Jens Eising, Line\303\246r Algebra, Kapitel 1 Preben Alsholm, 13/12 2007 restart; with(LinearAlgebra): libname:="g:/Diplomat/lib01905",libname; with(DiploMat);

Koefficientmatricen A:=Matrix([[1,3,2,4,5],[2,6,4,3,5],[3,8,6,7,6],[4,14,8,10,22]]); H\303\270jresiden b:=<9,3,5,32>; Totalmatricen T:=<A|b>; F\303\270rst lader vi Maple l\303\270se opgaven selv: LinearSolve(T); Vi ser, at der er 2 frie variable, hvis navne er en smule grimme. Det bliver lidt k\303\270nnere, hvis man selv bestemmer, hvad de frie variable skal hedde: LinearSolve(T,free=t); Gausselimination: GaussianElimination(T); Reduceret echelonform: ReducedRowEchelonForm(T); For at opn\303\245 den endelige form i Eksempel 1.2, m\303\245 s\303\270jle 3 og 4 ombyttes: ColumnOperation(%,[3,4]); Det tilh\303\270rende ligningssystem er (husk at x3 og x4 er byttet om!) GenerateEquations(%,[x1,x2,x4,x3,x5]); solve(%,[x1,x2,x4]); subs(op(%),<x1,x2,x3,x4,x5>); I det f\303\270lgende bruges to hjemmelavede procedurer (de ligger i DiploMat-pakken) . ExpandVector(%); En s\303\270jlevis Gausselimination (ingen s\303\270jleombytninger): Describe(Gauss); Gauss(T,1); Gauss(%,2); Gauss(%,4);

Koefficientmatricen A:=Matrix([[1,3,-2,0],[1,2,3,-a],[3,7,a+3,-2*a],[0,1,-5,a^2],[1,1,8,-2*a]]); H\303\270jresiden b:=<0,-1,a-3,a,-2>; Totalmatricen T:=<A|b>; F\303\270rst lader vi Maple l\303\270se opgaven selv: LinearSolve(T); Man skal v\303\246re opm\303\246rksom p\303\245, at Maple uden blusel dividerer med udtryk, som ikke \303\245benlyst er nul, alts\303\245 udtryk som ikke i Maple reducerer til 0. Vi ser eksempelvis her, at Maple \303\245benbart har divideret med a og a er jo heller ikke en variabel, der under de givne omst\303\246ndigheder reducerer til 0 i Maple. Konklusionen er, at undtagelsestilf\303\246lde m\303\245 vi selv tage vare p\303\245. Det er slet ikke nok blot at kigge p\303\245 resultatet og se om dette indeholder br\303\270ker med n\303\246vnere, der kunne v\303\246re nul for visse v\303\246rdier af en parameter (her a). Ved Gausseliminationen kan nemlig udm\303\246rket v\303\246re foretaget r\303\246kkeoperationer, der foruds\303\246tter noget om parametrene. Eksempel: a*x=a*(a-1); solve(%,x); Ligningen har uendeligt mange l\303\270sninger for a=0, men dette ses jo ikke i Mapleresultatet! Vi m\303\245 alts\303\245 g\303\245 forsigtigt til v\303\246rks og unders\303\270ge hver eneste r\303\246kkeoperation. I Maple kan de n\303\270dvendige r\303\246kkeoperationer afl\303\246ses i L-matricen, som er midterste output fra kommandoen LUDecomposition(T); Den f\303\245s ogs\303\245 alene s\303\245ledes L:=LUDecomposition(T,output='L'); De tal der st\303\245r under diagonalen er de konstanter k(i, j) = L[i, j], der optr\303\246der ved r\303\246kkeoperationen R(i):=R(i) - k(i, j)*R(j) RowOperation(T,[2,1],-L[2,1]); RowOperation(%,[3,1],-L[3,1]); RowOperation(%,[4,1],-L[4,1]); RowOperation(%,[5,1],-L[5,1]); Gausselimination: GaussianElimination(T); Reduceret echelonform: ReducedRowEchelonForm(T); For at opn\303\245 den endelige form i Eksempel 1.2, m\303\245 s\303\270jle 3 og 4 ombyttes: ColumnOperation(%,[3,4]); Det tilh\303\270rende ligningssystem er (husk at x3 og x4 er byttet om!) GenerateEquations(%,[x1,x2,x4,x3,x5]); solve(%,[x1,x2,x4]); subs(op(%),<x1,x2,x3,x4,x5>); I det f\303\270lgende bruges to hjemmelavede procedurer (de ligger i DiploMat-pakken) . ExpandVector(%); En s\303\270jlevis Gausselimination (ingen s\303\270jleombytninger): Describe(Gauss); Gauss(T,1); Gauss(%,2); Gauss(%,4);

