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{SECT 0 {SECT 1 {PARA 258 "" 0 "" {TEXT 257 40 "Chapter 19\n\nLinear A
lgebra: Applications" }{TEXT 258 1 "\n" }}{PARA 19 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 256 65 "\251 Copyright 1997, Latin American
 Maple Center, Petr\363polis, Brasil" }}{PARA 0 "" 0 "" {HYPERLNK 17 "
About the Worksheets" 1 "about.mws" "" }}}{SECT 1 {PARA 3 "" 0 "Kinema
tics" {TEXT -1 43 "19.1 Kinematics of the Stanford Manipulator" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; 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 "" {MPLTEXT 1 0 44 "alias( seq( c[i] = cos(theta
[i]), i=1..6 ), " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "  seq( s[i] = s
in(theta[i]), i=1..6 ) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
52 "M := (a,alpha,d,theta) -> matrix( 4, 4, [cos(theta)," }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 48 "  -sin(theta)*cos(alpha), sin(theta)*sin(alph
a)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "  a*cos(theta), sin(theta), \+
cos(theta)*cos(alpha)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "  -cos(th
eta)*sin(alpha), a*sin(theta), 0, sin(alpha)," }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "  cos(alpha), d, 0, 0, 0, 1 ] ):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 20 "M(a,alpha,d,theta); " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7&7&-%$cosG6#%&thetaG,$*&-%$sinGF*\"\"\"-F)
6#%&alphaGF0!\"\"*&F.\"\"\"-F/F2F0*&%\"aGF0F(F07&F.*&F(F6F1F6,$*&F(F6F
7F6F4*&F9F6F.F67&\"\"!F7F1%\"dG7&F@F@F@F0" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 47 "a := vector([0$6]):                            " }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "alpha := vector([-Pi/2,Pi/2
,0,-Pi/2,Pi/2,0]): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "d :=
 vector([0$6]): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "d[2] :=
 evaln(d[2]): d[3] := evaln(d[3]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "theta := vector(6): theta[3] := 0:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 15 "for i to 6 do  " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "  A[i] := M( a[i], alpha[i], d[i], theta[i] )  " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "od;  " }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%\"AG6#\"\"\"-%'matrixG6#7&7&&%\"cGF&\"\"!,$&%\"sGF&!\"\"F/7&F
1F/F-F/7&F/F3F/F/7&F/F/F/F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG
6#\"\"#-%'matrixG6#7&7&&%\"cGF&\"\"!&%\"sGF&F/7&F0F/,$F-!\"\"F/7&F/\"
\"\"F/&%\"dGF&7&F/F/F/F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#
\"\"$-%'matrixG6#7&7&\"\"\"\"\"!F.F.7&F.F-F.F.7&F.F.F-&%\"dGF&7&F.F.F.
F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"%-%'matrixG6#7&7&&%
\"cGF&\"\"!,$&%\"sGF&!\"\"F/7&F1F/F-F/7&F/F3F/F/7&F/F/F/\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"&-%'matrixG6#7&7&&%\"cGF&
\"\"!&%\"sGF&F/7&F0F/,$F-!\"\"F/7&F/\"\"\"F/F/7&F/F/F/F6" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"'-%'matrixG6#7&7&&%\"cGF&,$&%\"sGF&
!\"\"\"\"!F37&F0F-F3F37&F3F3\"\"\"F37&F3F3F3F6" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 46 "Tip := evalm( `&*`( seq( A[i], i=1..6 ) ) ):  \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "collect( Tip[1,1], [c[1
],c[2],s[1]] );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&*&,&*(&%\"c
G6#\"\"%\"\"\"&F*6#\"\"&F-&F*6#\"\"'F-F-*&&%\"sGF+F-&F6F2F-!\"\"F-&F*6
#\"\"#F-F-*(&F6F:F-&F6F/F-F1\"\"\"F8F-&F*6#F-F-F-*&,&*(F5F?F.F?F1F?F8*
&F)F?F7F?F8F-&F6FAF-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "T
 := subvector( Tip, 1..3, 4 ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"TG-%'vectorG6#7%,&*(&%\"cG6#\"\"\"F.&%\"sG6#\"\"#F.&%\"dG6#\"\"$F.F.
*&&F0F-F.&F4F1F.!\"\",&*(F8\"\"\"F/F=F3F=F.*&F+F=F9F=F.*&&F,F1F.F3F=" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "R := submatrix( Tip, 1..3
, 1..3 ):  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Jv := jacobi
an( T, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "  [theta[1],theta[2],the
ta[4],theta[5],theta[6]] );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Jv
G-%'matrixG6#7%7',&*(&%\"sG6#\"\"\"F/&F-6#\"\"#F/&%\"dG6#\"\"$F/!\"\"*
&&%\"cGF.F/&F4F1F/F7*(F9\"\"\"&F:F1F/F3F=\"\"!F?F?7',&*(F9F=F0F=F3F=F/
*&F,F=F;F=F7*(F,F=F>F=F3F=F?F?F?7'F?,$*&F0F=F3F=F7F?F?F?" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "R1 := map( diff, R, theta[1] ):   \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "R2 := map( diff, R, the
ta[2] ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "R3 := map( diff
, R, theta[4] ):   " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "R4 :
= map( diff, R, theta[5] ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "R5 := map( diff, R, theta[6] ):  " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "Rtranspose := transpose( R ):  " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 58 "evalm( R1 &* Rtranspose ):  omega := map( sim
plify, % );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG-%'matrixG6#
7%7%\"\"!!\"\"F*7%\"\"\"F*F*7%F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "Jomega[1] := vector([omega[3,2],-omega[3,1],omega[2,1
]]);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'JomegaG6#\"\"\"-%'vector
G6#7%\"\"!F,F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for i fro
m 2 to 5 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "  evalm( R.i &* Rtra
nspose ): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "  omega := map( simpl
ify, % ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "  Jomega[i] := vector(
 [ omega[3,2], -omega[3,1], " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "   \+
 omega[2,1] ] ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "  print( Jomega
[i]=eval(Jomega[i]) )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "od:  " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'JomegaG6#\"\"#-%'vectorG6#7%,$&%\"
sG6#\"\"\"!\"\"&%\"cGF/\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'Jo
megaG6#\"\"$-%'vectorG6#7%*&&%\"sG6#\"\"#\"\"\"&%\"cG6#F1F1*&F-\"\"\"&
F.F4F1&F3F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'JomegaG6#\"\"%-%'ve
ctorG6#7%,&*(&%\"cG6#\"\"\"F1&F/6#\"\"#F1&%\"sGF&F1!\"\"*&&F6F0F1&F/F&
F1F7,&*(F9\"\"\"F2F=F5F=F7*&F.F=F:F=F1*&&F6F3F1F5F=" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#/&%'JomegaG6#\"\"&-%'vectorG6#7%,(**&%\"sGF&\"\"\"&%
\"cG6#F0F0&F26#\"\"#F0&F26#\"\"%F0F0*(F1\"\"\"&F/F5F0&F2F&F0F0*(F.F;&F
/F3F0&F/F8F0!\"\",(*(F.F;F1F;F@F;F0*(F?F;F<F;F=F;F0**F.F;F?F;F4F;F7F;F
0,&*(F<F;F7F;F.F;FA*&F4F;F=F;F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 53 "Jomega := concat( seq( eval(Jomega[i]), i=1..5 ) ):  " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "J := stackmatrix( Jv, Jomega
 );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%'matrixG6#7(7',&*(&%
