{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Co urier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plo t" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "\n#\n# J\366rg Lem m\n#\n# Maple-Worksheet zur\n# Vorlesung Finanzmathematik und Risiko management\n# Universit\344t M\374nster\n# WS 2007/2008\n#\n# Ratin gmigrationsmatrizen und deren Generatoren\n#\n" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1147 "#\n# 8x8-Migrationsmatrix\n#\n# nach Israel, Rosenthal, Wei:\n# Finding Generators for Markov Chains \n# Via Emp irical Transistion Matrices,\n# With Applications to Credit Ratings\n # Mathematical Finance, Vol.11, No.2, 2001,\n# \374bernommen aus Jarr ow, Lando, Turnbull: \n# A Markov Model for the Term Structure of Cre dit Spreads, \n# Rev. Financial Stud., 1977,\n# basierend auf den emp irischen Daten aus\n# Standard and Poors's Credit Review (1993) f\374r die Jahre 1981 - 1991\n#\n\n# Anzahl Ratingklassen (mit Ausfallklasse ) \nn:= 8:\n# Ordnung Taylorentwicklung\nk := 16:\nk := 40:\n\n# Migra tionsmatrix\nP:= Matrix(n,n,\n[[0.8910,\0110.0963,\0110.0078,\0110.001 9,\0110.0030,\0110.0000,\0110.0000, 0.0000],\n [0.0086,\0110.9010,\011 0.0747,\0110.0099,\0110.0029,\0110.0029,\0110.0000,\0110.0000],\n [0.0 009,\0110.0291,\0110.8894,\0110.0649,\0110.0101,\0110.0045,\0110.0000, \0110.0009],\n [0.0006,\0110.0043,\0110.0656,\0110.8427,\0110.0644, 0. 0160,\0110.0018,\0110.0045],\n [0.0004,\0110.0022,\0110.0079,\0110.071 9,\0110.7764,\0110.1043,\0110.0127,\0110.0241],\n [0.0000, 0.0019,\011 0.0031,\0110.0066,\0110.0517,\0110.8246,\0110.0435,\0110.0685],\n [0.0 000, 0.0000,\0110.0116,\0110.0116,\0110.0203,\0110.0754,\0110.6493, \0110.2319],\n [0, 0, 0, 0, 0, 0, 0, \+ 1 ] \n]);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'RTABL EG6$\")%)QT8-%'MATRIXG6#7*7*$\"%5*)!\"%$\"$j*F0$\"#yF0$\"#>F0$\"#IF0$ \"\"!F:F9F97*$\"#')F0$\"%5!*F0$\"$Z(F0$\"#**F0$\"#HF0FDF9F97*$\"\"*F0$ \"$\"HF0$\"%%*))F0$\"$\\'F0$\"$,\"F0$\"#XF0F9FG7*$\"\"'F0$\"#VF0$\"$c' F0$\"%F%)F0$\"$W'F0$\"$g\"F0$\"#=F0FQ7*$\"\"%F0$\"#AF0$\"#zF0$\"$>(F0$ \"%kxF0$\"%V5F0$\"$F\"F0$\"$T#F07*F9F5$\"#JF0$\"#mF0$\"$<&F0$\"%Y#)F0$ \"$N%F0$\"$&oF07*F9F9$\"$;\"F0F[q$\"$.#F0$\"$a(F0$\"%$\\'F0$\"%>BF07*F :F:F:F:F:F:F:\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 262 "# E insvektor\nIVec := Vector(n,1):\n\n# Einheitsmatrix\nIMat := LinearAlg ebra:-DiagonalMatrix(Vector(n,1)):\n\n# Normierung der Zeilen muss jew eils eins sein\nZnorm := P . IVec;\n#Pnorm := Matrix(n,n,(i,j)->P[i,j] /Znorm[i]):\n#Znorm := Pnorm . IVec;\n#P := Matrix(Pnorm):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ZnormG-%'RTABLEG6$\")wSQ8-%'MATRIXG6#7*7# $\"\"\"\"\"!F-7#$\"35***********z***!