Koefficientmatricen A:=Matrix([[1,4,3,4],[3,1,-2,1],[-2,0,1,2],[2,1,-1,0]]); H\303\270jresiden b:=<2,6,-6,5>; Totalmatricen T:=<A|b>; Maplel\303\270sning direkte LinearSolve(T); Gausselimination til echelonform GaussianElimination(T); Helt til reduceret echelonform ReducedRowEchelonForm(T);

A:=Matrix([[a,a+1,-1],[1,0,1],[0,-(a+1)^2,1]]); b:=<0,0,2>; T:=<A|b>; Her indg\303\245r en parameter a, s\303\245 vi g\303\270r bedst i at g\303\245 forsigtigt til v\303\246rks: LUDecomposition(T); Fra midterste matrix ser vi, at vi ved Gausseliminationen bliver n\303\270dt til at skelne mellem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GV0Y5og LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=. F\303\270rst det 'generiske' tilf\303\246lde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=. GaussianElimination(T); Ved den fortsatte l\303\270sning optr\303\246der nu yderligere specialtilf\303\246ldene 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 og LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GUy1JI21uR0YkNiRRIjFGJ0Y5Rjk=. N\303\245r bortses fra de 3 specialtilf\303\246lde f\303\245s LinearSolve(T); Specialtilf\303\246ldet LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=: T1:=subs(a=0,T); LinearSolve(T1); Ingen l\303\270sning med andre ord! Specialtilf\303\246ldet LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GUy1JI21uR0YkNiRRIjJGJ0Y5Rjk=: T2:=subs(a=-2,T); LinearSolve(T2); Heller ingen l\303\270sning. Nu sidste specialtilf\303\246lde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GUy1JI21uR0YkNiRRIjFGJ0Y5Rjk=: T3:=subs(a=-1,T); LinearSolve(T3,free=t); Uendeligt mange l\303\270sninger.

b:='b': Da vi skal inds\303\246tte punkter om et \303\270jeblik, er det praktisk at g\303\270re ligningen til en funktion: ligning:=(x,y,z)->a*x+b*y+c*z+d=0; Vi inds\303\246tter de tre punkter: ligning(0,1,1),ligning(1,1,2), ligning(2,-1,-1); T:=GenerateMatrix([%],[a,b,c,d],augmented=true); LinearSolve(T,free=t); Vi v\303\246lger den frie parameter til -1: subs(t[4]=-1,%); zip(`=`,<a,b,c,d>,%); convert(%,list); L:=subs(%,ligning(x,y,z)); plots:-implicitplot3d(L,x=0..2,y=-1..1,z=-1..2,axes=normal); F:=solve(L,z); plot3d(F,x=-1..3,y=-2..2,axes=normal,style=patchnogrid); plots:-pointplot3d([[0,1,1],[1,1,2], [2,-1,-1]],symbol=solidsphere,symbolsize=25,color=maroon): plots:-display(%%,%);