\"sG6#\"\"\"F/&F-6#\"\"#F/&%\"dG6#\"\"$F/!\"\"*&&%\"cGF.F/&F4F1F/F7*(F
9\"\"\"&F:F1F/F3F=\"\"!F?F?7',&*(F9F=F0F=F3F=F/*&F,F=F;F=F7*(F,F=F>F=F
3F=F?F?F?7'F?,$*&F0F=F3F=F7F?F?F?7'F?,$F,F7*&F0F=F9F=,&*(F9F=F>F=&F-6#
\"\"%F/F7*&F,F=&F:FNF/F7,(**&F-6#\"\"&F/F9F=F>F=FQF=F/*(F9F=F0F=&F:FUF
/F/*(FTF=F,F=FMF=F77'F?F9*&F0F=F,F=,&*(F,F=F>F=FMF=F7*&F9F=FQF=F/,(*(F
TF=F9F=FMF=F/*(F,F=F0F=FXF=F/**FTF=F,F=F>F=FQF=F/7'F/F?F>*&F0F=FMF=,&*
(F0F=FQF=FTF=F7*&F>F=FXF=F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
25 "print( Jv = eval(Jv) );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#Jv
G-%'matrixG6#7%7',&*(&%\"sG6#\"\"\"F/&F-6#\"\"#F/&%\"dG6#\"\"$F/!\"\"*
&&%\"cGF.F/&F4F1F/F7*(F9\"\"\"&F:F1F/F3F=\"\"!F?F?7',&*(F9F=F0F=F3F=F/
*&F,F=F;F=F7*(F,F=F>F=F3F=F?F?F?7'F?,$*&F0F=F3F=F7F?F?F?" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs( s[2]=0, eval(Jv) );  " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7',$*&&%\"cG6#\"\"\"F-&%
\"dG6#\"\"#F-!\"\"*(F*\"\"\"&F+F0F-&F/6#\"\"$F-\"\"!F9F97',$*&&%\"sGF,
F-F.F4F2*(F=F4F5F4F6F4F9F9F97'F9F9F9F9F9" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "submatrix( %, 1..2, 1..2 );  " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7$7$,$*&&%\"cG6#\"\"\"F-&%\"dG6#\"\"#F-!\"
\"*(F*\"\"\"&F+F0F-&F/6#\"\"$F-7$,$*&&%\"sGF,F-F.F4F2*(F<F4F5F4F6F4" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "det(%);  " }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "
addrow( Jv, 3, 1, c[2]*c[1]/s[2] ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%'matrixG6#7%7',&*(&%\"sG6#\"\"\"F-&F+6#\"\"#F-&%\"dG6#\"\"$F-!\"\"
*&&%\"cGF,F-&F2F/F-F5\"\"!F:F:F:7',&*(F7\"\"\"F.F>F1F>F-*&F*F>F9F>F5*(
F*F>&F8F/F-F1F>F:F:F:7'F:,$*&F.F>F1F>F5F:F:F:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 36 "addrow( %, 3, 2, s[1]*c[2]/s[2] );  " }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7',&*(&%\"sG6#\"\"\"F-&F+6#\"
\"#F-&%\"dG6#\"\"$F-!\"\"*&&%\"cGF,F-&F2F/F-F5\"\"!F:F:F:7',&*(F7\"\"
\"F.F>F1F>F-*&F*F>F9F>F5F:F:F:F:7'F:,$*&F.F>F1F>F5F:F:F:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "\{ %[1,1]=0, %[2,1]=0 \};  " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/,&*(&%\"cG6#\"\"\"F*&%\"sG6#\"\"#F*
&%\"dG6#\"\"$F*F**&&F,F)F*&F0F-F*!\"\"\"\"!/,&*(F4\"\"\"F+F;F/F;F6*&F'
F;F5F;F6F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve( %, \{s
[1],c[1]\} ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/&%\"sG6#\"\"\"\"
\"!/&%\"cGF'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "Cadmium" {TEXT -1 46 "19.2 A 3-Compartment Model
 of Cadmium Transfer" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "rest
art; with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new defini
tion for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition \+
for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sequence := " 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "  seq(seq( a.i.j = a[i,j], i=0..3
 ), j=0..3 )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "  seq(seq( b.i.j =
 b[i,j], i=0..3 ), j=0..3 )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "  s
eq(seq( t.i.j = t[i,j], i=0..3 ), j=0..3 ), c3 = c[3]:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "(eval@subs)( S=sequence, 'macro(S)'
 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A := matrix( 3, 3, [
 -a21, a12, 0, a21, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "  -(a02+a12
+a32), a23, 0, a32, -(a03+a23) ] );  " }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"AG-%'matrixG6#7%7%,$&%\"aG6$\"\"#\"\"\"!\"\"&F,6$F/F.\"\"!7%F+,
(&F,6$F3F.F0F1F0&F,6$\"\"$F.F0&F,6$F.F:7%F3F8,&&F,6$F3F:F0F;F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "B := matrix( 3, 1, [0,1,0] )
;  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7%7#\"\"!7#\"
\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "C := matrix( 2, 3
, [0,0,c3,0,a02,a03] );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'
matrixG6#7$7%\"\"!F*&%\"cG6#\"\"$7%F*&%\"aG6$F*\"\"#&F16$F*F." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "AB := evalm( A &* B ):   A2B
 := evalm( A &* AB ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "MC
 := augment( B, AB, A2B );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MCG
-%'matrixG6#7%7%\"\"!&%\"aG6$\"\"\"\"\"#,&*&&F,6$F/F.F.F+F.!\"\"*&F+\"
\"\",(&F,6$F*F/F4F+F4&F,6$\"\"$F/F4F.F.7%F.F7,(F1F.*$)F7F/F6F.*&&F,6$F
/F<F.F:F.F.7%F*F:,&*&F:F6F7F6F.*&,&&F,6$F*F<F4FBF4F.F:F6F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "rank( MC ); " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ffg
ausselim( MC );  " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%
\"\"\",(&%\"aG6$\"\"!\"\"#!\"\"&F+6$F(F.F/&F+6$\"\"$F.F/,2*&&F+6$F.F(F
(F0F(F(*$)F*F.\"\"\"F(*&F0F;F*F(F.*&F*F;F2F(F.*$)F0F.F;F(*&F0F;F2F;F.*
$)F2F.F;F(*&&F+6$F.F4F(F2F;F(7%F-F0,*F6F/F<F/F>F/F@F/7%F-F-,(*(F0F;F2F
;&F+6$F-F4F(F/*(F0F;FDF;F2F;F/*(F2F;F7F;F0F;F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "map( factor, % );  " }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\",(&%\"aG6$\"\"!\"\"#!\"\"&F+6$F(F
.F/&F+6$\"\"$F.F/,2*&&F+6$F.F(F(F0F(F(*$)F*F.\"\"\"F(*&F0F;F*F(F.*&F*F
;F2F(F.*$)F0F.F;F(*&F0F;F2F;F.*$)F2F.F;F(*&&F+6$F.F4F(F2F;F(7%F-F0,$*&
F0F;,*F0F(F7F(F*F(F2F(F(F/7%F-F-*(F0F;F2F;,(&F+6$F-F4F/FDF/F7F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "factor( det( MC ) );  " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(&%\"aG6$\"\"\"\"\"#F(&F&6$\"\"$F)F
(,(&F&6$\"\"!F,!\"\"&F&6$F)F,F1&F&6$F)F(F(F(F1" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "CA := evalm( C &* A ):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "MO := stackmatrix( C, CA );  " }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%#MOG-%'matrixG6#7&7%\"\"!F*&%\"cG6#\"\"$7%F*&%\"
aG6$F*\"\"#&F16$F*F.7%F**&F+\"\"\"&F16$F.F3F8*&F+\"\"\",&F4!\"\"&F16$F
3F.F>F87%*&F0F8&F16$F3F8F8,&*&F0F<,(F0F>&F16$F8F3F>F9F>F8F8*&F9F<F4F8F
8,&*&F0F<F?F8F8*&F4F<F=F<F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "ffgausselim( MO );  " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'matrixG
6#7&7%*&&%\"aG6$\"\"!\"\"#\"\"\"&F*6$F-F.F.,**$)F)F-\"\"\"!\"\"*&&F*6$
F.F-F.F)F4F5*&F)F4&F*6$\"\"$F-F.F5*&F:F4&F*6$F,F<F.F.,(*&F)F4&F*6$F-F<
F.F.*$)F>F-F4F5*&F>F4FBF4F57%F,*&F3F4F/F4*(F>F4F)F4F/F47%F,F,*(&%\"cG6
#F<F.F3F4F/F47%F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "fa
ctor( det( transpose(MO) &* MO ) );  " }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#**)&%\"cG6#\"\"$\"\"#\"\"\")&%\"aG6$F)\"\"\"F)F*)&F-6$\"\"!F)F)F*,2
*$F0F*F/*&)&F-6$F(F)F)F*)&F-6$F3F(F)F*F/*(F0F*F;F/&F-6$F)F(F/F)*&F$F*F
7F*F/*&F0F*)F>F)F*F/*&F0F*F:F*F/**F1F/F8F/F;F*F>F*F)*(F1F*F8F*F:F*F)F/
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "collect( op(4,%), a32 )
; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,.*&,&*$)&%\"aG6$\"\"!\"\"$\"\"#
\"\"\"\"\"\"*$)&%\"cG6#F,F-F.F/F/)&F)6$F,F-F-F.F/*&,&*&&F)6$F+F-F/F'F.
