JSFHLine\303\241rn\303\255 algebrarestart;B\304\233\305\276n\303\251 operace line\303\241rn\303\255 algebry jsou dostupn\303\251 v bal\303\255ku LinearAlgebrawith(LinearAlgebra);Zad\303\241v\303\241n\303\255 skal\303\241r\305\257, vektor\305\257 a matica:=2; #skal\303\241ru:=Vector([1,2,3]); #sloupcov\303\275 vektorv:=Vector[row]([4,5,6]); #\305\231\303\241dkov\303\275 vektorA:=Matrix([[1,3,5],[2,1,6],[1,0,2]]); #matice 3x3, prvky zad\303\241ny po \305\231\303\241dc\303\255ch
B:=Matrix([[1,1,1],[1,2,3],[2,3,4]]); E:=DiagonalMatrix([1,1,1]); #jednotkov\303\241 matice 3x3
E:=IdentityMatrix(3,4); #jednotkov\303\241 matice 3x4Operace s vektory a maticemiAT:=Transpose(A); #transponov\303\241n\303\255 matice
vT:=Transpose(v); #transponov\303\241n\303\255 vektoruApAT:=A+AT; #s\304\215\303\255t\303\241n\303\255 matic
upvT:=u+vT; #s\304\215\303\255t\303\241n\303\255 vektor\305\257aA:=a*A; #n\303\241soben\303\255 matice skal\303\241rem
au:=a*u; #n\303\241soben\303\255 vektoru skal\303\241remATA:=Multiply(AT,A); #n\303\241soben\303\255 matic, z\303\241le\305\276\303\255 na po\305\231ad\303\255
AAT:=Multiply(A,AT);
uv:=Multiply(u,v); #skal\303\241rn\303\255 sou\304\215in vektor\305\257, z\303\241le\305\276\303\255 na po\305\231ad\303\255
vu:=Multiply(v,u);
Au:=Multiply(A,u); #n\303\241soben\303\255 matic vektorem, z\303\241le\305\276\303\255 na po\305\231ad\303\255 i na typu
uA:=Multiply(u,A);
Av:=Multiply(A,v);
vA:=Multiply(v,A);nu:=Norm(u,Euclidean); #norma vektoru (Euklidovsk\303\241)rA:=Rank(A); #hodnost matice
rB:=Rank(B);detA:=Determinant(A); #determinant matice
detB:=Determinant(B);lambda:=Eigenvalues(A); #vlastn\303\255 \304\215\303\255sla matice, p\305\231esn\304\233
lambda:=evalf(Eigenvalues(A)); #vlast\303\255 \304\215\303\255sla matice, vy\304\215\303\255slena
lambda,h:=evalf(Eigenvectors(A)); #vlastn\303\255 \304\215\303\255sla a vektory matice, vy\304\215\303\255slenansA:=NullSpace(A);
nsB:=NullSpace(B); #b\303\241zeAG:=GaussianElimination(A); #Gaussova eliminace, p\305\231evod na HT matici
BG:=GaussianElimination(B);AQ,AR:=QRDecomposition(A); #QR rozklad matice
A=Multiply(AQ,AR);AP,AL,AU:=LUDecomposition(A); #LU rozklad matice, AP-permutacni, AL-dolni trojuhelnikova, AU-horni trojuhelnikova
A=Multiply(Multiply(AP,AL),AU);x:=LinearSolve(A,u); #\305\231e\305\241en\303\255 soustavy Ax=uGaussova-Jordanova eliminace (bez pivot\303\241\305\276e)Algoritmus nen\303\255 p\305\231ipraven na situaci, kdy se na diagon\303\241le objev\303\255 0. V takov\303\251m p\305\231\303\255pad\304\233 sel\305\276e. \303\232prava algoritmu pro takov\303\251to situace je p\305\231edm\304\233tem jednoho z dom\303\241c\303\255ch \303\272kol\305\257.Ab:=<A|u>; #zad\303\241n\303\255 matice A roz\305\241\303\255\305\231en\303\251 o vektor un,m:=Dimensions(Ab); #n-po\304\215et \305\231\303\241dk\305\257, m-po\304\215et sloupc\305\257 matice A#dop\305\231edn\303\275 chod
for i from 1 to n do
Ab[i,1..m]:=Ab[i,1..m]/Ab[i,i]; #vyd\304\233len\303\255 \305\231\303\241dku diagon\303\241ln\303\255m prvkem
for j from i+1 to n do
Ab[j,1..m]:=Ab[j,1..m]-Ab[j,i]*Ab[i,1..m]; #ode\304\215ten\303\255 Ab[j,i]-n\303\241sobku i-t\303\251ho \305\231\303\241dku od j-t\303\251ho \305\231\303\241dku
od:
od:Ab; #matice Ab p\305\231evedena na HT-tvar#zp\304\233tn\303\275 chod
for i from n by -1 to 2 do
for j from 1 to i-1 do
Ab[j,1..m]:=Ab[j,1..m]-Ab[j,i]*Ab[i,1..m];
od:
od:
Ab;
x:=Ab[1..n,m];JSFHJSFHTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzE0NjIyWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiJCIkIiIiIiIjIiIkRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzE0ODYyWColKWFueXRoaW5nRzYiRiVbZ2whJCUhISEiJCIkIiIlIiImIiInRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzE0OTgyWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQiIiIiIiNGJiIiJEYmIiIhIiImIiInRidGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzE1MTAyWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQiIiJGJiIiI0YmRiciIiRGJkYoIiIlRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzE1NzAyWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQiIiIiIiFGJ0YnRiZGJ0YnRidGJkYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzE1ODIyWCwlKWFueXRoaW5nRzYjJSlpZGVudGl0eUc2IltnbCEiIiEhISMhIiQiJUYnTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzE2MTgyWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQiIiIiIiQiIiYiIiNGJiIiJ0YmIiIhRilGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzE2MzAyWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiJCIkIiIlIiImIiInRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzE2NDIyWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQiIiMiIiYiIidGJ0YmRihGKEYoIiIlRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzE2NTQyWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiJCIkIiImIiIoIiIqRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzE2OTAyWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQiIiMiIiVGJiIiJ0YmIiIhIiM1IiM3RidGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzE3MTQyWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiJCIkIiIjIiIlIiInRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzE3MjYyWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQiIiciIiYiIz5GJyIjNSIjQEYoRioiI2xGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzE3MzgyWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQiI05GJiIjNkYmIiNUIiM5RidGKSIiJkYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzE4MjIyWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQiIiUiIikiIzciIiYiIzUiIzoiIidGKCIjPUYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzE4NTgyWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiJCIkIiNBRiYiIihGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDM3NTY2WColKWFueXRoaW5nRzYiRiVbZ2whJCUhISEiJCIkIiM/IiM8IiNpRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDI4NTM0WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiJCIkLCgqJCIjPCMiIiMiIiQjIiIiRisqJEYoRixGKSMiIiVGK0YtLCpGJyMhIiIiIidGLiNGM0YrRi9GLSooXiMjRi1GKkYtRitGOCwmRidGLEYuIyEiI0YrRi1GLSwqRidGMkYuRjVGL0YtKiheIyNGM0YqRi1GK0Y4RjlGLUYtRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDMwNjk0WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiJCIkJCIrSzJOXl8hIipeJCQhKm9gbkQnRigkIitLT1VTVSEjNV4kRiokIStLT1VTVUYuRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDIyNTAyWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiJCIkJCIrSzJOXl8hIipeJCQhKm9gbkQnRigkIitLT1VTVSEjNV4kRiokIStLT1VTVUYuRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDIyNjIyWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQkIitBMk5eSyEiKiQiK0Z1KDMlSEYoJCIiIiIiIV4kJCErbWBuREVGKCQiK1VPVVNVISM1XiQkIStHMHNQSUYzJCErXGc8NGdGM0YrXiRGLyQhK1VPVVNVRjNeJEY1JCIrXGc8NGdGM0YrRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDEzOTUwWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiJCIkIiIiISIjRiZGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDE0NzkwWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQiIiIiIiFGJyIiJCEiJkYnIiImISIlIyEiJEYqRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDA3ODE0WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQiIiIiIiFGJ0YmRiZGJ0YmIiIjRidGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDA4NTM0WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQsJCokIiInIyIiIiIiIyNGKkYoLCRGJyNGKiIiJEYmLCQqJCIkNSNGKSMiIzhGMiwkRjEjISIjIiQwIiwkRjEjISIiIiNVLCQqJCIjTkYpI0Y7Rj8sJEY+I0YvRj8sJEY+I0Y7IiIoRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDA4NDE0WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQqJCIiJyMiIiIiIiMiIiFGKywkRiYjIiImRicsJCokIiQ1I0YoI0YpRidGKywkRiYjIiM+RicsJEYwIyIjSkYxLCQqJCIjTkYoIyIiJEY7RiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDA4ODk0WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQiIiIiIiNGJiIiJEYmIiIhIiImIiInRidGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MTk0NzMwNjU0WC8lKWFueXRoaW5nRzYiRiVbZ2wjIiEhISEjKiIkIiQkJCEiIyQhISEhISEiIyQhISEhISEiIiIhISEhISFGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDA5NDk0WCwlKWFueXRoaW5nRzYjJiUrdHJpYW5ndWxhckc2IyUmbG93ZXJHNiJbZ2whIikhISEjJyIkIiQiIiIiIiNGK0YrIyIiJCIiJkYrRio=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDA5NzM0WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQiIiIiIiFGJyIiJCEiJkYnIiImISIlIyEiJEYqRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDEwNDU0WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQiIiIiIiNGJiIiJEYmIiIhIiImIiInRidGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDEwOTM0WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiJCIkIyIjSCIiJCMiIilGKCMhIzVGKEYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDAyODc4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjLSIkIiUiIiIiIiFGJ0YnRiZGJ0YnRidGJiMiI0giIiQjIiIpRiojISM1RipGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MjM1MDAwOTQyWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiJCIkIyIjSCIiJCMiIilGKCMhIzVGKEYl