Newton-Cotes: n:=3: h:=(b-a)/(n-1); NC:=f->add(w[k]*f(a+(k-1)*h),k=1..n); NC(f); Vi forlanger, at ovenst\303\245ende formel skal give integralets eksakte v\303\246rdi for alle polynomier af grad h\303\270jst n - 1, men udregner af hensyn til en senere bem\303\246rkning ogs\303\245 de ligninger, der svarer til inds\303\246ttelse af polynomier to grader h\303\270jere: for i from 1 to n+2 do g:=x->x^(i-1); ligning[i]:=NC(g)=int(g(x),x=a..b); end do; Totalmatricen for systemet af ligningerne 1, 2, ... , n er GenerateMatrix([seq(ligning[i],i=1..n)],[seq(w[i],i=1..n)],augmented=true); L\303\270sningen for v\303\246gtene w er LinearSolve(%); zip(`=`,<seq(w[i],i=1..n)>,%); subs(convert(%,list),NC(f)); Den konkrete formel for numerisk integration er s\303\245 factor(%); Tager vi en ligning mere med (polynomier af n'te grad) f\303\245s GenerateMatrix([seq(ligning[i],i=1..n+1)],[seq(w[i],i=1..n)],augmented=true); GaussianElimination(%); S\303\245 det ses, at de v\303\246gte, der sikrede at polynomier af grad h\303\270jst n - 1 integreres eksakt ved vores approksimation, ogs\303\245 sikrer eksakt integration af polynomier af grad n, hvis n er ulige! Hvis n er lige, s\303\245 opn\303\245s kun eksakt resultat for polynomier af grad n - 1. For polynomier af grad n + 1 g\303\245r den dog ikke uanset om n er ulige eller ej: GenerateMatrix([seq(ligning[i],i=1..n+2)],[seq(w[i],i=1..n)],augmented=true); GaussianElimination(%); Student-pakken indeholder en underpakke Calculus1, hvor proceduren ApproximateInt kan give det Simpsonresultat, som vi fandt ovenfor: Student:-Calculus1:-ApproximateInt(f(x),x=a..b,method=simpson,partition=1); Konkret eksempel: Student:-Calculus1:-ApproximateInt(sin(x),x=0..Pi/2,method=simpson,partition=1); evalf(%); Deles intervallet [a, b] i flere (her to) og udnyttes Simpsons basale formel p\303\245 hvert af disse f\303\245s: Student:-Calculus1:-ApproximateInt(f(x),x=a..b,method=simpson,partition=2); I det konkrete tilf\303\246lde: Student:-Calculus1:-ApproximateInt(sin(x),x=0..Pi/2,method=simpson,partition=2); evalf(%); 16 delintervaller: Student:-Calculus1:-ApproximateInt(sin(x),x=0..Pi/2,method=simpson,partition=16): evalf(%); Hvis vi i vore beregninger tager n = 4 f\303\245s Student:-Calculus1:-ApproximateInt(f(x),x=a..b,method=simpson[3/8],partition=1); Maple kalder ogs\303\245 en Newton-Cotes af orden n - 1 Student:-Calculus1:-ApproximateInt(f(x),x=a..b,method=newtoncotes[3],partition=1); Student:-Calculus1:-ApproximateInt(f(x),x=a..b,method=newtoncotes[4],partition=1);

h:='h': d1:=f->add(alpha[k]*f(a+k*h),k=-1..1): d2:=f->add(beta[k]*f(a+k*h),k=-1..1): d1(f); d2(f); Vi forlanger, at ovenst\303\245ende formler skal give eksakte v\303\246rdi for x = a for den f\303\270rste og anden afledede (henholdsvis) af alle polynomier af grad h\303\270jst 2. Den f\303\270rste afledede: for i from 0 to 2 do g:=x->x^i; ligning1[i]:=d1(g)=D(g)(a); end do; Den anden afledede: for i from 0 to 2 do g:=x->x^i; ligning2[i]:=d2(g)=(D@@2)(g)(a); end do; Totalmatricen for systemet for den f\303\270rste afledede T1:=GenerateMatrix([seq(ligning1[i],i=0..2)],[seq(alpha[i],i=-1..1)],augmented=true); Da systemet for den anden afledede har samme koefficientmatrix udelader vi 'augmented=true': GenerateMatrix([seq(ligning2[i],i=0..2)],[seq(beta[i],i=-1..1)]); Vi sammenskriver de to systemer i \303\251n totalmatrix (to h\303\270jresider): T:=<T1|%[2]>; L:=ReducedRowEchelonForm(T); L\303\270sningen for v\303\246gtene alpha er zip(`=`,<seq(alpha[i],i=-1..1)>,Column(L,4)); Den konkrete formel for numerisk differentiation (f\303\270rste afledede) er s\303\245 normal(subs(convert(%,list),d1(f))); L\303\270sningen for v\303\246gtene beta er zip(`=`,<seq(beta[i],i=-1..1)>,Column(L,5)); Den konkrete formel for numerisk differentiation (anden afledede) er s\303\245 normal(subs(convert(%,list),d2(f)));

