A MATLAB Program to Implement the Jacobi Iteration

A MATLAB Program to Implement Jacobi Iteration to Solve System of Linear Equations:

The following MATLAB codes uses Jacobi iteration formula to solve any system of linear equations where the coefficient matrix is diagonally dominant to achieve desired convergence.

%-----------------------------------------------------------------%

function [x, rel_error] = jacobimethod(A, b, x0, tol, iteration)

% Inputs: A - Coefficient matrix

%         b - Input matrix

%       tol - Defining tolerance for solution

% iteration - Number of iterations

% Outputs: x - Solutions

%  rel_error - Relative error

D = diag(diag(A));   % Making coefficient matrix diagonal

R = A - D;           % Construction of another matrix "R"

N = 1;               % iteration counter

x = x0;

rel_error = tol * 2; % norm(x - x0)/norm(x);

% Implementation of Jacobi method to solve Ax = b

while (rel_error>tol && N <= iteration)

   

    xprev = x;   

    x = inv(D)*(b - R*xprev);

    rel_error = norm(x - xprev)/norm(x);

    Z1=x(1);

    disp(Z1);

    Z2=x(10);

    disp(Z2);

    Z3=x(16);

    disp(Z3);

   

    fprintf('\n Iteration %i: Relative error =%d ',N, rel_error)   

    N = N + 1;       

end

%----------------------------------------------------------------%

Now, define your input matrices in the MATLAB command window and use the above developed function to compute Jacobi iteration.

MATLAB Command Prompt:

>> A = [-2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

    1 -2 1 0 0 0 0 0 0 0 0 0 0 0 0 0;

    0 1 -2 1 0 0 0 0 0 0 0 0 0 0 0 0;

    0 0 1 -2 1 0 0 0 0 0 0 0 0 0 0 0;

    0 0 0 1 -2 1 0 0 0 0 0 0 0 0 0 0;

    0 0 0 0 1 -2 1 0 0 0 0 0 0 0 0 0;

    0 0 0 0 0 1 -2 1 0 0 0 0 0 0 0 0;

    0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 0;

    0 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0;

    0 0 0 0 0 0 0 0 1 -2 1 0 0 0 0 0;

    0 0 0 0 0 0 0 0 0 1 -2 1 0 0 0 0;

    0 0 0 0 0 0 0 0 0 0 1 -2 1 0 0 0;

    0 0 0 0 0 0 0 0 0 0 0 1 -2 1 0 0;

    0 0 0 0 0 0 0 0 0 0 0 0 1 -2 1 0;

    0 0 0 0 0 0 0 0 0 0 0 0 0 1 -2 1;

    0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -2];

>> b = [-1; -2; -2; -2; -2; -2; -2; -2; -2; -2; -2; -2; -2; -2; -2; 1];

>> x0 = [0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0];

>> tol=10^-5;

>> iteration=350;

>> jacobimethod(A, b, x0, tol, iteration)

ans =

   14.8499

   28.7009

   40.5541

   50.4104

   58.2708

   64.1360

   68.0066

   69.8830

   69.7653

   67.6537

   63.5477

   57.4473

   49.3516

   39.2600

   27.1715

   13.0852