#=7#$\"35++++++****F47#$\"3A,++++ +****F4F57#$\"3************4+5!# " 0 "" {MPLTEXT 1 0 78 "# Logarithmus von P\n# Taylorentwicklung bis zur Ordn ung k\nQ0 := Matrix(n,n,0):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {MPLTEXT 1 0 99 " for i from 1 to k do; \n Q0 := Q0 + ((-1)^(i+1) )*(P-IMat)^i/i;\n od:\n #printf(\"Q0=\\n%f\",Q0);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "# Korrektur negativer Nichtdiagonal eintr\344ge\nQ := Matrix(Q0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "f or i from 1 to n do;\n for j from 1 to n do;\n if not(i = j) then Q[i ,j] := max(0,Q[i,j]);fi;\n od;\nod;\n#printf(\"Q0=\\n%f\",Q0);\n#print f(\"Q=\\n%f\",Q);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "# N ormierung der Zeilensumme auf Null\nZnorm := Q . IVec;\nQ := Matrix(n, n,(i,j)->Q[i,j]- IMat[i,j]*Znorm[i]):\nZnorm := Q . IVec;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ZnormG-%'RTABLEG6$\"(;n>$-%'MATRIXG6#7*7# $\"3MZbW0QV!\\%!#@7#$\"3a)3yq'G[E ]'*F77#$!3Ie1^%RDG-\"F07#$!3O2-BH'z([#)F77#$\"3-mf!*=39vcF07#$\"\"!FF " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ZnormG-%'RTABLEG6$\")cTQ8-%'MAT RIXG6#7*7#$\"39O*y))*)****f$!#G7#$!3+pC&\\)*)***R%F07#$!3!)*)G@=,+gyF0 7#$!3bz/KH.+SAF07#$!3\"e&ex$****\\,\"!#F7#$\"3g@1.!o-+?#F07#$!3+U2[l,+ +?F07#$\"\"!FF" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "# Exponen tial von Q\n# Zeit\nt := 1:\n# Taylorentwicklung bis zur Ordnung k\nPT := IMat:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 59 " for i f rom 1 to k do; \n PT := PT + (t*Q)^i/i!;\n od:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "# Ausgabe\nprintf(\"P=\\n%f\",P); \nprintf(\"Q0=Log(P)=\\n%f\",Q0);\nprintf(\"Q=\\n%f\",Q);\nprintf(\"PT =exp(Q)=\\n%f\",PT);\nprintf(\"P-PT=\\n%f\",P-PT);" }}{PARA 6 "" 1 "" {TEXT -1 2 "P=" }}{PARA 6 "" 1 "" {TEXT -1 66 ".891000 .096300 .007800 .001900 .003000 0.000000 0.000000 0.000000" }}{PARA 6 "" 1 "" {TEXT -1 65 ".008600 .901000 .074700 .009900 .002900 .002900 0.000000 0.0000 00" }}{PARA 6 "" 1 "" {TEXT -1 64 ".000900 .029100 .889400 .064900 .01 0100 .004500 0.000000 .000900" }}{PARA 6 "" 1 "" {TEXT -1 63 ".000600 \+ .004300 .065600 .842700 .064400 .016000 .001800 .004500" }}{PARA 6 "" 1 "" {TEXT -1 63 ".000400 .002200 .007900 .071900 .776400 .104300 .012 700 .024100" }}{PARA 6 "" 1 "" {TEXT -1 64 "0.000000 .001900 .003100 . 