F-*(F;F.F(F/&F)6$F-F,F/F-F/F6F/F/*$)F;F-F.F/*&FAF.F'F.F/*(FAF.F(F.F>F.
F-*&FAF.)F>F-F.F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "discri
m(%,a32);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,,*$)&%\"cG6#\"\"$
\"\"#\"\"\"\"\"\"*(F'F-&%\"aG6$\"\"!F+F.&F16$F,F+F.F,*&F'F-)F4F,F-F.*&
F'F-)F0F,F-F.*$F9F-F.F.)&F16$F3F,F,F-!\"%" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 17 "n := rowdim( A );" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"nG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "CA0B := ev
alm( C&*B ): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "for i from
 1 to 5 do CA.i.B := evalm( C &* A^i &* B ) od: " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 45 "J := stackmatrix( seq(CA.i.B, i=0..2*n-1) ): \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "J12 := map( diff, J, a1
2):  J21 := map( diff, J, a21): " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "J23 := map( diff, J, a23):  J32 := map( diff, J, a32)
: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "J02 := map( diff, J, \+
a02):  J03 := map( diff, J, a03): " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "JM := augment( J21, J12, J02, J23, J32, J03 ): " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ffgausselim( JM ): " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "map( factor, % ); " }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7.7(*&&%\"aG6$\"\"\"\"\"#F,&F*6$
\"\"!F-F,,,*&F.\"\"\"&F*6$F-F,F,F,*$)F.F-F3F-F(F-*&F.F3&F*6$\"\"$F-F,F
-*&F9F3&F*6$F0F;F,!\"\",4*&F4F3F)F3F,F6F;F(\"\"%F8FB*$)F)F-F3F,*&F)F3F
9F3F-*$)F9F-F3F,*&&F*6$F-F;F,F9F3F,F<F?,$*&F9F3,&F.F?F=F,F,F?,4F6F-F(F
-F8F-*&F.F3FIF3F,*&F=F3F.F3F?*&F=F3F)F3F?F<!\"#*$)F=F-F3F?*&F=F3FIF3F?
,$*&F9F3,,F9F,F.F,F)F,F=F-FIF,F,F?7(F0,$*&F)F3F7F3F?,$*(,(F.F-F)F,F9F,
F,F)F3F.F3F?F0*(FMF3F)F3F.F3*(F.F3F)F3F9F37(F0F0FZF0F0F07(F0F0F0**&%\"
cG6#F;F,F7F3F)F3F9F3**F^oF3F)F3F.F3,.F6F,F(F,FPF,FOF,F8F,F<F,F,*,F^oF3
F.F3F)F3F9F3,&F9F,F.F,F,7(F0F0F0F0**)F^oF-F3F9F3F)F3F7F3F07(F0F0F0F0F0
*.FgoF3F7F3F)F3,(FIF?F=F?F4F,F,FGF3,*F4F,F)F,FIF?F=F?F,7(F0F0F0F0F0F0F
\\pF\\pF\\pF\\pF\\p" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "H :=
 evalm( C &* (s-A)^(-1) &* B ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 56 "evalm( denom(H[1,1]) * H ) / collect(denom(H[1,1]),s);  " }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#*&-%'matrixG6#7$7#*(&%\"cG6#\"\"$\"\"
\",&&%\"aG6$\"\"#F.F.%\"sGF.F.&F16$F-F3F.7#*&,H*(F0F.&F16$\"\"!F-F.&F1
6$F=F3F.F.*(F4F.F>\"\"\"&F16$F3F-F.F.*(F;FAF5FAF0FAF.*(F0FAF5FAF4FAF.*
(F4FAFBFA&F16$F.F3F.F.*(F0FAF4FAF;FAF.*(F0FAF4FAFBFAF.*(F4FAF;FAF>FAF.
*(F4FAF;FAFGFAF.*&F0FA)F4F3FAF.*&F>FAFNFAF.*&FGFAFNFAF.*&F5FAFNFAF.*&F
NFAF;FAF.*&FNFAFBFAF.*$)F4F-FAF.*(F4FAF5FAF;FAF.*(F0FAF>FAF4FAF.*(F0FA
F>FAFBFAF.F.,&*&*(F>FAF/FA,(F;F.FBF.F4F.F.FAF9!\"\"F.*&*(F;FAF/FAF5FAF
AF9FgnF.F.FA,.FTF.*&,.FGF.F0F.F>F.F5F.F;F.FBF.F.FNFAF.*&,4*&F5FAF0FAF.
*&FBFAFGFAF.*&F>FAFBFAF.*&FBFAF0FAF.*&F;FAF>FAF.*&F;FAFGFAF.*&F>FAF0FA
F.*&F;FAF0FAF.*&F5FAF;FAF.F.F4FAF.F:F.FXF.FDF.Fgn" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 30 "collect( numer(H[1,1]), s );  " }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,&*(&%\"cG6#\"\"$\"\"\"&%\"aG6$F(\"\"#F)%\"sGF)F
)*(F%\"\"\"F*F0&F+6$F-F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "collect( numer(H[2,1]), s );  " }}{PARA 12 "" 1 "" {XPPMATH 20 "
6#,(*&&%\"aG6$\"\"!\"\"#\"\"\")%\"sGF)\"\"\"F**&,**&F%F-&F&6$F)F*F*F**
&&F&6$F(\"\"$F*F%F-F**&F%F-&F&6$F)F6F*F**&&F&6$F6F)F*F4F-F*F*F,F*F**&F
1F-,(F3F*F7F*F:F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "cf
s1 := \{ coeffs( expand(numer(H[1,1])), s ) \};  " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%cfs1G<$*(&%\"cG6#\"\"$\"\"\"&%\"aG6$F*\"\"#F+&F-6$F/
F+F+*&F'\"\"\"F,F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "cfs2 \+
:= \{ coeffs( expand(numer(H[2,1])), s ) \};  " }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%%cfs2G<%,**&&%\"aG6$\"\"!\"\"#\"\"\"&F)6$F,F-F-F-*&&F
)6$F+\"\"$F-F(\"\"\"F-*&F(F4&F)6$F,F3F-F-*&&F)6$F3F,F-F1F4F-,(*(F.F4F1
F4F(F4F-*(F.F4F(F4F6F4F-*(F1F4F9F4F.F4F-F(" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 48 "cfs3 := \{ coeffs( expand(denom(H[1,1])), s ) \}  \+
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "  minus \{1\};  " }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>%%cfs3G<%,4*&&%\"aG6$\"\"$\"\"#\"\"\"&F)6$F,F-F
-F-*&&F)6$F,F+F-&F)6$F-F,F-F-*&&F)6$\"\"!F,F-F1\"\"\"F-*&F1F9F.F9F-*&&
F)6$F8F+F-F6F9F-*&F<F9F3F9F-*&F6F9F.F9F-*&F<F9F.F9F-*&F(F9F<F9F-,.F3F-
F.F-F6F-F(F-F<F-F1F-,(*(F.F9F<F9F6F9F-*(F.F9F6F9F1F9F-*(F<F9F(F9F.F9F-
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "cfs := `union`( cfs.(1.
.3) ): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "eqns := map( x -
> x = subs( a12=b12, a21=b21, a23=b23,  " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "  a32=b32, a23=b23, a03=b03, a02=b02, x ), cfs );  " 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<)/*&&%\"cG6#\"\"$\"\"\"&%\"
aG6$F+\"\"#F,*&F(\"\"\"&%\"bGF/F,/&F.6$\"\"!F0&F4F7/,4*&F-F2&F.6$F0F,F
,F,*&&F.6$F0F+F,&F.6$F,F0F,F,*&F6F,F@F2F,*&F@F2F=F2F,*&&F.6$F8F+F,F6F2
F,*&FGF2FBF2F,*&F6F2F=F2F,*&FGF2F=F2F,*&F-F2FGF2F,,4*&F3F2&F4F>F,F,*&&
F4FAF,&F4FCF,F,*&F9F,FQF2F,*&FQF2FOF2F,*&&F4FHF,F9F2F,*&FVF2FRF2F,*&F9
F2FOF2F,*&FVF2FOF2F,*&F3F2FVF2F,/,.FBF,F=F,F6F,F-F,FGF,F@F,,.FRF,FOF,F
9F,F3F,FVF,FQF,/,*FJF,FFF,FDF,FLF,,*FXF,FUF,FSF,FZF,/,(*(F=F2FGF2F6F2F
,*(F=F2F6F2F@F2F,*(FGF2F-F2F=F2F,,(*(FOF2FVF2F9F2F,*(FOF2F9F2FQF2F,*(F
VF2F3F2FOF2F,/*(F(F2F-F2F=F2*(F(F2F3F2FOF2" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 35 "vars := \{b12,b21,b23,b32,b02,b03\}: " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(eqns,vars);  " }}{PARA 12 "" 
1 "" {XPPMATH 20 "6$<(/&%\"bG6$\"\"!\"\"#&%\"aGF'/&F&6$F(\"\"$&F+F./&F
&6$F/F)&F+F3/&F&6$\"\"\"F)&F+F7/&F&6$F)F/&F+F</&F&6$F)F8&F+F@<(F$/F-,$
*&,,*&F0F8F*F8!\"\"*&F4F8F0\"\"\"FH*&F9F8F*FJF8*&F*FJFAF8F8*&F*FJF=F8F
HFJF4!\"\"FHF1/F6,(F0F8F=F8FAFH/F;*&,0FGFHFIFHFKF8FLF8FMFH*&F9FJF4FJF8
*&F4FJFAFJF8FJF4FNF>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "AA \+
:= subs( a12=b12, a21=b21, a23=b23, a32=b32,                         \+
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "  a23=b23, a03=b03, a02=b02, ev
al(A) ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AAG-%'matrixG6#7%7%,$&
%\"bG6$\"\"#\"\"\"!\"\"&F,6$F/F.\"\"!7%F+,(&F,6$F3F.F0F1F0&F,6$\"\"$F.
F0&F,6$F.F:7%F3F8,&&F,6$F3F:F0F;F0" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "BB := copy(B):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 42 "CC := subs( a02=b02, a03=b03, eval(C) );  " }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#CCG-%'matrixG6#7$7%\"\"!F*&%\"cG6#\"\"$7%F*&%\"bG6
$F*\"\"#&F16$F*F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "T := m
atrix( 3, 3, (i,j)->t[i,j] );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"TG-%'matrixG6#7%7%&%\"tG6$\"\"\"F-&F+6$F-\"\"#&F+6$F-\"\"$7%&F+6$F0F
-&F+6$F0F0&F+6$F0F37%&F+6$F3F-&F+6$F3F0&F+6$F3F3" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 21 "evalm( T&*BB - B );  " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7#&%\"tG6$\"\"\"\"\"#7#,&&F)6$F,F,F+!\"\"
F+7#&F)6$\"\"$F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "t12 := \+
0:  t32:= 0:  t22 := 1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "
evalm( CC - C&*T );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7
$7%,$*&&%\"cG6#\"\"$\"\"\"&%\"tG6$F-F.F.!\"\"\"\"!,&F*F.*&F*\"\"\"&F06
$F-F-F.F27%,&*&&%\"aG6$F3\"\"#F.&F06$F?F.F.F2*&&F=6$F3F-F.F/F6F2,&F<F2
&%\"bGF>F.,(&FGFDF.*&F<F6&F06$F?F-F.F2*&FCF6F7F6F2" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 19 "map( factor, % );  " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7$7%,$*&&%\"cG6#\"\"$\"\"\"&%\"tG6$F-F.F.!
\"\"\"\"!,$*&F*\"\"\",&F2F.&F06$F-F-F.F.F27%,&*&&%\"aG6$F3\"\"#F.&F06$
F@F.F.F2*&&F>6$F3F-F.F/F6F2,&F=F2&%\"bGF?F.,(&FHFEF.*&F=F6&F06$F@F-F.F
2*&FDF6F8F.F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "t31 := 0: \+
 t33 := 1:  b02 := a02:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "
evalm( CC - C&*T ):  map( factor, % );  " }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7$7%\"\"!F(F(7%,$*&&%\"aG6$F(\"\"#\"\"\"&%\"tG6$F/F0
F0!\"\",&F,F4&%\"bGF.F0,(&F76$F(\"\"$F0*&F,\"\"\"&F26$F/F;F0F4&F-F:F4
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "t21 := 0:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn1 := evalm(CC - C&*T)[2,3];  " }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G,(&%\"bG6$\"\"!\"\"$\"\"\"*&&
%\"aG6$F)\"\"#F+&%\"tG6$F0F*F+!\"\"&F.F(F4" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "evalm( T&*AA - A&*T );  " }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7%,&*&&%\"tG6$\"\"\"F-F-&%\"bG6$\"\"#F-F-
!\"\"*&&%\"aGF0F-F*\"\"\"F-,(*&F*F6&F/6$F-F1F-F-*&&F+6$F-\"\"$F-&F/6$F
>F1F-F-&F5F:F2,(*&F<F6,&&F/6$\"\"!F>F2&F/6$F1F>F2F-F-*&F4F6F<F6F-*&FAF
-&F+FIF-F27%,&F.F-F3F2,,F9F2F?F2*&FLF6F?F6F-FAF-&F5F@F-,,FHF-*&FLF6FDF
6F-FJF2*&,(&F56$FGF1F2FAF2FQF2F-FLF6F2&F5FIF27%FG,&FQF2F?F-,,FEF2FHF2*
&FQF-FLF6F2&F5FFF-FXF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "e
qn2 := map( op, convert(%,'listlist') ): " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 39 "eqns := \{ eqn1, op(eqn2) \} minus \{0\};  " }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<+,,&%\"bG6$\"\"#\"\"$\"\"\"*&
&%\"tGF)F,,&&F(6$\"\"!F+!\"\"F'F4F,F,*&&%\"aG6$F*F,F,&F/6$F,F+F,F4*&,(
&F76$F3F*F4&F76$F,F*F4&F76$F+F*F4F,F.\"\"\"F4&F7F)F4,&FAF4&F(FBF,,&*&&
F/6$F,F,F,&F(F8F,F4*&F6FCFIFCF,,&FKF,FLF4,(*&FIFC&F(F@F,F,*&F9FCFFF,F,
F?F4,(*&F9FCF0FCF,F5F,*&F?F,F.FCF4,,FPF4FFF4*&F.FCFFFCF,F?F,FAF,,,F1F4
F'F4*&FAF,F.FCF4&F7F2F,FDF,,(F1F,*&F=F,F.FCF4FYF4" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 50 "solve( eqns, \{b12,b32,b23,b03,b21,t11,t13,t
23\} ); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6$<*/&%\"tG6$\"\"\"\"\"$\"\"
!/&F&6$\"\"#F)F*/&%\"bG6$F)F.&%\"aGF2/&F16$F.F(&F4F7/&F16$F*F)&F4F;/&F
16$F(F.&F4F?/&F1F-&F4F-/&F&6$F(F(F(<*/F,,$*&,*F<!\"\"FCFLF8F(F@F(\"\"
\"F3!\"\"FL/F%FJF/F5/FB*&,0*&F<F(&F46$F*F.F(FL*&FTFMFCF(FL*&FTFMF8F(F(
*&F@F(FTFMF(*&F3F(F<FMFL*&F3FMF8FMF(*&F@FMF3FMF(FMF3FN/F:,$*&,,FSFLFVF
LFWF(FXF(FYFLFMF3FNFL/F>,(F<F(FCF(F8FLFD" }}}}{SECT 1 {PARA 3 "" 0 "Hu
eckel" {TEXT -1 37 "19.3 Molecular-Orbital Hueckel Theory" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(linalg):" }}{PARA 7 "
" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "
" {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 26 "`number of C-atoms` := 10:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "Azulene_skeleton := " }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 47 "  [ [1,2], [1,9], [2,3], [3,10], [4,5], [4,10]," }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "    [4,10], [5,6], [6,7], [7,8], [8
,9], [9,10] ]:  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Hueckel
_matrix := array( sparse," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "  1..`
number of C-atoms`, 1..`number of C-atoms` ):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 43 "for i from 1 to nops( Azulene_skeleton ) do" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "  Hueckel_matrix[ op( Azulene_skele
ton[i] ) ] := beta" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "od: " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Hueckel_matrix := evalm( alp
ha + Hueckel_matrix  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "  + transp
ose( Hueckel_matrix ) ): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
46 "topological_matrix := subs( alpha=0, beta=1,  " }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 29 "  eval( Hueckel_matrix ) ):  " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 46 "factor( charpoly( topological_matrix, x ) ); \+
 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,,*$)%\"xG\"\"%\"\"\"\"\"\"*$)F
'\"\"$F)F**$)F'\"\"#F)!\"$F'!\"\"F*F*F*,0*$)F'\"\"'F)F**$)F'\"\"&F)F2F
%!\"(F+F9F.\"#8F'!\"'!\"%F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 35 "charmat( topological_matrix, x );  " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7,7,%\"xG!\"\"\"\"!F*F*F*F*F*F)F*7,F)F(F)F*
F*F*F*F*F*F*7,F*F)F(F*F*F*F*F*F*F)7,F*F*F*F(F)F*F*F*F*F)7,F*F*F*F)F(F)
F*F*F*F*7,F*F*F*F*F)F(F)F*F*F*7,F*F*F*F*F*F)F(F)F*F*7,F*F*F*F*F*F*F)F(
F)F*7,F)F*F*F*F*F*F*F)F(F)7,F*F*F)F)F*F*F*F*F)F(" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 19 "factor( det(%) );  " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&,,*$)%\"xG\"\"%\"\"\"\"\"\"*$)F'\"\"$F)F**$)F'\"\"#F)
!\"$F'!\"\"F*F*F*,0*$)F'\"\"'F)F**$)F'\"\"&F)F2F%!\"(F+F9F.\"#8F'!\"'!
\"%F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eigenvalues( top
ological_matrix );  " }}{PARA 12 "" 1 "" {XPPMATH 20 "6$-%'RootOfG6#,,
*$)%#_ZG\"\"%\"\"\"\"\"\"*$)F)\"\"$F+F,*$)F)\"\"#F+!\"$F)!\"\"F,F,-F$6
#,0*$)F)\"\"'F+F,*$)F)\"\"&F+F4F'!\"(F-F=F0\"#8F)!\"'!\"%F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "eigenvectors( topological_ma
trix );  " }}{PARA 12 "" 1 "" {XPPMATH 20 "6$7%-%'RootOfG6#,0*$)%#_ZG
\"\"'\"\"\"\"\"\"*$)F*\"\"&F,!\"\"*$)F*\"\"%F,!\"(*$)F*\"\"$F,F0*$)F*
\"\"#F,\"#8F*!\"'!\"%F-F-<#-%'vectorG6#7,F-,.F$#F<F;!\"$F-*$)F$F;F,#F0
F;*$)F$F4F,#F1F;*$)F$F0F,#F-F;*$)F$F8F,#F5F;F-,.F=F-FG#\"\"(F;FJFLFMFO
FPFRF$#\"#6F;,.F$FIFP#!\"&F;F1F-FG#F8F;FJFLFMFO,.FG#!#:F;FJFenF$FRFPFI
\"\")F-FMFLFXFS,.F8F-FGFYFJFOFMFLFPFTF$#!#6F;Fjn7%-F%6#,,F2F-F6F-F9FFF
*F1F-F-F-<#-FA6#7,,*FFF-*$)F^oF;F,F;F^o!\"#*$)F^oF8F,F-\"\"!,*F8F-F^oF
;FioF1FfoFhoF1,*F^oFFFioF-FfoF-F1F-F[p,*F-F-F^oF8FfoF1FioF1F-,*F^oFhoF
ioF-FfoF-F1F-,*F^oF;FioF1FfoF1F-F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "%[2];  " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%-%'RootOf
G6#,,*$)%#_ZG\"\"%\"\"\"\"\"\"*$)F*\"\"$F,F-*$)F*\"\"#F,!\"$F*!\"\"F-F
-F-<#-%'vectorG6#7,,*F4F-*$)F$F3F,F3F$!\"#*$)F$F0F,F-\"\"!,*F0F-F$F3F?
