*(F4F.FgoF.F_pF6/F0F," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 34 "and ask Maple to solve the problem" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "result:=solve(AllEqns,Vars);" }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#>%'resultG6*<6/&%\"CG6$%\"XG\"\"!F,/&F)6$%\"DGF,F,/&
F)6$%\"ZGF,F,/&F)6$F)F,F,/&F)6$%\"YGF,F,/&%\"KG6#%#R1G$\"\"\"F,/&F>6#%
#R3G$\"\"&F,/&F)6#F4$\"+Kh\\O6!\"*/&F)6#F;$\"+ZGkd\")!#5/&F)6#F0$\"+iJ
>s5FN/&F)6$%\"BGF,$\"+++++:FN/&F>6#%#R2G$\"++++IEFN/&F)6$%\"AGF,Fhn/&F
)6#Fco$!+TH*o3(FT/&F)6#F)$\"+x()GkDFT/&%\"eGFEFL/&F)6#Fgn$!+qWdzQFT/&F
)6#F+$!+r%=t?$FT/&F`pF?FX/&F`pF\\oFR<6F'F-F1F5F8F<FC/Feo,&$!+VzE'H%FTF
B%\"IG$!+j]!*HT!#6/F\\q,&$\"+P;%p,#FTFBFdq$!+<9p'o(Fgq/Fgp,&$!+*fAE-)F
TFBFdq$\"+$p*46AFgq/FJ,&$\"+JE$zs\"FNFBFdq$\"+['f;=\"FT/FVFiq/FP,&$\"+
2Pqc#*FTFBFdq$\"+<'pFS\"FT/Fbp,&$\"+cYNEPFTFBFdq$!+cZ+TjFgq/F^qF\\s/Fj
o,&$!+q?wRsFTFBFdq$!+f(Q9<#FT/F_pFerFZFjnF`o<6F'F-F1F5F8F<FC/F_p,&FfrF
BFdq$!+['f;=\"FT/Fbp,&FcsFBFdq$\"+cZ+TjFgq/Fjo,&FjsFBFdq$\"+f(Q9<#FT/F
JFat/FV,&FjqFBFdq$\"+<9p'o(Fgq/FP,&F]sFBFdq$!+<'pFS\"FT/Feo,&FbqFBFdq$
\"+j]!*HTFgq/F\\qF^u/F^qFbu/Fgp,&F`rFBFdq$!+$p*46AFgqFZFjnF`o<6F'F-F1F
5F8F<FC/F_p$\"+B-!**[#FN/FJFav/FV$!+c0sD')FT/FP$!+0l4@9FN/Feo$!+q;Gt7F
T/F\\qFev/F^qFhv/Fgp$!+Gn*4\"RFN/Fbp$\"+h&oOy$FN/Fjo$\"+%\\W_e&FTFZFjn
F`o<6F'F-F1F5F8F<FC/Fgp,&$\"+Fx4]DFTFBFdq$\"+Zxr$p$FT/Feo,&$\"+Rl/o9FT
FBFdq$!+wM]dEFT/F\\q,&$\"+9&)3_nFTFBFdq$\"+'[vYL$FT/Fbp,&$!+)=^?3\"FTF
BFdq$!+B7A^jFT/Fjo,&$!+gT(yd#FTFBFdq$\"+%\\(H\"=$Fgq/F_p,&$\"+Z\\')znF
TFBFdq$!+,,srnFgq/FP,&$\"+uE'*H$*FTFBFdq$\"+Pda;IFT/FJFhy/FVFfx/F^qF^z
FZFjnF`o<6F'F-F1F5F8F<FCFZFjnF`o/F\\q,&FgxFBFdq$!+'[vYL$FT/Feo,&FaxFBF
dq$\"+wM]dEFT/Fgp,&F[xFBFdq$!+Zxr$p$FT/F^q,&F_zFBFdq$!+Pda;IFT/Fbp,&F]
yFBFdq$\"+B7A^jFT/Fjo,&FcyFBFdq$!+%\\(H\"=$Fgq/F_p,&FiyFBFdq$\"+,,srnF
gq/FJF`\\l/FVFhz/FPFd[l<6F'F-F1F5F8F<FCFZFjnF`o/Fjo$!+G%G+j\"FN/FV$\"+
U2lA_Fgq/FP$\"+O\\D#o\"FN/F_p$\"+%49Z3\"FN/Fgp$\"+?%3a(fFT/F^qF^]l/Feo
$\"+()RfIOFT/F\\qF[]l/Fbp$!+LW\"[M#FT/FJFa]l<6F'F-F1F5F8F<FCFZFjnF`o/F
gp$\"+sq#pw\"FT/FP$\"+@,<9bFT/FV$\"+4-A6qFT/FJ$\"+\\ICZPFT/F_pFj^l/Fbp
$\"+q'4YZ#FT/Fjo$\"+*3]q\\\"FT/Feo$\"+Un`TUFT/F\\qFg^l/F^qFd^l" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 187 "Maple found several solutions to \+
this problem without much effort. The number of solutions can be count
ed using the nops command (note that in this case we must enclose resu
lt in braces)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "nops(\{re
sult\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 361 "It is easy to see, however, that not all of the s
olutions are physically meaningful; some of them contain negative conc
entrations, while others are complex. Only one of the solutions is phy
sically meaningful (all quantities being non-negative). We can pick th
is solution from the set of 8 by hand or we can be more sophisticated \+
and ask Maple to find it for us." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 105 "for k from 1 to nops(\{result\}) do    \n   if not h
astype(result[k],negative) then print(result[k]) fi;\nod;" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#<6/&%\"CG6$%\"XG\"\"!F)/&F&6$%\"DGF)F)/&F&6$%\"
ZGF)F)/&F&6$F&F)F)/&F&6$%\"YGF)F)/&%\"KG6#%#R1G$\"\"\"F)/&F;6#%#R3G$\"
\"&F)/&F&6$%\"BGF)$\"+++++:!\"*/&F;6#%#R2G$\"++++IEFL/&F&6$%\"AGF)FJ/&
F&6#F($\"+sq#pw\"!#5/&F&6#F8$\"+@,<9bFfn/&F&6#F-$\"+4-A6qFfn/&F&6#F1$
\"+\\ICZPFfn/&%\"eGFBFdo/&F&6#FI$\"+q'4YZ#Ffn/&F&6#F&$\"+*3]q\\\"Ffn/&