006600 .051700 .824600 .043500 .068500" }}{PARA 6 "" 1 "" {TEXT -1 65 "0.000000 0.000000 .011600 .011600 .020300 .075400 .649300 .231900" }} {PARA 6 "" 1 "" {TEXT -1 71 "0.000000 0.000000 0.000000 0.000000 0.000 000 0.000000 0.000000 1.000000" }}{PARA 6 "" 1 "" {TEXT -1 10 "Q0=Log( P)=" }}{PARA 6 "" 1 "" {TEXT -1 75 "-0.115931 0.107466 0.004208 0.0013 34 0.003372 -0.000409 -0.000014 -0.000025" }}{PARA 6 "" 1 "" {TEXT -1 74 "0.009566 -0.106131 0.083233 0.008115 0.002567 0.002942 -0.000114 - 0.000168" }}{PARA 6 "" 1 "" {TEXT -1 73 "0.000831 0.032362 -0.121382 0 .074626 0.009028 0.004013 -0.000274 0.000589" }}{PARA 6 "" 1 "" {TEXT -1 72 "0.000623 0.003572 0.075556 -0.177517 0.079050 0.013991 0.001350 0.003277" }}{PARA 6 "" 1 "" {TEXT -1 72 "0.000440 0.002181 0.005768 0 .088535 -0.261178 0.129535 0.013816 0.020800" }}{PARA 6 "" 1 "" {TEXT -1 73 "-0.000027 0.002086 0.002711 0.004655 0.063962 -0.199781 0.05901 6 0.067268" }}{PARA 6 "" 1 "" {TEXT -1 74 "-0.000015 -0.000420 0.01444 7 0.013639 0.024547 0.101298 -0.435344 0.281980" }}{PARA 6 "" 1 "" {TEXT -1 71 "0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.0 00000 0.000000" }}{PARA 6 "" 1 "" {TEXT -1 2 "Q=" }}{PARA 6 "" 1 "" {TEXT -1 67 "-.116380 .107466 .004208 .001334 .003372 0.000000 0.00000 0 0.000000" }}{PARA 6 "" 1 "" {TEXT -1 66 ".009566 -.106423 .083233 .0 08115 .002567 .002942 0.000000 0.000000" }}{PARA 6 "" 1 "" {TEXT -1 65 ".000831 .032362 -.121448 .074626 .009028 .004013 0.000000 .000589 " }}{PARA 6 "" 1 "" {TEXT -1 64 ".000623 .003572 .075556 -.177420 .079 050 .013991 .001350 .003277" }}{PARA 6 "" 1 "" {TEXT -1 64 ".000440 .0 02181 .005768 .088535 -.261075 .129535 .013816 .020800" }}{PARA 6 "" 1 "" {TEXT -1 65 "0.000000 .002086 .002711 .004655 .063962 -.199699 .0 59016 .067268" }}{PARA 6 "" 1 "" {TEXT -1 66 "0.000000 0.000000 .01444 7 .013639 .024547 .101298 -.435911 .281980" }}{PARA 6 "" 1 "" {TEXT -1 71 "0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 \+ 0.000000" }}{PARA 6 "" 1 "" {TEXT -1 10 "PT=exp(Q)=" }}{PARA 6 "" 1 " " {TEXT -1 71 "0.890600 0.096265 0.007799 0.001901 0.003011 0.000351 0 .000026 0.000048" }}{PARA 6 "" 1 "" {TEXT -1 71 "0.008597 0.900737 0.0 74687 0.009900 0.002901 0.002906 0.000097 0.000175" }}{PARA 6 "" 1 "" {TEXT -1 71 "0.000900 0.029095 0.889343 0.064903 0.010103 0.004511 0.0 00211 0.000935" }}{PARA 6 "" 1 "" {TEXT -1 71 "0.000600 0.004300 0.065 601 0.842781 0.064406 0.016002 0.001808 0.004501" }}{PARA 6 "" 1 "" {TEXT -1 71 "0.000402 0.002203 0.007900 0.071907 0.776479 0.104310 0.0 12698 0.024101" }}{PARA 6 "" 1 "" {TEXT -1 71 "0.000024 0.001911 0.003 100 0.006601 0.051705 0.824668 0.043491 0.068502" }}{PARA 6 "" 1 "" {TEXT -1 71 "0.000014 0.000323 0.011611 0.011599 0.020296 0.075383 0.6 48934 0.231839" }}{PARA 6 "" 1 "" {TEXT -1 71 "0.000000 0.000000 0.000 000 0.000000 0.000000 0.000000 0.000000 1.000000" }}{PARA 6 "" 1 "" {TEXT -1 5 "P-PT=" }}{PARA 6 "" 1 "" {TEXT -1 76 "0.000400 0.000035 0. 000001 -0.000001 -0.000011 -0.000351 -0.000026 -0.000048" }}{PARA 6 " " 1 "" {TEXT -1 75 "0.000003 0.000263 0.000013 0.000000 -0.000001 -0.0 00006 -0.000097 -0.000175" }}{PARA 6 "" 1 "" {TEXT -1 76 "0.000000 0.0 00005 0.000057 -0.000003 -0.000003 -0.000011 -0.000211 -0.000035" }} {PARA 6 "" 1 "" {TEXT -1 78 "-0.000000 0.000000 -0.000001 -0.000081 -0 .000006 -0.000002 -0.000008 -0.000001" }}{PARA 6 "" 1 "" {TEXT -1 78 " -0.000002 -0.000003 -0.000000 -0.000007 -0.000079 -0.000010 0.000002 - 0.000001" }}{PARA 6 "" 1 "" {TEXT -1 78 "-0.000024 -0.000011 -0.000000 -0.000001 -0.000005 -0.000068 0.000009 -0.000002" }}{PARA 6 "" 1 "" {TEXT -1 74 "-0.000014 -0.000323 -0.000011 0.000001 0.000004 0.000017 \+ 0.000366 0.000061" }}{PARA 6 "" 1 "" {TEXT -1 71 "0.000000 0.000000 0. 000000 0.000000 0.000000 0.000000 0.000000 0.000000" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 242 "# variierende Zeit\ntmax := 20: # maximale Zeit in Jahren\ntmin := 0: # minimale Zeit in Jahren \nm := 41: # Anzahl Gitterpunkte f\374r t \ndel := (tmax-tm in)/(m-1): # Delta-t zwischen zwei Gitterpunkten\nPD := Matrix(n,m): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "\n# \334bergangsmatrizen\nfor \+ j from 1 to m do;\n PT := IMat: \n for i from 1 to k do; \n PT \+ := PT + ((tmin+(j-1)*del)*Q)^i/i!:\n od:\n PD[1..n,j] := Vector(n,i- >PT[i,n]);\nod:\n#printf(\"PD=\\n%f\",PD);\n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "# Plots\nwith(plots):with(CurveFitting):\nx:=[seq(tm in+(i-1)*del,i=1..m)]: # x-Werte\ny:=array(1..m): # y -Werte \ns:=array(1..n): # f\374r jede Kurve ein Arr ay " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "for i from 1 to n do; \n y := [seq(PD[i,j],j=1..m)];\n s[i] := BSplineCurve(x,y,v):\nend do:\np lot(s); \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6,-%'CURVESG6$7S7$ $\"3++++++++]!#=$\"3V*****4G;c^\"!#A7$$\"3/nmm\"Hf99*F*$\"3;6Yh>^1^WF- 7$$\"3WL$3x3#\\u7!#<$\"3)Q!)fR%*4-v)F-7$$\"3gmmTI!R(z;F6$\"3YQmh(\\EFe \"!#@7$$\"3%pm;H@sw3#F6$\"3Ue9$Ru]`c#F>7$$\"3AL$3-cmO\\#F6$\"3kbe9$G7C %QF>7$$\"3Em;/\"Hv+(GF6$\"3j17,qSm<`F>7$$\"3c**\\PHZ#)fKF6$\"3M`c-yT*Q ;(F>7$$\"3'fmTg:/Hm$F6$\"3;Cv%)4)e!Q%*F>7$$\"3;+](=!4pkSF6$\"3tges@4u4 7!#?7$$\"3[LL$e&z(zZ%F6$\"3XkL)[5Gl_\"Fgn7$$\"3Qmm\"zT,?%[F6$\"3b$*RtF -qV=Fgn7$$\"3_++]kF6$\"3Y <#ze!>!zm$Fgn7$$\"37LL$esH\"[oF6$\"3w=7R,IF)H%Fgn7$$\"3;MLL.e'3@(F6$\" 3omrn`BN!)[Fgn7$$\"3.+]P%49Gj(F6$\"34\"=Xh\\nWh&Fgn7$$\"3\\KLLG)4j+)F6 $\"3s:&)*[0%p;jFgn7$$\"3>+]PWQ4;%)F6$\"3CEqK\"=$4XrFgn7$$\"3&3+DJ80j!) 