F5F<F>F5,*F$F4F?F-F<F-F5F-FA,*F-F-F$F0F<F5F?F5F-,*F$F>F?F-F<F-F5F-,*F$
F3F?F5F<F5F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "allvalues
(%,'dependent')[1];   " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%,(#!\"\"\"
\"%\"\"\"*$-%%sqrtG6#\"\"&\"\"\"#F(F'*$-F+6#,&\"#AF(F)!\"#F.F/F(<#-%'v
ectorG6#7,,,#!\"&\"\"#F(*$)F$F>F.F>F)#F&F>F0FA*$)F$\"\"$F.F(\"\"!,,#F-
F>F(F)#F(F>F0FHFBF&F?F5F&,,F%F(F)#!\"$F'F0FJFBF(F?F(FE,,F/F(F)#FDF'F0F
MF?F&FBF&F(,,FAF(F)FAF0FAFBF(F?F(,,FHF(F)FHF0FHFBF&F?F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eigenvalues := evalf( " }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 54 "  Eigenvals( topological_matrix, 'eigenvecto
rs' ) );  " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,eigenvaluesG-%'vector
G6#7,$!+&)RH&4#!\"*$!+%)R@p=F+$!+*4=#z:F+$!+HISwt!#5$!+)=BR+%F2$\"+w**
fsZF2$\"+?Cvp))F2$\"+$HucN\"F+$\"+9Bd^;F+$\"+UoF5BF+" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 54 "subvector( eigenvectors, 1..`number of C-
atoms`, 1 ); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7,$!++jy!
f#!#5$\"%\")=!#7$\"+!G'y!f#F)$\"+<@(\\N$F)$!+iR>,;F)$!%!>#F,$\"+/S>,;F
)$!+Q@(\\N$F)$\"+]*e%GaF)$!+B*e%GaF)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "evalm( ( topological_matrix  " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "- eigenvalues[1] ) &* % );  " }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7,$\"#=!#5$\"#>F)$\"\"&F)$\"#7F)$\"%5=!#7$!
\"%F)F,$\"#8F)$\"#HF)$\"\"#!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 50 "map( fnormal, % ); # floating-point normalization " }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7,\"\"!F'F'F'F'F'F'F'F'F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "occupation_numbers := [0,0,0
,0,0,2,2,2,2,2,2]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "pi_el
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{MPLTEXT 1 0 37 "  * ( alpha + eigenvalues[i]*beta )'," }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 33 "  'i'=1..`number of C-atoms` );  " }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%3pi_electron_energyG,&%&alphaG\"#5%%betaG$\"+
P<NO8!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eigenvectors :
= map( normalize,  [ seq(" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "  subv
ector( eigenvectors, 1..`number of C-atoms`, i )," }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 35 "  i=1..`number of C-atoms` ) ] ):  " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "electron_density := seq( sum( 'occu
pation_numbers[i] *  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "  (eigenve
ctors[i][j])^2', 'i'=1..`number of C-atoms` ), " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "  j=1..`number of C-atoms` ); " }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%1electron_densityG6,$\"+S$zG<\"!\"*$\"+3(*fY5F($\"+R$
zG<\"F($\"+cmX\\&)!#5$\"+b%oW')*F/$\"+-Q,+()F/$\"+$[oW')*F/$\"+KmX\\&)
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" }}}}{SECT 1 {PARA 3 "" 0 "VectorAnalysis" {TEXT -1 20 "19.4 Vector A
nalysis" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "restart; \ninterf
ace(showassumed=2): \nwith(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Wa
rning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning
, new definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
55 "alias( f=f(x,y,z), g=g(x,y,z), h=h(x,y,z), c=[x,y,z] ):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grad( f, c );  " }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%-%%diffG6$%\"fG%\"xG-F(6$F*%\"yG-
F(6$F*%\"zG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "diverge( [f,
g,h], c );  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%%diffG6$%\"fG%\"xG
\"\"\"-F%6$%\"gG%\"yGF)-F%6$%\"hG%\"zGF)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "laplacian( f, c ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,(-%%diffG6$%\"fG-%\"$G6$%\"xG\"\"#\"\"\"-F%6$F'-F)6$%\"yGF,F--F%6$F
'-F)6$%\"zGF,F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "jacobian
( [f,g,h], c ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%-%
%diffG6$%\"fG%\"xG-F)6$F+%\"yG-F)6$F+%\"zG7%-F)6$%\"gGF,-F)6$F6F/-F)6$
F6F27%-F)6$%\"hGF,-F)6$F>F/-F)6$F>F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "hessian( f, c ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'matrixG6#7%7%-%%diffG6$%\"fG-%\"$G6$%\"xG\"\"#-F)6%F+F/%\"yG-F)6%F+
F/%\"zG7%F1-F)6$F+-F-6$F3F0-F)6%F+F3F67%F4F<-F)6$F+-F-6$F6F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "curl( [f,g,h], c ); " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,&-%%diffG6$%\"hG%\"yG\"
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{XPPMATH 20 "6#-%'vectorG6#7%\"\"!F'F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "plots[coordplot3d]( prolatespheroidal );" }}{PARA 13 
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "alias( f=f(u,v,w), g=g(u,v,w
), h=h(u,v,w), c=[u,v,w] ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
44 "grad( f, c, coords = prolatespheroidal(a) );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7%*&-%%diffG6$%\"fG%\"uG\"\"\"*&%\"aG\"\"\"
-%%sqrtG6#,&*$)-%%sinhG6#F,\"\"#F-\"\"\"*$)-%$sinG6#%\"vGF:F-F;F-!\"\"
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oseproc = 2 ): # make Maple communicative" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 56 "readlib( `plots/changecoords` ): # load library fun
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gecoords` ); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#R6$%\"pG%&coordG6&%'c
_nameG%\"aG%\"bG%\"cG6#%aoCopyright~(c)~1994~by~the~University~of~Wate
rloo.~All~rights~reserved.G6\"C'>8%\"\"\">8&#F2\"\"#>8'#F2\"\"$@%-%%ty
peG6$9%%)functionGC$>8$-%#opG6$\"\"!F?@)/-%%nopsG6#F?F2>F1-FE6$F2F?/FJ
F6C$>F1FN>F4-FE6$F6F?/FJF:C%>F1FN>F4FT>F8-FE6$F:F?-%&ERRORG6#%DInappro
priate~number~of~parameters.G>FCF?@'-%'memberG6$FC--%(readlibG6#%1plot
3d/coordset2GF.-%<plots/changecoords/twotransG6%9$FCF1-F^o6$FC--Fbo6#%
0plot3d/coordsetGF.-%>plots/changecoords/threetransG6'FhoFCF1F4F8-Fhn6
$%DCannot~convert~to~coordinate~systemGF?F.F.F." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 26 "setcoords := proc( coord )" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "  local c_name;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
40 "  global curl, diverge, grad, laplacian," }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "         scalefactors, `setcoords/cname`;" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 30 "  if type(coord,function) then" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 30 "        c_name := op(0,coord);" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 28 "     if nops(coord) > 3 then" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 52 "        ERROR(`Inappropriate number of parameter
