matrix gauss-seidel method

how to solve this problem? ...

clear
clc

%A*Xi=B matrix
%i want to know Xi.. Thank you

A=[20 15 10; -3 -2.249 7; 5 1 3];
B=[45; 1.751; 9];

Xi=[1; 0; 1];
e1=100; e2=100; e3=100;
n=3;
f=0;
Xu=Xi

while e1>0.1 && e2>0.1 && e3>0.1
    for i=1:n
        for j=1:n
            if j==i
                f=f
            else
                f=f+A(i,j)*Xi(j)
            end
        end
        Xi(i)=(B(i)-f)/A(i,i)
        e1=abs((Xu(1)-Xi(1))/Xu(1))*100
        e2=abs((Xu(2)-Xi(2))/Xu(2))*100
        e3=abs((Xu(3)-Xi(3))/Xu(3))*100
        Xu=Xi
        f=0
    end
end

Xu

[Sam Chak]

Can you show the Gauss–Seidel equations or execution steps (algorithm) here for investigation purposes?

Answer 1

[Piyush Patil]

Hello,

You may refer to the following code to perform Gauss-Seidel Method -

clear
clc

A = [27 6 -1 ; 6 15 2 ; 1 1 54]
B = [85 ; 72 ; 110]

% You can try with this example also
% A = [12 3 -5 ; 1 5 3 ; 3 7 13]
% B = [1 ; 28 ; 76]

% A = [20 15 10 ; -3 -2.249 7 ; 5 1 3];
% B = [45 ; 1.751 ; 9];

x = linspace(0,0,length(A))';  % setting initial guess for 'x' values as 0
% x = [1;0;1]  % we can also set initial guess to values of our own choice

n = size(x, 1);  
normVal = Inf;
nmax = 10000; % setting maximum number of iterations to 10000
maxErr = 1e-3; % setting maximum error to 10^(-3)

itr = 0;
while normVal > maxErr && itr < nmax
    x_old = x;
        
    for i = 1:n
        
        sum = 0;
        
        for j = 1:i-1
            sum = sum + A(i,j)*x(j);
        end
        
        for j = i+1:n
            sum = sum + A(i,j)*x_old(j);
        end
        
        x(i) = (1/A(i,i)) * (B(i)-sum);
        
    end
    itr = itr + 1;
    normVal = norm(x_old - x);
    
end

disp(x)

Please note that not all systems of equations will converge using the Gauss-Seidel Method.

Gauss-Seidel Convergence Theorem - If "A" is Diagonally Dominant, then the Gauss-Seidel method converges for any starting vector x. A sufficient, but not necessary condition.

Diagonally Dominant - A nxn matrix "A" is diagonally dominant if - and for atleast one "i".

In simple terms, the coefficient on the diagonal must be at least equal to the sum of the other coefficients in that row and at least one row with a diagonal coefficient greater than the sum of the other coefficients in that row.

In your case, the matrix "A" is not diagonally dominant matrix. So, either the matrix has no unique solution, or we need a better starting guess for our variables so that the matrix will converge.