F&6#FV$\"+Un`TUFfn/&FhoF<F_o/&FhoFOFjn" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 456 "Note that we did not have to rearrange the equations int
o a form more suitable for numerical computation as was done in the Po
lymath solution. We did not have to provide initial guesses of the unk
nown variables, and we did not have to work out the independent equati
ons since we could do that with Maple as well. Of course, it is possib
le to compute a purely numerical solution using the Newton's method pa
ckage in the share library and demonstrated below." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 36 "read `c:/maple/numerics/newton.mpl`:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 
1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" 
{TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 190 "Newton requires two arguments, a list of equations \+
to be solved and a list of equations that provide the initial values. \+
The list of eqns is constructed from the set used in the last example
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Eqns:=[op(AllEqns)];" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%EqnsG76/&%\"CG6$%\"BG\"\"!$\"#:!\"
\"/&F(6$%\"AGF+,(&F(6#F2\"\"\"&%\"eG6#%#R1GF6&F86#%#R3GF6/&F(6$%\"XGF+
F+/&F(6$%\"DGF+F+/&%\"KGF9F6/&FH6#%#R2G$\"$j#!\"#/&FHF<\"\"&/&F(6$%\"Z
GF+F+/&F(6$F(F+F+/F',(&F(6#F*F6F7F6&F8FKF6/FX,(&F(6#F(F6F7F.FhnF6/&F(6
$%\"YGF+F+/FC,&&F(6#FEF6F7F./F?,(&F(6#FAF6FhnF.F;F6/F^o,&&F(6#F`oF6Fhn
F./FT,&&F(6#FVF6F;F./FG**F4F.FfnF.F[oF6FcoF6/FJ**FfnF.F[oF.FgoF6F[pF6/
FQ*(F4F.FgoF.F_pF6/F0F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "The s
tartingpoint is constructed in a rather simple-minded way (that is eff
ective here). We make a list of the variables in the equations:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Vars := [op(indets(Eqns,name
))];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%VarsG76&%\"CG6$%\"AG\"\"!&F
'6#F)&%\"eG6#%#R1G&F.6#%#R3G&F'6$%\"BGF*&F'6#F6&F'6$F'F*&F'6#F'&F.6#%#
R2G&F'6$%\"DGF*&F'6#FB&F'6$%\"XGF*&F'6#FG&F'6$%\"YGF*&F'6#FL&F'6$%\"ZG
F*&F'6#FQ&%\"KGF/&FUF>&FUF2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "an
d give all of them the initial value of 0.1." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 29 "Start := [seq(x=0.1,x=Vars)];" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%&StartG76/&%\"CG6$%\"AG\"\"!$\"\"\"!\"\"/&F(6#F*F,/
&%\"eG6#%#R1GF,/&F46#%#R3GF,/&F(6$%\"BGF+F,/&F(6#F>F,/&F(6$F(F+F,/&F(6
#F(F,/&F46#%#R2GF,/&F(6$%\"DGF+F,/&F(6#FOF,/&F(6$%\"XGF+F,/&F(6#FVF,/&
F(6$%\"YGF+F,/&F(6#FgnF,/&F(6$%\"ZGF+F,/&F(6#F^oF,/&%\"KGF5F,/&FdoFJF,
/&FdoF9F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "result:=Newton
(Eqns,Start);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'resultG76/&%\"CG6$
%\"AG\"\"!$\"#:!\"\"/&F(6#F*$\"+Un`TU!#5/&%\"eG6#%#R1G$\"+7-A6qF4/&F76
#%#R3G$\"+\\ICZPF4/&F(6$%\"BGF+F,/&F(6#FE$\"+r'4YZ#F4/&F(6$F(F+F+/&F(6
#F($\"+!4]q\\\"F4/&F76#%#R2G$\"+A,<9bF4/&F(6$%\"DGF+F+/&F(6#FfnF:/&F(6
$%\"XGF+F+/&F(6#F]o$\"+tq#pw\"F4/&F(6$%\"YGF+F+/&F(6#FfoFW/&F(6$%\"ZGF
+F+/&F(6#F]pF@/&%\"KGF8$\"#5F./&FcpFU$\"$j#!\"#/&FcpF>$\"#]F." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 263 "This is the result obtained earli
er. As an exercise we suggest that readers try different starting poin
ts. It may also be advisable in some cases to arrange the equations in
 a form that does not involve potential divisions by zero (see the equ
ilibrium equations)." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }