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00000000-%+AXESLABELSG6%%\"xG%\"yGQ!F/" 1 2 0 1 10 0 2 1 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 38 "Berechnun g der Rekursionskoeffizienten" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 73 "Wir betrachten die drei h\366chsten Koeffozienten des orthogonalen Po lynoms:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "p:=k[n]*x^n+kstrich[n]*x^(n-1)+kstrichstrich[n]*x^(n-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,(*&&%\"kG6#%\"nG\"\"\")%\"xGF*F+F+*&& %(kstrichGF)F+)F-,&F*F+F+!\"\"F+F+*&&%.kstrichstrichGF)F+)F-,&F*F+\"\" #F3F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 25 "Wir erkl\344ren die P olynome" }{TEXT -1 1 " " }{TEXT 257 0 "" }{XPPEDIT 18 0 "sigma;" "6#%& sigmaG" }{TEXT -1 1 " " }{TEXT 259 3 "und" }{TEXT -1 1 " " }{XPPEDIT 18 0 "tau;" "6#%$tauG" }{TEXT -1 1 " " }{TEXT 260 39 "mit beliebigen K oeffizienten a,b,c,d,e:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "sigma:=a*x^2+b*x+c;\ntau:=d*x+e;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&sigmaG,(*&%\"aG\"\"\")%\"xG\"\"#F(F(*&%\"bGF(F*F(F (%\"cGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tauG,&*&%\"dG\"\"\"%\"x GF(F(%\"eGF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 55 "Das Polynom erf \374llt die Differentialgleichung DE=0 mit:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "DE:=sigma*diff(p,x$2)+tau*diff(p,x)+lambda[n]*p;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#DEG,(*&,(*&%\"aG\"\"\")%\"xG\"\"# F*F**&%\"bGF*F,F*F*%\"cGF*F*,.**&%\"kG6#%\"nGF*)F,F6F*F6F-F,!\"#F***F3 F*F7F*F6F*F,F8!\"\"**&%(kstrichGF5F*)F,,&F6F*F*F:F*F?F-F,F8F***FF *F?F*F,F8F:**&%.kstrichstrichGF5F*)F,,&F6F*F-F:F*FEF-F,F8F***FBF*FDF*F EF*F,F8F:F*F**&,&*&%\"dGF*F,F*F*%\"eGF*F*,(**F3F*F7F*F6F*F,F:F***FF*F?F*F,F:F***FBF*FDF*FEF*F,F:F*F*F**&&%'lambdaGF5F*,(*&F3F*F7F*F**&F F*F**&FBF*FDF*F*F*F*" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 32 "Wi r sortieren nach Potenzen von " }{TEXT 269 1 "x" }{TEXT -1 1 ":" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "de:=col lect(simplify(DE/x^(n-4)),x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#de G,0*&,**(%\"aG\"\"\"&%\"kG6#%\"nGF*)F.\"\"#F*F**(%\"dGF*F+F*F.F*F**&&% 'lambdaGF-F*F+F*F**(F)F*F+F*F.F*!\"\"F*)%\"xG\"\"%F*F**&,4*(F)F*&%(kst richGF-F*F/F*F**&F2F*F>F*F7*(%\"bGF*F+F*F.F*F7*&F4F*F>F*F**(FBF*F+F*F/ F*F***\"\"$F*F)F*F>F*F.F*F7*(%\"eGF*F+F*F.F*F**(F2F*F>F*F.F*F**(F0F*F) F*F>F*F*F*)F9FFF*F**&,<*&F4F*&%.kstrichstrichGF-F*F**(F0F*FBF*F>F*F**( F2F*FOF*F.F*F**(F)F*FOF*F/F*F**(\"\"'F*F)F*FOF*F***\"\"&F*F)F*FOF*F.F* F7*(%\"cGF*F+F*F/F*F**(FYF*F+F*F.F*F7*(FBF*F>F*F/F*F***FFF*FBF*F>F*F.F *F7*(FHF*F>F*F.F*F**(F0F*F2F*FOF*F7*&FHF*F>F*F7F*)F9F0F*F**&,2*(F0F*FY F*F>F*F**(FYF*F>F*F/F*F***FFF*FYF*F>F*F.F*F7*(FHF*FOF*F.F*F**(FUF*FBF* FOF*F**(FBF*FOF*F/F*F***FWF*FBF*FOF*F.F*F7*(F0F*FHF*FOF*F7F*F9F*F**(FU