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P . vecs - dvals); # Test diagonal\nLinearAlgebra:- MatrixNorm(vecs . dvals . lvecs - P); # Test Rekonstrukton" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"+DrUGG!\"*$\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"34XmP.dGGG!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" 3ERgh#p>>4\"!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3w***z.NA.<\"!#< " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "# Logarithmus von P \+ \374ber Eigenwerte\nlnvals := LinearAlgebra:-Map(ln,Vector(vals)); # M ap arbeitet \"inplace\"\ndlnvals := LinearAlgebra:-DiagonalMatrix(lnva ls):\nQ2 := vecs . dlnvals . lvecs:\nLinearAlgebra:-MatrixNorm(Q2-Q0); # Vergleich beider Varianten\nLinearAlgebra:-MatrixNorm(lvecs . Q2 . vecs - dlnvals); # Test Q2 diagonal\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lnvalsG-%'RTABLEG6$\")wUQ8-%'MATRIXG6#7*7#^$$!3q*****Rm[I\\)! #>$\"\"!F37#^$$!3#*******RP\"f*=F1F27#^$$!3-+++_EmF7!#=F27#^$$!35+++G> Wl:F'R]-NF " 0 "" {MPLTEXT 1 0 227 "# Exponentiieren exp(Q) = P ?\nvalsj := Li nearAlgebra:-Map(exp,Vector(lnvals)); \ndvalsj := LinearAlgebra:-Diago nalMatrix(valsj):\nPT := vecs . dvalsj . lvecs:\nPD[1..n,1] := Vector( n,i->PT[i,n]);\nLinearAlgebra:-MatrixNorm(P-PT,1);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&valsjG-%'RTABLEG6$\");VQ8-%'MATRIXG6#7*7#^$$\"3Y**** *\\Nhd=*!#=$\"\"!F37#^$$\"3%)*****fc%>7)*F1F27#^$$\"3`+++*p.Z%))F1F27# ^$$\"3W+++zs$4b)F1F27#^$$\"3]*****p%\\:%)zF1F27#^$$\"3M+++1m6XqF1F27#^ $$\"3l*****zaJ6K'F1F27#^$$\"\"\"F3F2" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%#PDG6$;\"\"\"\"\")F(-%'RTABLEG6$\")cVQ8-%'MATRIXG6#7*7#^$$\"3K\"* H@V%*[bg!#H$\"\"!F87#^$$!3Z`PwpJECV!#GF77#^$$\"3S\"G[Du******)!#@F77#^ $$\"3G)3A)p*****\\%!#?F77#^$$\"3ROO;&*****4C!#>F77#^$$\"3/d,6-++]oFLF7 7#^$$\"3;4,`******=B!#=F77#^$$\"2))***************!# " 0 "" {MPLTEXT 1 0 60 "# Korrektur negativer Nichtdiagonaleintr\344ge\nQ := \+ Matrix(Q2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "for i from 1 to n d o;\n for j from 1 to n do;\n if not(i = j) then Q[i,j] := max(0,Re(Q[ i,j]));fi;\n od;\nod;\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "# Normi erung der Zeilensumme auf Null\nZnorm := Q . IVec:\nQ := Matrix(n,n,(i ,j)->Q[i,j]- IMat[i,j]*Znorm[i]):\nZnorm := Q . IVec;\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&ZnormG-%'RTABLEG6$\")OWQ8-%'MATRIXG6#7*7#^$$ \"3GXIB(o****H#!#G$\"\"!F37#^$$\"32,V+<&4++'!