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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "In" }{TEXT 32 27 "stitution
en f\366r matematik, " }{TEXT 256 3 "KTH" }}{PARA 0 "" 0 "" {TEXT -1 
9 "Mats Boij" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 18 "" 0 "" {TEXT -1 9 "Exempel 1" }}{PARA 256 "" 0 "" 
{TEXT -1 75 "Vi ska se p\345 ett linj\344rt system av f\366rsta ordnin
gens ODE som ges av X'=AX. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protect
ed names norm and trace have been redefined and unprotected\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "A:=matrix(3,3, [[-2.0,1.0,0.
0],[1.0,-2.0,1.0],[0.0,1.0,-1.0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"AG-%'matrixG6#7%7%$!#?!\"\"$\"#5F,$\"\"!F07%F-F*F-7%F/F-$!#5F," }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Egenv\344rdena kan  f\345s genom
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "eigenvals(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$!+/'zpC$!\"*$!+K
\"e\\b\"F%$!+UEi!)>!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "F\366r \+
att diagonalisera matrisen A kan vi anv\344nda kommandot jordan. Basby
tesmatrisen l\344ggs i P.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "B:=jordan(A,'P');" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"BG-%'matrixG6#7%7%$!+/'zpC$!\"*\"\"!F-7%F-$!+K\"e\\
b\"F,F-7%F-F-$!+UEi!)>!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "Vi \+
f\345r en fundamentalmatris f\366r det transformerade problemet genom \+
att applicera exponentialfunktionen p\345 diagonalen efter att ha mult
iplicerat med t. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 45 "E:=multiply(map(x->x*exp(x*t),B),inverse(B));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG-%'matrixG6#7%7%,$-%$expG6#,$%\"
tG$!+/'zpC$!\"*$\"+++++5F2$\"\"!F6F57%F5,$-F,6#,$F/$!+K\"e\\b\"F2F3F57
%F5F5,$-F,6#,$F/$!+UEi!)>!#5F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
100 "En fundamentalmatris f\366r det ursprungliga systemet f\345s geno
m att multiplicera med basbytesmatrisen. " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "X:=multiply(P,E);" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG-%'mat
rixG6#7%7%,$-%$expG6#,$%\"tG$!+/'zpC$!\"*$\"+]p\"H\\$!#5,$-F,6#,$F/$!+
K\"e\\b\"F2$\"+5'R8V&F5,$-F,6#,$F/$!+UEi!)>F5$\"+IMuv5F57%,$F+$!+%>'fb
VF5,$F7$\"+IN<<CF5,$F?$\"+kEUQ>F57%,$F+$\"+mEUQ>F5,$F7$!+&>'fbVF5,$F?$
\"+HN<<CF5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "F\366r att kontroll
era att det verkligen \344r en l\366sning till systemet X'=AX derivera
r vi X" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "map(x->diff(x,t),X);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%,$-%$expG6#,$%\"tG$!+/
'zpC$!\"*$!+5I9M6F0,$-F*6#,$F-$!+K\"e\\b\"F0$!+%p0bW)!#5,$-F*6#,$F-$!+
UEi!)>F;$!+%zT18#!#67%,$F)$\"+,KD99F0,$F4$!+POgePF;,$F=$!+=QGRQFD7%,$F
)$!+g)=SH'F;,$F4$\"+B(pFx'F;,$F=$!+@'3vy%FD" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 23 "och j\344mf\366r detta med AX" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "multiply(A,X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'
matrixG6#7%7%,$-%$expG6#,$%\"tG$!+/'zpC$!\"*$!+4I9M6F0,$-F*6#,$F-$!+K
\"e\\b\"F0$!+!p0bW)!#5,$-F*6#,$F-$!+UEi!)>F;$!*'>kI@F;7%,$F)$\"++KD99F
0,$F4$!+XOgePF;,$F=$!*p$GRQF;7%,$F)$!+g)=SH'F;,$F4$\"+D(pFx'F;,$F=$!*l
3vy%F;" }}}{EXCHG {PAGEBK }{PARA 18 "" 1 "" {TEXT -1 10 "Exempel 2." }
}{PARA 0 "" 0 "" {TEXT -1 110 "Vi vill nu f\366rs\366ka med ett st\366
