Question

Write a function to solve a system of linear equations of the form Ax= b using the iterative Gauss-Seidel method. You are free to use any basic MATLAB operation to implement the algorithm (i.e. you may use any combination of loops, indexing, math, etc.), but avoid “built-in” solution methods — you would not be allowed to use the GaussSeidel function if such a function existed. The function must also test for a number of possible issues. If an issue is encountered, you should use the error() command to issue a reasonable error message. Use distinct error messages for each of the following cases: • If the dimensions of the input matrices do not conform with each other • If the matrix of coefficients is not square

Answer 1

We need to solve a system of linear equations represented as

The general Gauss-Seidel algorithm for this system of equation is written as,

where for k =0, vector is known. a is the element of matrix A. b_i is the element of vector b. i represents the row and j represents the column.

MATLAB code.

clc;clear; close all;

%% Take the input
A = input('Enter the matrix of coefficients, A: ')
% Check for error
[row_A, column_A] = size(A); 
if row_A ~= column_A
    error('Matrix of coeffecient is not square.')
end

b = input('Enter the column vector, b: ')
% Check for error
[row_b, column_b] = size(b)
if column_A ~=size(b)
    error('Dimension of input matrices do not confirm with each other.')
end

x0 = input('Enter the initial guess vector, x0 :')
nItr = input('Enter the max no. of iterations :')

%% Gauss-Seidel iterations
for k=1:nItr % k=>iteration
    for i=1:length(b) % i=>unknown
        x(i,1) = (b(i)/A(i,i)) - (A(i,[1:i-1, i+1:length(b)])*x0([1:i-1, i+1:length(b)]))/...
            A(i,i);
        x0(i,1) = x(i,1); % update to use for next unknown(xi)
    end
    fprintf('Iteration no. %d\n', k)
    x
end