F*FYF*FOF*F***FWF*FYF*FOF*F.F*F7*(FYF*FOF*F/F*F*" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 262 93 "Koeffizientenvergleich beim h\366chsten Koeffizie nten liefert die bereits erw\344hnte Gleichung f\374r" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 263 1 ":" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "rule1:=lambda[n]=s olve(coeff(de,x,4),lambda[n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&r ule1G/&%'lambdaG6#%\"nG,$*&F)\"\"\",(*&%\"aGF,F)F,F,%\"dGF,F/!\"\"F,F1 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 20 "Wir setzen dies ein:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "de:=exp and(subs(rule1,de));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#deG,X**)%\" xG\"\"$\"\"\"%\"eGF*&%\"kG6#%\"nGF*F/F*F***)F(\"\"#F*%\"bGF*&%(kstrich GF.F*)F/F2F*F**,\"\"%F*F1F*%\"aGF*&%.kstrichstrichGF.F*F/F*!\"\"**F1F* %\"cGF*F,F*F6F*F***F2F*F'F*F9F*F4F*F***\"\"'F*F1F*F9F*F:F*F***F2F*F1F* F3F*F4F*F***FAF*F(F*F3F*F:F*F***F2F*F(F*F>F*F4F*F**(FAF*F>F*F:F*F**(F' F*%\"dGF*F4F*F<**F2F*F1F*FGF*F:F*F<*(F1F*F+F*F4F*F<**F2F*F(F*F+F*F:F*F <*,F2F*F'F*F9F*F4F*F/F*F<**F'F*F3F*F,F*F6F*F***F'F*F3F*F,F*F/F*F<*,F)F *F1F*F3F*F4F*F/F*F<**F(F*F3F*F:F*F6F*F**,\"\"&F*F(F*F3F*F:F*F/F*F<**F1 F*F>F*F,F*F/F*F<**F(F*F>F*F4F*F6F*F**,F)F*F(F*F>F*F4F*F/F*F<*(F>F*F:F* F6F*F***FQF*F>F*F:F*F/F*F<**F1F*F+F*F4F*F/F*F***F(F*F+F*F:F*F/F*F*" }} }{EXCHG {PARA 257 "" 0 "" {TEXT -1 122 "und machen Koeffizientenvergle ich beim zweith\366chsten Koeffizienten. Dies liefert k'[n] als ration ales Vielfaches von k[n]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "rule2:=kstrich[n]=solve(coeff(de,x,3),kstrich[n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rule2G/&%(kstrichG6#%\"nG**&%\"kGF(\"\"\"F)F-,( %\"eGF-%\"bG!\"\"*&F0F-F)F-F-F-,(*(\"\"#F-%\"aGF-F)F-F-%\"dGF-*&F5F-F6 F-F1F1" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 113 "Koeffizientenverglei ch beim zweith\366chsten Koeffizienten gibt k''[n] ebenfalls als ratio nales Vielfaches von k[n]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "rule3:=kstrichstrich[n]=solve(coeff(subs(rule2,de),x,2),kstrichstr ich[n]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&rule3G/&%.kstrichstrich G6#%\"nG,$*&#\"\"\"\"\"#F-*,&%\"kGF(F-F)F-,>**\"\"%F-%\"cGF-%\"aGF-F)F -!\"\"*&F5F-%\"dGF-F7*(F.F-F5F-F6F-F-**F.F-F5F-)F)F.F-F6F-F-*(F5F-F)F- F9F-F-*(\"\"$F-%\"bGF-%\"eGF-F-*&F.F-)F@F.F-F7*(\"\"&F-FCF-F)F-F-*$)FA F.F-F7**FEF-FAF-F@F-F)F-F7*(F4F-FCF-F " 0 "" {MPLTEXT 1 0 8 "k[n]:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%\"kG6#%\"nG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 13 "so das s gilt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "rule2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%(kstrichG6#%\"nG*(F'\"\"\",(%\"eGF)%\"bG! \"\"*&F,F)F'F)F)F),(*(\"\"#F)%\"aGF)F'F)F)%\"dGF)*&F1F)F2F)F-F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "rule3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%.kstrichstrichG6#%\"nG,$*,\"\"#!