using Gauss Jacobi method to solve a system of equations

I'm new to Matlab so don't have the best understanding here, i need to use the jacobi method to find the values of C but i dont really know the code that i need to use.

My equations are:

(837*C3)/2000 - (6851*C2)/10000 - (1091*C1)/2500 - (821*C4)/2000

(1337*C1)/2000 + (437*C2)/2500 + (3899*C3)/10000 - (6087*C4)/10000

(1601*C2)/2500 - (2939*C1)/5000 - (477*C3)/10000 - (4921*C4)/10000

(1313*C1)/10000 - (3*C2)/10 - (8209*C3)/10000 - (292*C4)/625

A = [-0.4364 -0.6851 0.4185 -0.4105; 0.6685 0.1748 0.3899 -0.6087; -0.5878 0.6404 -0.0477 -0.4921; 0.1313 -0.3000 -0.8209 -0.4672];

B = [0;40;0;0];

please note that this is not diaganally dominant but ive been told to do it anyway - it is part of an assignment.

cheers.

Answer 1

[Vidhi Agarwal]

To solve Ax = B using Gauss-Jacobi method, we need to first transform the system of equations into an iterative form, as follows:

x1(k+1) = ( B1 - A12*x2(k) - A13*x3(k) - A14*x4(k) ) / A11

x2(k+1) = ( B2 - A21*x1(k) - A23*x3(k) - A24*x4(k) ) / A22

x3(k+1) = ( B3 - A31*x1(k) - A32*x2(k) - A34*x4(k) ) / A33

x4(k+1) = ( B4 - A41*x1(k) - A42*x2(k) - A43*x3(k) ) / A44

where x1(k), x2(k), x3(k), and x4(k) are the current approximations of the solution, and x1(k+1), x2(k+1), x3(k+1), and x4(k+1) are the next approximations of the solution.

We can start the iteration with an initial guess for the values of x1(k), x2(k), x3(k), and x4(k). Let's choose x1(0) = x2(0) = x3(0) = x4(0) = 0 as our initial guess.

Then, we can plug in the values of x1(0), x2(0), x3(0), and x4(0) into the iterative equations to obtain the next approximations of the solution, as follows:

Here's the Implementation of Jacobi Method:

A = [-0.4364 -0.6851 0.4185 -0.4105; 0.6685 0.1748 0.3899 -0.6087; -0.5878 0.6404 -0.0477 -0.4921; 0.1313 -0.3000 -0.8209 -0.4672];
B = [0;40;0;0];
% Set the initial guess for the solution values
x0 = [0;0;0;0];
% Set the maximum number of iterations
max_iter = 100;
% Set the convergence criteria
tolerance = 1e-6;
% Perform Gauss-Jacobi iteration
x = x0;
for i = 1:max_iter
    x_old = x;
    for j = 1:length(B)
        x(j) = (B(j) - sum(A(j,:) .* x_old) + A(j,j)*x_old(j)) / A(j,j);
    end
    
    % Check for convergence
    if norm(x - x_old) < tolerance
        fprintf('Solution converged after %d iterations\n', i);
        break;
    end
end
% Display the solution vector x
fprintf('Solution:\n');
disp(x);