u1:=x->(u(x+h)-u(x-h))/2/h; u2:=x->(u(x+h)-2*u(x)+u(x-h))/h^2; ligning:=diff(u(x),x,x)=-f(x); Randv\303\246rdiproblemet kan l\303\270ses p\303\245 simpel m\303\245de ved to integrationer. Her er blot brugt Maple: dsolve({ligning,u(0)=0,u(1)=0}); En approksimativ l\303\270sning kan findes s\303\245ledes: d_ligning:=subs(diff(u(x),x,x)=u2(x),ligning); n:=4: seq(subs(x=k*h,d_ligning),k=1..n-1); T:=GenerateMatrix(subs(u(n*h)=0,u(0)=0,[%]),[seq(u(k*h),k=1..n-1)],augmented=true); Rank(T); LinearSolve(T); Vi tager et konkret eksempel (lav selv om p\303\245 n): n:=6: h:=1/n: f:=x->(1+x)*(2-x); seq(subs(x=k*h,d_ligning),k=1..n-1); T:=GenerateMatrix(subs(u(n*h)=0,u(0)=0,[%]),[seq(u(k*h),k=1..n-1)],augmented=true); U:=LinearSolve(T); Sammenligning med det eksakte resultat: res:=dsolve({ligning,u(0)=0,u(1)=0}); plot([[[0,0],seq([k/n,U[k]],k=1..n-1),[1,0]],rhs(res)],x=0..1);

Float(89,7); addressof(%); d:=disassemble(%); kernelopts(dagtag=d[1]); pointto(d[2]); pointto(d[3]); dismantle(Float(89,7)); HFloat(89,7); addressof(%); d:=disassemble(%); kernelopts(dagtag=d[1]); pointto(d[2]); pointto(d[3]); dismantle(HFloat(89,7));

n:=5: P:=x->add(alpha[k]*x^k,k=0..n); P(x); m:=8: Lx:=[seq(x[k],k=1..m)]: Ly:=[seq(y[k],k=1..m)]: ligninger:=[seq(P(Lx[k])=Ly[k],k=1..m)]; A,b:=GenerateMatrix(ligninger,[seq(alpha[k],k=0..n)]);

n:=4: P:=x->add(alpha[k]*phi[k](x),k=1..n); P(x); m:=n: Lx:=[seq(x[k],k=1..m)]: Ly:=[seq(y[k],k=1..m)]: ligninger:=[seq(P(Lx[k])=Ly[k],k=1..m)]; A,b:=GenerateMatrix(ligninger,[seq(alpha[k],k=1..n)]);

f1:=x->1/(1+x^2)-2; f2:=x->1/(1+(x-0.1)^2)-1; f3:=x->0.5/(0.1+(x-0.2)^2); plot([f1,f2,f3],-1..1,thickness=3); p1:=%: Lx:=[-0.5,0,0.2,0.6]: Ly1:=map(f1,Lx); Ly2:=map(f2,Lx); Ly3:=map(f3,Lx); pkt1:=zip((x,y)->[x,y],Lx,Ly1); pkt2:=zip((x,y)->[x,y],Lx,Ly2); pkt3:=zip((x,y)->[x,y],Lx,Ly3); plot(pkt1,style=point,symbol=circle,symbolsize=20): plot(pkt2,style=point,symbol=circle,symbolsize=20): plot(pkt3,style=point,symbol=circle,symbolsize=20): plots:-display(p1,%,%%,%%%); pp:=%: expand(ChebyshevT(2,x)); n:=nops(Lx): P:=x->add(alpha[k]*ChebyshevT(k,x),k=0..n-1); P(x); ligninger1:=[seq(P(Lx[k])=Ly1[k],k=1..n)]; ligninger2:=[seq(P(Lx[k])=Ly2[k],k=1..n)]; ligninger3:=[seq(P(Lx[k])=Ly3[k],k=1..n)]; A,b1:=GenerateMatrix(ligninger1,[seq(alpha[k],k=0..n-1)]); A:=simplify(A); GenerateMatrix(ligninger2,[seq(alpha[k],k=0..n-1)]); b2:=%[2]; GenerateMatrix(ligninger3,[seq(alpha[k],k=0..n-1)]): b3:=%[2]; T:=<A|b1|b2|b3>; ReducedRowEchelonForm(T); res:=map(fnormal,%); P(x); zip(`=`,<seq(alpha[k],k=0..n-1)>,Column(res,n+1)); F1:=subs(convert(%,list),P(x)); zip(`=`,<seq(alpha[k],k=0..n-1)>,Column(res,n+2)); F2:=subs(convert(%,list),P(x)); zip(`=`,<seq(alpha[k],k=0..n-1)>,Column(res,n+3)); F3:=subs(convert(%,list),P(x)); plot([F1,F2,F3],x=-1..1,-2..6); plots:-display(pp,%);

