# Gauss-Seidel method Help

45 views (last 30 days)
kemgi on 18 Apr 2020
Commented: Prabhan Purwar on 23 Apr 2020
Hello,
I am hoping someone can assist me with this problem. I created a Gauss-Seidel code that will allow me to solve a set of linear equations, finding x1, x2 x3 and x4. I made two matrices; A=[4 -21 -7 1; -4 0 -3 11 ; 4 -1 10 -1; 151/8 5 8 -3] and b =[11; 15; 19; -12;]. Through using the equatio Ax=b i would be able to find the unknows, which worked using the backslash built in solver. But my code below give 4 values for x but they do not match the reuslts of the built in solver and do not equate when substitued back into a line of the linear equations.
A major aspect of the code is that it is meant to make your matrix diagonally dominant to solve.
My code is as follows:
function gauss-seidel
A=input('write matrix a')
b=input('write matrix b')
x=linspace(0,0,length(A))';
n=size(x,1);
normVal=Inf;
nmax=1000; %number of maximum iterations which can be reached%
tol=0.000001; % Tolerence for method%
iter=0;
while normVal>tol && iter<nmax
x_old=x;
[V,A]=eig(A);
TF=isdiag(A);
if TF==0
fprintf('the matrix is not diagonally dominant')
else
for i=1:n
guess=0;
for j=1:i-1
guess=guess+A(i,j)*x(j);
end
for j=i+1:n
guess=guess+A(i,j)*x_old(j);
end
x(i)=(b(i)+guess)/(A(i,i));
end
iter=iter+1;
normVal=norm(x_old-x);
end
end
fprintf('Solution of the system is : ')
for i=1:length(x)
fprintf(' %1.4f ',x(i));
end
fprintf('in %d iterations ',iter)

John D'Errico on 22 Apr 2020
kemgi on 23 Apr 2020
I tried to delete the first one as I made a mistake, but I didn’t do it properly. There aren’t multiple questions there, just two within hundreds of posts. If you knew the answer I would have appreciated an answer. If you didn’t I would prefer you kept that to yourself as I doubt two posts affected you that much.

Prabhan Purwar on 22 Apr 2020
Hi,
Refer to the following code:
The following code performs Gauss-Seidel...
clc
clear
close all
A=[5 -2 3 0 6; -3 9 1 -2 7.4; 2 -1 -7 1 6.7; 4 3 -5 7 9; 2 3.5 6.1 -4 -8.1];
b=[-1 2 3 0.5 3.1]';
x=linspace(0,0,length(A))';
n=size(x,1);
normVal=Inf;
nmax=1000; %number of maximum iterations which can be reached%
tol=1e-3; % Tolerence for method%
iter=0;
while normVal>tol && iter<nmax
x_old=x;
for i=1:n
guess=0;
for j=1:i-1
guess=guess+A(i,j)*x(j);
end
for j=i+1:n
guess=guess+A(i,j)*x_old(j);
end
x(i)=(1/A(i,i))*(b(i)-guess);
end
iter=iter+1;
normVal=norm(x_old-x);
end
fprintf('Solution of the system is : \n%f\n%f\n%f\n%f\n%f in %d iterations',x,iter);
Output:
0.551479
0.469276
-0.595161
-0.649082
-0.171448 in 86 iterations
Tips:
The convergence properties of the Gauss-Seidel method are dependent on the matrix A. Namely, the procedure is known to converge if either:
https://www.mathworks.com/help/matlab/ref/chol.html (chol, only for symmetric matrix) [~,p] = chol(A)
OR
(Diagonal Dominant)
The Gauss-Seidel method sometimes converges even if these conditions are not satisfied.
Although you can prove that the methods converge if A has all eigenvalues positive from the above conditions (but not the only convergence criterion).
Hope it helps!!

kemgi on 23 Apr 2020
Thank you for taking your time to respond to this ?. I was meant to get 4 answers. I have managed to work out the answer.
Prabhan Purwar on 23 Apr 2020