# Aufgabe 3)
Digits = 6:
with(LinearAlgebra):
A:=Matrix([ [-4, 1, 0, 1, 0, 0, 0, 0, 0],
[ 1,-4, 1, 0, 1, 0, 0, 0, 0],
[ 0, 1,-4, 0, 0, 1, 0, 0, 0],
[ 1, 0, 0,-4, 1, 0, 1, 0, 0],
[ 0, 1, 0, 1,-4, 1, 0, 1, 0],
[ 0, 0, 1, 0, 1,-4, 0, 0, 1],
[ 0, 0, 0, 1, 0, 0,-4, 1, 0],
[ 0, 0, 0, 0, 1, 0, 1,-4, 1],
[ 0, 0, 0, 0, 0, 1, 0, 1,-4] ]);Ainverse:= MatrixInverse(A):
Kondition:=MatrixNorm( A, infinity)*MatrixNorm(Ainverse,infinity);
Seidel:= proc( A , B , eps)
local Tj,maxi,X,Xold,i,j,k,m,n;
n,m := Dimension(A) :
Tj:=Matrix(1..n,1..n,0):
for j from 1 by 1 while j <= n do
Tj[j,j]:= A[j,j] :
B[j]:=B[j]/A[j,j] :
end do :
for i from 1 by 1 while i <= n do
for k from 1 by 1 while k <= n do
Tj[i,k]:= (Tj[i,k] - A[i,k]) / A[i,i] :
end do :
end do :
maxi:=1000: X:=Vector(1..n,0) : Xold:=Vector(1..n,0) :
for j from 1 by 1 while (j <= 25 and maxi>=eps) do
for i from 1 by 1 while i<=n do
X[i]:= B[i] :
for k from 1 by 1 while k<=n do
X[i]:= X[i] + Tj[i,k]*X[k] :
end do:
end do:
print( j , evalf(Transpose(X)) ) ;
maxi:=0 :
for m from 1 by 1 while m <= n do
maxi:= max(abs(X[m]-Xold[m]), maxi ) :
Xold[m]:= X[m] :
end do :
end do :
if (j<=25) then " Gauss-Seidel converged " else " Gauss-Seidel NOT converged ! " end if ;
end proc:C:=< evalf(-sqrt(2)/2), -1, evalf(-sqrt(2)/2), 0, 0, 0, evalf(-sqrt(2)*exp(Pi)/2), evalf(-exp(Pi) ), evalf( -sqrt(2)*exp(Pi)/2) >;Nullvector:=<0,0,0,0,0,0,0,0,0>;Seidel ( A, Nullvector , 10^(-3) );Seidel ( A, C , 10^(-3) );TTdSMApJNFJUQUJMRV9TQVZFLzExMzAyNDRYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISNccCIqIiohIiUiIiIiIiFGKEYpRilGKUYpRilGKEYnCkYoRilGKEYpRilGKUYpRilGKEYnRilGKUYoRilGKUYpRihGKUYpRidGKEYpRihGKUYpRilGKEYpRihGJ0YoRilGKEYpRilGKUYoRilGKEYKJ0YpRilGKEYpRilGKUYoRilGKUYnRihGKUYpRilGKUYpRihGKUYoRidGKEYpRilGKUYpRilGKEYpRihGJ0YmCg==TTdSMApJNVJUQUJMRV9TQVZFLzIwNTUyMjYwWColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiKiIqJCIrX3B3bjwhIzUjIiIiIiIlRiciIiFGLUYtCiQiK3FedCE0JSEiKiQiK2dKPCZ5JkYwRi5GJgo=TTdSMApJNVJUQUJMRV9TQVZFLzIwMDc5MjgwWColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiKiIqIiIhRidGJ0YnRidGJ0YnRidGJ0YmCg==TTdSMApJNVJUQUJMRV9TQVZFLzE5Mjk3OTI0WColKWFueXRoaW5nRzYiNiJbZ2whJCUhISEiKiIqJCIiIUYoRidGJ0YnRidGJ0YnRidGJ0YmCg==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TTdSMApJNVJUQUJMRV9TQVZFLzIwNTUzMDIwWColKWFueXRoaW5nRzYiNiJbZ2whJCUhISEiKiIqJCIrOjowLigqISM1JCIrLWdPTTshIiokIis+CkVxLjhGLCQiK1lNUyopR0YsJCIraFE7IVIlRiwkIisoeWI0RSRGLCQiKzJKKFJOKEYsJCIrZCdwZTAiISIpJCIrZiNbY2EoRixGJgo=TTdSMApJNVJUQUJMRV9TQVZFLzIwNTUzMTAwWColKWFueXRoaW5nRzYiNiJbZ2whJCUhISEiKiIqJCIrZCE+eEkiISIqJCIrJVEnUis/RikkIitRClo2I1wiRikkIisxUydIRSRGKSQiKyg9YDJ4JUYpJCIrWiFIQFgkRikkIis5LjpZdkYpJCIrNTYzdjUhIikkIisyLVpUd0YpRiYKTTdSMApJNVJUQUJMRV9TQVZFLzIwNTUzMTgwWColKWFueXRoaW5nRzYiNiJbZ2whJCUhISEiKiIqJCIrJHo7RVwiISIqJCIrIT1yKSk9I0YpJCIrCl9uLShlIkYpJCIrdCtRX01GKSQiK115L2hcRikkIistaVFaTkYpJCIrakhgVHdGKSQiKyNmPVkzIiEiKSQiKysjRyIqbyhGKUYmCg==TTdSMApJNVJUQUJMRV9TQVZFLzIwNTUzMjYwWColKWFueXRoaW5nRzYiNiJbZ2whJCUhISEiKiIqJCIrMyYqMyhlIiEiKiQiK0Y1eiRHI0YpJCIrCi01ZE07RikkIishZTx1YSRGKSQiKyM9Jj5jXUYpJCIrJ2Z0XGYkRikkIitYZzgqbyhGKSQiKy0peSQqMyIhIikkIit1YiNIcihGKUYmCg==TTdSMApJNVJUQUJMRV9TQVZFLzIwNTUzMzQwWColKWFueXRoaW5nRzYiNiJbZ2whJCUhISEiKiIqJCIrWil5WGoiISIqJCIrZWlMSkJGKSQiK2ZUCk5lO0YpJCIrPnYoXGYkRikkIitbKW9QNSZGKSQiK1lAdz1PRikkIitibCNIcihGKSQiK1N5diI0IiEiKSQiKzEuI1tzKEYpRiYKTTdSMApJNVJUQUJMRV9TQVZFLzIwNTUzNDIwWColKWFueXRoaW5nRzYiNiJbZ2whJCUhISEiKiIqJCIrU15OZTshIiokIitQJj5eTiNGKSQiKztyCkNxO0YpJCIrT0V3PU9GKSQiKyFvYnY3JkYpJCIrdmRsSU9GKSQiK0gvI1tzKEYpJCIrRXMlSDQiISIpJCIrenJ3SXhGKUYmCg==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TTdSMApJNVJUQUJMRV9TQVZFLzIwNTUzNzQwWColKWFueXRoaW5nRzYiNiJbZ2whJCUhISEiKiIqJCIrKnlgMW8iISIqJCIrJT0+dVAjRikkIitHCnJSIm8iRikkIitZRDFUT0YpJCIrT2UmKVxeRikkIismKWUhPWskRikkIit1MChmdChGKSQiK1RBMSU0IiEiKSQiK1ZBTU94RilGJgo=TTdSMApJM1JUQUJMRV9TQVZFLzc1NTI2OFgqJSlhbnl0aGluZ0c2IjYiW2dsISQlISEhIioiKiQiK0dyUiJvIiEiKiQiK0JEO3lCRikkIisoem88Cm8iRikkIismKWUhPWskRikkIit2IipmXV5GKSQiK2F2PFVPRikkIitWQU1PeEYpJCIrMyUqNCU0IiEiKSQiK3khR2x0KEYpRiYK