rre system av samma slag. F\366rst skapar vi matrisen A av storlek tio
 g\345nger tio. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "A:=mul
tiply(LinearAlgebra[BandMatrix]([[1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0]
,[-2.0,-2.0,-2.0,-2.0,-2.0,-2.0,-2.0,-2.0,-2.0,-1.0],[1.0,1.0,1.0,1.0,
1.0,1.0,1.0,1.0,1.0]]),diag(1,1,1,1,1,1,1,1,1,1));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7,7,$!#?!\"\"$\"#5F,$\"\"!F0F/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-$!#5F,
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "Vi beh\366ver nu diagonalise
ra matrisen A. Detta kan g\366ras med kommandot \"jordan\", och basbyt
esmatrisen l\344ggs in i matrisen P." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 17 "B:=jordan(A,'P');" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7,7,$!+/'zpC$!\"*\"
\"!F-F-F-F-F-F-F-F-7,F-$!+K\"e\\b\"F,F-F-F-F-F-F-F-F-7,F-F-$!+UEi!)>!#
5F-F-F-F-F-F-F-7,F-F-F-$!+7c96RF,F-F-F-F-F-F-7,F-F-F-F-$!+jD'*Q`F4F-F-
F-F-F-7,F-F-F-F-F-$!+bZ$QB#!#6F-F-F-F-7,F-F-F-F-F-F-$!+(=g%\\@F,F-F-F-
7,F-F-F-F-F-F-F-$!+\\?oIFF,F-F-7,F-F-F-F-F-F-F-F-$!+\\vZ_OF,F-7,F-F-F-
F-F-F-F-F-F-$!\"\"F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 222 "F\366r d
et diagonala systemet x'=Bx kan vi l\344tt hitta en fundamentalmatris,
 E, genom att applicera exponentialfunktionen p\345 varje element  i B
. Vi f\345r d\345 ettor utanf\366r diagonalen, och d\344rf\366r multip
licerar vi resultatet med x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "E:=map(x->x*exp(x*t),B);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"EG-%'matrixG6#7,7,,$-%$expG6#,$%\"tG$!+/'zpC$!
\"*F0\"\"!F3F3F3F3F3F3F3F37,F3,$-F,6#,$F/$!+K\"e\\b\"F2F9F3F3F3F3F3F3F
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7c96RF2FHF3F3F3F3F3F37,F3F3F3F3,$-F,6#,$F/$!+jD'*Q`FBFOF3F3F3F3F37,F3F
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g%\\@F2FhnF3F3F37,F3F3F3F3F3F3F3,$-F,6#,$F/$!+\\?oIFF2F_oF3F37,F3F3F3F
3F3F3F3F3,$-F,6#,$F/$!+\\vZ_OF2FfoF37,F3F3F3F3F3F3F3F3F3,$-F,6#,$F/$!
\"\"F3F]p" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "X:=multiply(P,
E):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Vi definierar ett h\366ge
rled som ges av en puls fr\345n t=0 till t=1. Det \344r bara i f\366rs
ta raden som insignalen kommer in. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
72 "f:=matrix(10,1,[t*Heaviside(t)*(1-t)*Heaviside(1-t),0,0,0,0,0,0,0,
0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%'matrixG6#7,7#**%\"t
G\"\"\"-%*HeavisideG6#F+F,,&F,F,F+!\"\"F,-F.6#F0F,7#\"\"!F4F4F4F4F4F4F
4F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "F\366r att f\345 en partik
ul\344rl\366sningen anv\344nder vi variation av parametrar och mulitip
licerar  " }}{PARA 0 "" 0 "" {TEXT -1 83 "fundamentalmatrisen, X,  med
 integralen av inversen till X g\345nger h\366gerledet f, dvs" }}
{PARA 257 "" 0 "" {XPPEDIT 18 0 "X(t)*int(X(t)^(-1)*f(t),t);" "6#*&-%
\"XG6#%\"tG\"\"\"-%$intG6$*&)-F%6#F',$F(!\"\"F(-%\"fG6#F'F(F'F(" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "F:=multiply(inverse(X),f):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "u:=multiply(X,map(x->int(x,t),F)):" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 53 "Vi kan rita upp l\366sningarna i de olika m\344
tpunkterna. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot([f[1,1],u[1,1]
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