\"\"F'\"\"\",>**\"\" %F,%\"cGF,%\"aGF,F'F,F+*&F0F,%\"dGF,F+*(F*F,F0F,F1F,F,**F*F,F0F,)F'F*F ,F1F,F,*(F0F,F'F,F3F,F,*(\"\"$F,%\"bGF,%\"eGF,F,*&F*F,)F:F*F,F+*(\"\"& F,F=F,F'F,F,*$)F;F*F,F+**F?F,F;F,F:F,F'F,F+*(F/F,F=F,F6F,F+**F*F,F:F,F 6F,F;F,F,*&F=F,)F'F9F,F,*&FAF,F'F,F,F,,(*(F*F,F1F,F'F,F,F3F,*&F*F,F1F, F+F+,(*&F9F,F1F,F+F3F,*(F*F,F1F,F'F,F,F+F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 32 "Wir wollen nun die Koeffizienten" }{TEXT -1 1 " " } {XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT 267 7 "(n) und" }{TEXT -1 1 " " }{XPPEDIT 18 0 "gamma;" "6#%&gammaG" }{TEXT 268 46 "(n) in der Reku rsionsgleichung RE=0 bestimmen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "RE:=P(n+1)-(x-beta[n])*P(n)+gamma[n]*P(n-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG,(-%\"PG6#,&%\"nG\"\"\"F+F+F+*&,&%\"xGF+&%%bet aG6#F*!\"\"F+-F'F1F+F2*&&%&gammaGF1F+-F'6#,&F*F+F+F2F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "RE:=subs(\{P(n)=p,P(n+1)=subs(n=n+1 ,p),P(n-1)=subs(n=n-1,p)\},RE);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%# REG,,)%\"xG,&%\"nG\"\"\"F*F*F**&&%(kstrichG6#F(F*)F'F)F*F**&&%.kstrich strichGF.F*)F',&F)F*F*!\"\"F*F**&,&F'F*&%%betaG6#F)F5F*,(F/F**&&F-F:F* F3F*F**&&F2F:F*)F',&F)F*\"\"#F5F*F*F*F5*&&%&gammaGF:F*,(F3F**&&F-6#F4F *F@F*F**&&F2FIF*)F',&F)F*\"\"$F5F*F*F*F*" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 43 "Wie substitutieren die berechneten Formeln:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "RE:=subs(\{rule2,s ubs(n=n+1,rule2),subs(n=n-1,rule2),rule3,subs(n=n+1,rule3),subs(n=n-1, rule3)\},RE);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#REG,,)%\"xG,&%\"nG \"\"\"F*F*F***F(F*,(%\"eGF*%\"bG!\"\"*&F.F*F(F*F*F*,(*(\"\"#F*%\"aGF*F (F*F*%\"dGF**&F3F*F4F*F/F/)F'F)F*F**.F3F/F(F*,>**\"\"%F*%\"cGF*F4F*F(F *F/*&F**F;F*F**F;F*F " 0 "" {MPLTEXT 1 0 40 "re:=simplify(numer(normal (RE))/x^(n-3));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#reG,hhn*.\"$O$\" \"\")%\"xG\"\"$F(%\"eGF()%\"aG\"\"%F(%\"dGF()%\"nGF+F(!\"\"*,\"%#>#F(F )F(&%%betaG6#F2F()F.\"\"'F()F2\"\"#F(F(*0\"#'*F()F*FF(F?F(FIF(FenF(F-F(F0F(F(*0\"$'\\F(FMF(F@F(FIF (F,F(F0F(FFF(F3*,F/F(F?F(F1F(FEF(F\\qF(F(*0\"$g\"F(F*F(FMF(F,F(FQF(FVF (FFF(F(*,\"#BF(FMF(FEF(F;F(F\\qF(F(*,FcoF(F)F(F,F(F-F(F0F(F(*,\"%'4\"F (F)F(F@F(F;F(FfnF(F(*.\"$/(F(F*F(F6F(FQF(FenF(FfnF(F3*0\"#*)F(F*F(F6F( F;F(FinF(FVF(FTF(F(*.\"$?\"F(F)F(F6F(F\\qF(FTF(F;F(F(*0\"#5F(F*F(F6F(F 1F(FenF(F\\qF(F.F(F(*,\"#qF(FMF(FenF(FTF(FAF(F(*0\"$9'F(F*F(FMF(F,F(FV F(FFF(F2F(F3*,\"%g8F(F)F(F@F(FQF(FfnF(F(*,\"#IF(F)F(F,F(F\\qF(F.F(F3*. 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