write a script to solve the linear system of equations using gauss seidel interation solver in matlab

Here's an example MATLAB script to solve a linear system of equations using Gauss-Seidel iteration solver:

main.m

% Define matrix A and vector b
A = [4 -1 0; 1 4.5 -2; 0 -2 4];
b = [2; 1; 2];

% Choose an initial guess for x
x0 = [0; 0; 0];

% Set the maximum number of iterations
maxit = 100;

% Set the tolerance
tol = 1.0e-6;

% Perform Gauss-Seidel iteration until convergence or maximum number of iterations is reached
x = x0;
for k = 1:maxit
    for i = 1:length(b)
        x(i) = (b(i) - A(i,1:i-1)*x(1:i-1) - A(i,i+1:end)*x0(i+1:end)) / A(i,i);
    end
    if norm(x - x0) < tol % check for convergence
        break;
    end
    x0 = x;
end

% Display the solution and number of iterations needed
fprintf('Solution:\n');
disp(x);
fprintf('Number of iterations: %d\n', k);

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In this script, matrix A represents the coefficients of the linear system of equations, while vector b represents the right-hand side of the equations. The initial guess for the solution x is set to [0; 0; 0], the maximum number of iterations is set to 100, and the tolerance is set to 1.0e-6. The script then performs the Gauss-Seidel iteration until convergence or maximum number of iterations is reached.

At each iteration, the script updates each component of x one at a time, using the current values of the other components, according to the Gauss-Seidel formula. The iteration stops when the difference between x and the previous estimate x0 is less than the tolerance tol.

Finally, the script displays the solution x and the number of iterations needed to achieve convergence.

Note that in practice, the convergence of the Gauss-Seidel iteration depends on the properties of the matrix A, and may not always occur or be very slow, in which case other iterative solvers or direct methods should be used instead.