Gauss-Seidel method Help

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]

Please stop asking the same question day after day.

[kemgi]

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.

[John D'Errico]

If you made a mistake, it is trivial to edit your question to fix it. But many people seem to post the same question as many as 5 or 6 times. If they don't get an answer quickly enough to suit their needs, they just keep posting the same question.

[Mahmoud]

Writing the negative exponent to give the same result as the calculator. Example: 0.3000 * 10^-3 gives a result of 3 * 10^-4

But in MATLAB it gives a very different result. What is the problem here or how to write the equation correctly to give the same value as the calculator

Answer 1

[Prabhan Purwar]

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:

• A is symmetric positive-definite

https://www.mathworks.com/help/matlab/ref/issymmetric.html (issymmetric)

https://www.mathworks.com/help/matlab/ref/chol.html (chol, only for symmetric matrix) [~,p] = chol(A)

OR

• A is strictly or irreducibly diagonally dominant

https://www.mathworks.com/matlabcentral/fileexchange/28225-dominant-diagonal?focused=5167817&tab=function

(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]

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]

Kindly accept the answer if its helpful...

[ahmed al-khazraji]

thanks for the hard work

Answer 2

[Teresa Matthews]

You should update the values of x for all elements within the loop. However, to avoid overwriting the current element being updated, you can store the geometry dash world temporarily and then assign them to x after the loop completes.

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_new = 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_new(j);
        end
        
        for j=i+1:n
                guess=guess+A(i,j)*x(j);
        end
        
        x_new(i)=(b(i)+guess)/(A(i,i));
    
  end
    x = x_new;
    iter=iter+1;
    normVal=norm(x-x_new);
  end
end

fprintf('Solution of the system is : ')
for i=1:length(x)
fprintf(' %1.4f  ',x(i));

end
fprintf('in %d iterations ',iter)