#HF27#^$$!3S%)R5&f,+='F1F 27#^$$\"3Kn`t&f,+l(F1F27#^$$!3#49SU-)**p\\F1F27#^$$\"3*fAUk\\+++'F1F27 #^$$!3+U2[l,++?F1F27#^$F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "# Ohne Korrekturen \+ k\366nnten die EV von Q weiterverwendet werden\n# nach Korrekturen ist neu zu diagonalisieren\n(vals,vecs) := LinearAlgebra:-Eigenvectors(Q) :\nlvecs := LinearAlgebra:-MatrixInverse(vecs): # Linkseigenwerte\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "\n# variierende Zeit\ntm ax := 20: # maximale Zeit in Jahren\ntmin := 0: # minim ale Zeit in Jahren\nm := 41: # Anzahl Gitterpunkte f\374r t \+ \ndel := (tmax-tmin)/(m-1): # Delta-t zwischen zwei Gitterpunkten\n PD := Matrix(n,m):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 263 "# \334bergan gsmatrizen\nfor j from 1 to m do;\n valsj := LinearAlgebra:-Map(exp,(t min+(j-1)*del)*Vector(vals)): \n dvalsj := LinearAlgebra:-DiagonalMatr ix(valsj):\n PT := vecs . dvalsj . lvecs:\n PD[1..n,j] := Vector(n,i-> PT[i,n]);\nod:\n#printf(\"PD=\\n%f\",PD);\n#evalm(PD); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "# Plots\nwith(plots):with(CurveFit ting):\nx:=[seq(tmin+(i-1)*del,i=1..m)]: # x-Werte\ny:=array(1..m): \+ # y-Werte \ns2:=array(1..n): # f\374r \+ jede Kurve ein Array " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "for i fro m 1 to n do; \n y := [seq(PD[i,j],j=1..m)];\n s2[i] := BSplineCurve( x,y,v):\nend do:\nplot(s2); \n" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6,-%'CURVESG6$7S7$$\"3++++++++]!#=$\"35nm;'H=c^\"!#A7 $$\"3/nmm\"Hf99*F*$\"3C'4)yMb2^WF-7$$\"3WL$3x3#\\u7!#<$\"3/9Sfr>@]()F- 7$$\"3gmmTI!R(z;F6$\"39\"Q(\\Ris#e\"!#@7$$\"3%pm;H@sw3#F6$\"3a7$$\"3AL$3-cmO\\#F6$\"3Uf`.C;TUQF>7$$\"3Em;/\"Hv+(GF6$\"3[D'z+YjwJ&F >7$$\"3c**\\PHZ#)fKF6$\"3C,btdO*Q;(F>7$$\"3'fmTg:/Hm$F6$\"3*4CT-\"y0Q% 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$\"35vV[5tfB;Fgs$\"331N+++++5F67$$\"3+[*z78ggi\"Fgs$\"3+AZ+++++5F67$$ \"3*3_v?&H_G;Fgs$\"3Q!>1+++++\"F67$$\"3z$4rGx&)4j\"Fgs$\"3)e$z+++++5F6 7$Fax$\"3!=)*4+++++\"F67$$\"36HKk$=nfj\"Fgs$\"335C,++++5F67$$\"3`\"z>O x&[Q;Fgs$\"3=._,++++5F67$$\"3vs!3'o]uR;Fgs$\"3#[u;+++++\"F67$$\"3'RN'f jV+T;Fgs$\"31(Q=+++++\"F67$$\"3`NYeeOEU;Fgs$\"3oL,-++++5F67$$\"3u;Hd`H _V;Fgs$\"33()>-++++5F67$$\"3&z>h&[AyW;Fgs$\"3G^R-++++5F67$$\"3_z%\\NaT gk\"Fgs$\"3'*Gg-++++5F67$$\"3ugx`Q3IZ;Fgs$\"3kB#G+++++\"F67$$\"3%>/EN8 g&[;Fgs$\"3!y`I+++++\"F67$$\"3;BV^G%>)\\;Fgs$\"3%e(H.++++5F67$$\"3O/E] B(y5l\"Fgs$\"3m-++++++5F67$$\"3e&)3\\=!QBl\"FgsF`ct7$$\"3!o;zMJ(f`;Fgs $\"3m9++++++5F67$$\"3[;aQt;nj;Fgs$\"3!Ro++++++\"F67$Ffx$\"3#Hd.+++++\" F67$$\"3o9;2oPCw;Fgs$\"3OA[+++++5F67$$\"3[i:&G]T(y;Fgs$\"3gLj+++++5F67 $$\"3G5:jP#R7o\"Fgs$\"3%>83+++++\"F67$$\"31e9Tspt$o\"Fgs$\"3-U-,++++5F 67$$\"3'eS\">2ZB'o\"Fgs$\"3'*)o7+++++\"F67$$\"3+a8(>WK()o\"Fgs$\"3Q(\\ :+++++\"F67$$\"3EG8O48)**o\"Fgs$\"3/Xq,++++5F67$$\"39-8vw,B\"p\"Fgs$\" 3g#p=+++++\"F67$$\"3/w79W!zCp\"Fgs$\"3)HW?