Lx:=[seq(-0.8+k*0.2,k=0..8)]; Ly1:=map(f1,Lx); Ly2:=map(f2,Lx); Ly3:=map(f3,Lx); pkt1:=zip((x,y)->[x,y],Lx,Ly1); pkt2:=zip((x,y)->[x,y],Lx,Ly2); pkt3:=zip((x,y)->[x,y],Lx,Ly3); plot(pkt1,style=point,symbol=circle,symbolsize=20): plot(pkt2,style=point,symbol=circle,symbolsize=20): plot(pkt3,style=point,symbol=circle,symbolsize=20): plots:-display(p1,%,%%,%%%); pp:=%: n:=nops(Lx): P:=x->add(alpha[k]*ChebyshevT(k,x),k=0..n-1); P(x); ligninger1:=[seq(P(Lx[k])=Ly1[k],k=1..n)]: ligninger2:=[seq(P(Lx[k])=Ly2[k],k=1..n)]: ligninger3:=[seq(P(Lx[k])=Ly3[k],k=1..n)]: A,b1:=GenerateMatrix(ligninger1,[seq(alpha[k],k=0..n-1)]): A:=simplify(A): GenerateMatrix(ligninger2,[seq(alpha[k],k=0..n-1)]): b2:=%[2]: GenerateMatrix(ligninger3,[seq(alpha[k],k=0..n-1)]): b3:=%[2]: T:=<A|b1|b2|b3>; ReducedRowEchelonForm(T); res:=map(fnormal,%); zip(`=`,<seq(alpha[k],k=0..n-1)>,Column(res,n+1)): F1:=subs(convert(%,list),P(x)); zip(`=`,<seq(alpha[k],k=0..n-1)>,Column(res,n+2)): F2:=subs(convert(%,list),P(x)); zip(`=`,<seq(alpha[k],k=0..n-1)>,Column(res,n+3)): F3:=subs(convert(%,list),P(x)); plot([F1,F2,F3],x=-1..1,-2..6); plots:-display(pp,%);

with(CurveFitting); Lx; Ly1; PolynomialInterpolation(Lx,Ly1,x); map(fnormal,%); simplify(F1);

Lx:=[1,2,3,4.5,6,8]: Ly:=[0.87,1.14,1.71,2.9,3.78,4.61]: with(CurveFitting); res:=LeastSquares(Lx,Ly,x,curve=alpha*x); pkt:=zip((x,y)->[x,y],Lx,Ly): plot(pkt,style=point,symbol=solidcircle,symbolsize=20): plot(res,x=0..8,color=blue): plots:-display(%%,%);

L:=[x1-x3+x4 = 0,x1+x2+x3+x4=1,4*x1+4*x2+4*x3+3*x4 = 5]; A,b:=GenerateMatrix(L,[x1,x2,x3,x4]); T:=<A|b>; Alternativt T:=GenerateMatrix(L,[x1,x2,x3,x4],augmented=true); LinearSolve(T,free=t); ExpandVector(%); Detaljer: GaussianElimination(T); Vi ser, at x3 er fri. GenerateEquations(%,[x1,x2,x3,x4]); solve(%,[x1,x2,x4]); subs(op(%),<x1,x2,x3,x4>); ExpandVector(%);