s.`)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "     fi" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "  else c_name := coord" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "  fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "  if membe
r(c_name,readlib(`plot3d/coordset2`)()) or" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "     member(c_name,readlib(`plot3d/coordset`)())" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "  then" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "     `setcoords/cname` := coord," }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 48 "     unprotect( curl, diverge, grad, laplacian, " }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "                scalefactors ):" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "     curl := proc() linalg[curl]( \+
args," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "             coords = `set
coords/cname` ) end;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "     diverg
e := proc() linalg[diverge]( args," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
49 "                coords = `setcoords/cname` ) end;" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 39 "     grad := proc() linalg[grad]( args," }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "             coords = `setcoords/cn
ame` ) end;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "     laplacian := pr
oc() linalg[laplacian]( args," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "  \+
                coords = `setcoords/cname` ) end;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 57 "     scalefactors := proc() readlib(`linalg/scalefc
ts`)( " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "                     args
, `setcoords/cname` ) end;              " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "     protect( curl, diverge, grad, laplacian, " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "               scalefactors )" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "  else" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "     ERROR(`Cannot convert to coordinate system`,coor
d)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "  fi" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "set
coords( prolatespheroidal(a) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "laplacian( f, c );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,,**%
\"aG\"\"\"-%%coshG6#%\"uGF'-%$sinG6#%\"vGF'-%%diffG6$%\"fGF+F'F'**F&\"
\"\"-%%sinhGF*F'F,F5-F16$F3-%\"$G6$F+\"\"#F'F'**F&F5F6F5-%$cosGF.F'-F1
6$F3F/F'F'**F&F5F6F5F,F5-F16$F3-F;6$F/F=F'F'*&*(F&F5,&*$)F6F=F5F'*$)F,
F=F5F'F'-F16$F3-F;6$%\"wGF=F'F5*&F6\"\"\"F,\"\"\"!\"\"F'F5**)F&\"\"$F5
FJ\"\"\"F6\"\"\"F,\"\"\"FW" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
18 "scalefactors( c );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*&%\"aG\"
\"\"-%%sqrtG6#,&*$)-%%sinhG6#%\"uG\"\"#\"\"\"F&*$)-%$sinG6#%\"vGF1F2F&
F2F$*(F%F2F-F&F5F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "dV :=
 convert( %, `*` ); # the volume element" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#dVG**)%\"aG\"\"$\"\"\",&*$)-%%sinhG6#%\"uG\"\"#F)\"\"\"*$)-%$
sinG6#%\"vGF1F)F2F2F-F2F5F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
51 "readlib( addcoords ):   # load the library function" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "addcoords(" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "  myprolatespheroidal,  # name of coordinate system" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "  [xi,eta,phi],         # list of
 variables" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "  [a*sqrt((xi^2-1)*(1
-eta^2))*cos(phi), " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "   a*sqrt((x
i^2-1)*(1-eta^2))*sin(phi), " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "   \+
a*xi*eta ],        # the cartesian coordinates x,y,z" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 55 "                      # in terms of the new coor
dinates" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "  [a],                # \+
list of names of constants" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "  [[1
.1],[sqrt(2)/2],[Pi/3],        # default values for" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 51 "   [1..1.2,-1..1,0..2*Pi],          # coordplot or
 " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "   [-1.2..1.2,-1.2..1.2,-1.7..
1.7]] # coordplot3d" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "assume( a>0, xi>=1, -1<=eta,
 eta<=1, 0<=phi, phi<=2*Pi );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 45 "alias( f = f(xi,eta,phi), c = [xi,eta,phi] ):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 36 "setcoords( myprolatespheroidal(a) ):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grad( f, c );" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%*&-%%diffG6$%\"fG%$xi|irG\"\"\"*$-
%%sqrtG6#,(*&)%#a|irG\"\"#F-)-%$absG6#*&*(F,\"\"\"-%$cosG6#%%phi|irGF=
,&!\"\"F=*$)%%eta|irGF6F-F=F-F-*$-F06#,$*&,&*$)F,F6F-F=FCF=F=FBF=FCF-!
\"\"F6F-F=*&F4F-)-F96#*&*(-%$sinGF@F=F,F-FBF-F-*$-F06#FJF-FOF6F-F=*&F4
F-)-F96#FFF6F-F=F-FO*&-F)6$F+FFF-*$-F06#,(*&F4F-)-F96#*&*(F>F-FLF-FFF=
F-*$-F06#FJF-FOF6F-F=*&F4F-)-F96#*&*(FVF-FLF-FFF-F-*$-F06#FJF-FOF6F-F=
*&F4F-)-F96#F,F6F-F=F-FO*&-F)6$F+FAF-*$-F06#,&*&F4F-)-F96#*&-F06#FJF-F
VF-F6F-F=*&F4F-)-F96#*&FbqF-F>F-F6F-F=F-FO" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 34 "map( simplify@combine@factor, % );" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'vectorG6#7%*&*&-%%diffG6$%\"fG%$xi|irG\"\"\"-%%
sqrtG6#,&*$)F-\"\"#\"\"\"F.!\"\"F.F6F6*&%#a|irG\"\"\"-F06#,&*$)%%eta|i
rGF5F6F7F3F.F6!\"\",$*&*&-F*6$F,F@F.,&F7F.F>F.F.F6*(F9\"\"\"-F06#F=F6-
F06#,&F.F.F>F7F6FAF7*&-F*6$F,%%phi|irGF6*(F9\"\"\"-F06#F2F6-F06#FNF6FA
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "scalefactors( c ):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "map( simplify@combine@factor
, % );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*&*&%#a|irG\"\"\"-%%sqrtG6
#,&*$)%%eta|irG\"\"#\"\"\"!\"\"*$)%$xi|irGF/F0F'F0F0*$-F)6#,&F2F'F1F'F
0!\"\",$*&*(F&F0F(F0-F)6#,&F'F'F,F1F0F0,&F1F'F,F'F9F1*&F&F0-F)6#*&F8F'
F?F'F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "dV := simplify( c
onvert( %, `*` ) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dVG,$*&,&*$)
%%eta|irG\"\"#\"\"\"\"\"\"*$)%$xi|irGF+F,!\"\"F-)%#a|irG\"\"$F,F1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; 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 "" {MPLTEXT 1 0 57 "assume( a>0, xi>=1, -1<=eta,