+++++\"F67$$\"3%*\\7`6zs$p\" Fgs$\"3G*HA+++++\"F67$$\"3#QA@*yn(\\p\"Fgs$\"3ekU-++++5F67$$\"3s(>6jkD ip\"Fgs$\"3EUj-++++5F67$$\"3ir6q8XZ(p\"Fgs$\"3iN&G+++++\"F67$$\"3'e9\" 4\"QB()p\"Fgs$\"3)o%3.++++5F67$$\"37?6[[A(**p\"Fgs$\"3/!GL+++++\"F67$$ \"3,%4re6@7q\"FgsFjc\\l7$$\"3\"z1hK)*pCq\"FgsFaa[l7$$\"3\"=/^1&)=Pq\"F gs$\"3)f,++++++\"F67$$\"3RP4@?Vr3hSe&=DHu![FiOMksr8[.++++5F67$F`yFchw7$$\"3!f*)fD %G1`ON$pbkZry\"Fgs$\"3AR(3+++++\"F67$$\" 3yX'y_(*GYy\"Fgs$\"3-w5,++++5F67$$\"3gT&Qj[fsy\"Fgs$\"33'z8+++++\"F67$ $\"3M*[o=uu&)y\"Fgs$\"3')3`,++++5F67$$\"3UP%)R(***))*y\"Fgs$\"3oGp,+++ +5F67$$\"3]&QGHD07z\"Fgs$\"3ie'=+++++\"F67$Fey$\"3q-0-++++5F67$$\"3[$f $o))op$z\"Fgs$\"3)>DA+++++\"F67$$\"31a)3*oK([z\"Fgs$\"3i(4C+++++\"F67$ $\"3k9T8\\'\\gz\"Fgs$\"3sUg-++++5F67$$\"3)[Pf$HgA(z\"Fgs$\"3)**3G+++++ \"F67$$\"37NYe4CS)z\"Fgs$\"3OT-.++++5F67$$\"3q&*)4)*yy&*z\"Fgs$\"3A+D. ++++5F67$$\"3Gc^.q^v+=FgsF\\fx7$$\"3^;/E]:$>!=Fgs$\"3W/++++++5F67$$\"3 LP4r5VG/=FgsFi\\y7$$\"3:e9;rqj1=Fgs$\"3;!3++++++\"F67$$\"3(*y>hJ)*)*3= Fgs$\"31'>++++++\"F67$$\"3z*\\i?fU8\"=Fgs$\"3Y\"R++++++\"F67$$\"3VTN'H 6[g\"=Fgs$\"3W/6+++++5F67$$\"32$ekQj`2#=Fgs$\"33'Q-+++++\"F67$$\"3rCcw a\"fa#=Fgs$\"3#HS/+++++\"F67$Fjy$\"3q@t+++++5F67$$\"3$QMF!4KnK=Fgs$\"3 m.$4+++++\"F67$$\"3h?!)QU<=N=Fgs$\"3q9;,++++5F67$$\"3v(p[dF!pP=Fgs$\"3 9!G9+++++\"F67$$\"3)[P4\"4))>S=Fgs$\"3![K<+++++\"F67$$\"3F8(*yvIXT=Fgs $\"3C(**=+++++\"F67$$\"3-_+ZUtqU=Fgs$\"3Eu2-++++5F67$$\"3w!R]\"4;'R%=F gs$\"3weE-++++5F67$$\"3;H2$e(e@X=Fgs$\"3i`Y-++++5F67$$\"3an5^U,ZY=Fgs$ \"3gin-++++5F67$$\"3G19>4WsZ=Fgs$\"3e))*G+++++\"F67$$\"3/X<(eny*[=Fgs$ \"3EM8.++++5F67$$\"3U$3_D%HB]=FgsF\\fx7$$\"3cgF\"fZTF&=Fgs$\"3w2++++++ 5F67$$\"3oPMF4+Db=Fgs$\"3'3/++++++\"F67$$\"3#[6MEaex&=Fgs$\"3+F,+++++5 F67$$\"3'>z%*f2n-'=Fgs$\"3)3H++++++\"F67$$\"3AYhrUTGl=Fgs$\"3Ma4+++++5 F67$F_z$\"3KLA+++++5F67$$\"3MDJX#=.^(=Fgs$\"3%3A/+++++\"F67$$\"3=](oa: 0*z=Fgs$\"3IMr+++++5F67$$\"3gil(>91B)=Fgs$\"3q$**3+++++\"F67$$\"3-vV[G rq%)=Fgs$\"33^6,++++5F67$$\"3W(=#*\\63r)=Fgs$\"3mGO,++++5F67$$\"3')*** *\\,\"4&*)=Fgs$\"3=[k,++++5F67$$\"3C1Rv%f42*=Fgs$\"3g$*z,++++5F67$$\"3 E7y+)35>*=Fgs$\"3_K'>+++++\"F67$$\"3I=FgsFchw7$$\"3Ii!RS.9 :!>FgsFdas7$$\"3ooHHFXr->FgsFe]^l7$$\"32voa?]\"R!>Fgs$\"3U=++++++5F67$ $\"3\\(oaq+;j!>Fgs$\"3]p++++++5F67$FdzFej\\l7$$\"3C(=n$pt(Q\">Fgs$\"3) \\r++++++\"F67$$\"3$\\(=>Fgs$\"3IU=+++++5F67$$\"3iil(47)>C>Fgs$\" 3w!y.+++++\"F67$$\"3'*\\7y'\\e$H>Fgs$\"3Q]n+++++5F67$$\"3i$f$o%oQ>$>Fg s$\"3;!p3+++++\"F67$$\"3HPfes)=X$>Fgs$\"3yq4,++++5F67$$\"3J\"G)[g!*4P> Fgs$\"3m=O,++++5F67$$\"3)\\i!R[#z'R>Fgs$\"3yhm,++++5F67$$\"3*pzTBMp4%> Fgs$\"3+S$=+++++\"F67$$\"3loHHO%fA%>Fgs$\"3YF,-++++5F67$$\"3mSTCI&\\N% >Fgs$\"3[F?-++++5F67$$\"3n7`>C'R[%>Fgs$\"3WVS-++++5F67$$\"3K%[Y\"=(Hh% >Fgs$\"3iyh-++++5F67$$\"3Mcw47)>u%>Fgs$\"3!oVG+++++\"F67$$\"3OG)[g!*4( [>Fgs$\"3M@3.++++5F67$FizFchwF\\el-%+AXESLABELSG6$Q!6\"Fgi^l-%%VIEWG6$ %(DEFAULTGF\\j^l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 16 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 13413884 13384076 3196716 13384156 13384236 13384276 13384316 13384356 13384436 } {RTABLE M7R0 I5RTABLE_SAVE/13413884X,%)anythingG6"6"[gl!"%!!!#[o")")$"%5*)!"%$"#')F)$""*F)$" "'F)$""%F)$""!F3F2F3$"$j*F)$"%5!*F)$"$"HF)$"#VF)$"#AF)$"#>F)F2F3$"#yF)$"$Z(F)$" %%*))F)$"$c'F)$"#zF)$"#JF)$"$;"F)F3F>$"#**F)$"$\'F)$"%F%)F)$"$>(F)$"#mF)FLF3$"# IF)$"#HF)$"$,"F)$"$W'F)$"%kxF)$"$<&F)$"$.#F)F3F2FZ$"#XF)$"$g"F)$"%V5F)$"%Y#)F)$ "$a(F)F3F2F2F2$"#=F)$"$F"F)$"$N%F)$"%$\'F)F3F2F2F,F`o$"$T#F)$"$&oF)$"%>BF)"""6" } {RTABLE M7R0 I5RTABLE_SAVE/13384076X*%)anythingG6"6"[gl'#%!!!")")3FF00000000000003FF00000000 000003FEFFE5C91D14E3B3FEFFF2E48E8A71E3FEFFF2E48E8A71F3FEFFF2E48E8A71E3FF00068DB 8BAC713FF00000000000006" } {RTABLE M7R0 I4RTABLE_SAVE/3196716X*%)anythingG6"6"[gl'#%!!!")")3F3D6DB2A248DA503F33233CB23F 30803F116661916FAD58BF194C2EDF37E200BF1AD010A5C79500BF159FA98CB128003F4298A735F B900000000000000000006" } {RTABLE M7R0 I5RTABLE_SAVE/13384156X*%)anythingG6"6"[gl'#%!!!")")3DC3CA8CA8000000BDC83072B40 00000BDD59AFBBE800000BDB8A10A58000000BDDBE66D700000003DB8307500000000BDB5FD8000 00000000000000000000006" } {RTABLE M7R0 I5RTABLE_SAVE/13384236X*%)anythingG6"6"[gl(#%!!!")")3FED64F9C79B4D1800000000000 000003FEF66265886853000000000000000003FEC4D94CE20A7ED00000000000000003FEB5CED85 8A829400000000000000003FE98C9EA6CBCB6000000000000000003FE68B5C09BCE07D000000000 00000003FE43A455DEB0ECB00000000000000003FF000000000000000000000000000006" } {RTABLE M7R0 I5RTABLE_SAVE/13384276X*%)anythingG6"6"[gl(#%!!!")")BFB5BE011E8D5AAE00000000000 00000BF936A062C5F9C820000000000000000BFBF6D9C261608380000000000000000BFC409A3DE 3A34390000000000000000BFCCD0EEFF6B745C0000000000000000BFD66A80A345C342000000000 0000000BFDD5B20201732DE0000000000000000000000000000000000000000000000006" } {RTABLE M7R0 I5RTABLE_SAVE/13384316X*%)anythingG6"6"[gl(#%!!!")")3FED64F9C79B795C00000000000 000003FEF6626588977F100000000000000003FEC4D94CE200C7A00000000000000003FEB5CED85 8B0C1100000000000000003FE98C9EA6CB65AC00000000000000003FE68B5C09BDAF53000000000 00000003FE43A455DE798A900000000000000003FF000000000000000000000000000006" } {RTABLE M7R0 I5RTABLE_SAVE/13384356X*%)anythingG6"6"[gl!#%!!!")")^$$"3K"*H@V%*[bg!#H$""!F,^$ $!3Z`PwpJECV!#GF+^$$"3S"G[Du******)!#@F+^$$"3G)3A)p*****\%!#?F+^$$"3ROO;&*****4 C!#>F+^$$"3/d,6-++]oF