 eta<=1, 0<=phi, phi<=2*Pi );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 45 "alias( f = f(xi,eta,phi), c = [xi,eta,phi] ):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 43 "T := [a*sqrt((xi^2-1)*(1-eta^2))*cos(phi)
, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "      a*sqrt((xi^2-1)*(1-eta^
2))*sin(phi), a*xi*eta ];  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG7
%*(%#a|irG\"\"\"-%%sqrtG6#*&,&*$)%$xi|irG\"\"#\"\"\"F(!\"\"F(F(,&F(F(*
$)%%eta|irGF1F2F3F(F2-%$cosG6#%%phi|irGF(*(F'F2F)F2-%$sinGF:F(*(F'F2F0
F(F7F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "J := jacobian(T,[
xi,eta,phi]);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%'matrixG6#7
%7%*&**%#a|irG\"\"\"-%$cosG6#%%phi|irGF-%$xi|irGF-,&F-F-*$)%%eta|irG\"
\"#\"\"\"!\"\"F8F8*$-%%sqrtG6#*&,&*$)F2F7F8F-F9F-F-F3F-F8!\"\",$*&**F,
F8F.F8F?F8F6F-F8*$-F<6#F>F8FBF9,$*(F,F8-F<6#F>F8-%$sinGF0F-F97%*&**F,F
8FMF8F2F8F3F8F8*$-F<6#F>F8FB,$*&**F,F8FMF8F?F8F6F8F8*$-F<6#F>F8FBF9*(F
,F8FKF8F.F87%*&F,F8F6F8*&F,F8F2F8\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "g := evalm( transpose(J) &* J ):  g := map(simplify,g
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG-%'matrixG6#7%7%*&*&)%#a|i
rG\"\"#\"\"\",&*$)%%eta|irGF.F/!\"\"*$)%$xi|irGF.F/\"\"\"F8F/,&F5F8F4F
8!\"\"\"\"!F;7%F;,$*&*&F,F/F0F/F/,&F4F8F1F8F:F4F;7%F;F;,**&F,F/F6F/F8*
(F,F/F6F/F2F/F4*$F,F/F4*&F,F/F2F/F8" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "h1 := sqrt( g[1,1] );" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#h1G*&%#a|irG\"\"\"-%%sqrtG6#*&,&*$)%%eta|irG\"\"#\"\"\"!\"\"*
$)%$xi|irGF0F1F'F1,&F3F'F2F'!\"\"F1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "h2 := sqrt( g[2,2] );" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#h2G*&%#a|irG\"\"\"-%%sqrtG6#*&,&*$)%%eta|irG\"\"#\"\"\"!\"\"*
$)%$xi|irGF0F1F'F1,&F'F'F-F2!\"\"F1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "h3 := sqrt( factor(g[3,3]) );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#h3G*&%#a|irG\"\"\"-%%sqrtG6#**,&F'F'%%eta|irG!\"\"F'
,&F-F'F'F'F',&%$xi|irGF'F.F'F',&F1F'F'F'F'\"\"\"" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 42 "sftable := readlib( `linalg/scaletable` ):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sftable[myprolatespheroida
l] := subs(" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 " \{ a=_a, xi=_x, eta
=_y, phi=_z \}, [h1,h2,h3] );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%(s
ftableG6#%4myprolatespheroidalG7%*&%#_aG\"\"\"-%%sqrtG6#*&,&*$)%#_xG\"
\"#\"\"\"F+*$)%#_yGF4F5!\"\"F5,&F1F+F9F+!\"\"F5*&F*F5-F-6#*&F0F5,&F+F+
F6F9F;F5*&F*F5-F-6#**,&F+F+F8F9F+,&F8F+F+F+F+,&F3F+F9F+F+,&F3F+F+F+F+F
5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "rtable := readlib(`plo
t3d/coordSystems`):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "rtab
le[myprolatespheroidal] := subs( \{a=_a, eta=_x, psi=_y, z=_z\}, T );
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%'rtableG6#%4myprolatespheroidal
G7%*(%#_aG\"\"\"-%%sqrtG6#*&,&*$)%$xi|irG\"\"#\"\"\"F+!\"\"F+F+,&F+F+*
$)%#_xGF4F5F6F+F5-%$cosG6#%%phi|irGF+*(F*F5F,F5-%$sinGF=F+*(F*F5F3F+F:
F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "laplacian( f, c, coor
ds=myprolatespheroidal(a) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 18 "map( combine, % ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 
"collect( %, [ diff(f,xi$2), diff(f,eta$2), diff(f,xi)," }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 43 "  diff(f,eta), f ], distributed, simplify):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "map(combine, %):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "map(normal, %, expanded);" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,*&,&*&-%%diffG6$%\"fG-%\"$G6$%$xi|i
rG\"\"#\"\"\")F.F/\"\"\"F0F'!\"\"F2,&*&)%#a|irGF/F2)%%eta|irGF/F2F3*&F
6F2F1F2F0!\"\"F0*&,&-F(6$F*-F,6$F9F/F0*&F>F0F8F2F3F2F4F;F0*&*&F.F0-F(6
$F*F.F0F2F4F;F/*&*&F9F0-F(6$F*F9F0F2F4F;!\"#*&-F(6$F*-F,6$%%phi|irGF/F
2,**(F6F2F1F2F8F2F0F5F3F:F3*$F6F2F0F;F3" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "collect( %, [ diff(f,xi$2), diff(f,eta$2), diff(f,xi)
," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "  diff(f,eta), f ], distribute
d, simplify);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,*&*&%$xi|irG\"\"\"-
%%diffG6$%\"fGF&F'\"\"\"*&)%#a|irG\"\"#F,,&*$)%%eta|irG\"\"#F,!\"\"*$)
F&F5F,F'\"\"\"!\"\"F5*&*&,&F7F'F6F'F'-F)6$F+-%\"$G6$F&F5F'F,*&)F/\"\"#
F,F1\"\"\"F:F'*&*&,&F6F'F2F'F'-F)6$F+-FA6$F4F5F'F,*&)F/\"\"#F,F1\"\"\"
F:F6*&*&F4F'-F)6$F+F4F'F,*&)F/\"\"#F,F1\"\"\"F:!\"#*&-F)6$F+-FA6$%%phi
|irGF5F,*&)F/\"\"#F,,**&F8F,F3F,F'F2F6F7F6F'F'\"\"\"F:F6" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "MoorePen
rose" {TEXT -1 26 "19.5 Moore-Penrose Inverse" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "restart; with(linalg):" }}{PARA 7 "" 1 "" {TEXT 
-1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 
33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "MPinv := A -> map( limit, evalm( " }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 46 "  evalm( (linalg[transpose](A)&*A+x^2)^(-1) ) " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "  &* linalg[transpose](A) ), x=0 );
  " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&MPinvGR6#%\"AG6\"6$%)operator
G%&arrowGF(-%$mapG6%%&limitG-%&evalmG6#-%#&*G6$-F16#*&\"\"\"F9,&-F46$-
&%'linalgG6#%*transposeG6#9$FC\"\"\"*$)%\"xG\"\"#F9FD!\"\"F=/FG\"\"!F(
F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "M := randmatrix(3,2
);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7%7$!#&)!#b7
$!#P!#N7$\"#(*\"#]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "MPinv
(M);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7%#\"$F%\"&d*)
)#\"%zDF*#\"%vAF*7%#!&$49\"'q&*))#!&`f%F2#!&R\\\"\"'&yW%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "poly := proc() Randpoly(2,y) mod 2 \+
end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "M := randmatrix(3,2
,entries=poly);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6
#7%7$,&*$)%\"yG\"\"#\"\"\"\"\"\"F-F0,&F+F0F0F07$F+F+7$F1F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "MPinv(M);  " }}{PARA 259 "" 0 "" 
{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7%*&*&)%
\"yG\"\"#\"\"\",(*$)F+\"\"$F-F,*$F*F-!\"$!\"\"\"\"\"F5F-,.*$)F+\"\"'F-
F,*$)F+\"\"%F-F9F2F<*$)F+\"\"&F-!\"'F/!\"#F5F5!\"\",$*&*&F*F-,(F4F5F/F
5F+F5F5F-F6FBF4,$*&,,F:F3F2F3F4F5F=F5F/F5F-F6FBF47%,$*&,,F=F,F:F3F/F5F
2F3F4F5F-F6FBF4*&*&F*F-,*F2F5F/F5F+F4F5F5F5F-F6FB*&*&F+F5,,F/FAF+F4F:F
5F2FAF4F5F5F-F6FB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}
{MARK "0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }