Question

Problem 1:

Carry out the first three iterations of the solution of the following set of equations using the Gauss Seidel iterative method.Provide the solution using a program, you are free to use any language including MATLAB.

8x₁+2x₂+3x₃ = 51

2x₁+5x₂+x₃ = 23

-3x₁+x₂+6x₃ = 20

Answer 1

clc

clear all

close all

clear;clc

format compact

A = [8 2 3;

2 5 1;

-3 1 6];% coefficients matrix

C = [51;23;20];% constants vector

n = length(C);

X = zeros(n,1);

Error_eval = ones(n,1);

for i = 1:n

j = 1:n;

j(i) = [];

B = abs(A(i,j));

Check(i) = abs(A(i,i)) - sum(B); % Is the diagonal value greater than the remaining row values combined?

if Check(i) < 0

fprintf('The matrix is not strictly diagonally dominant at row %2i\n\n',i)

end

end

iteration = 0;

while max(Error_eval) > 0.001

iteration = iteration + 1;

Z = X; % save current values to calculate error later

for i = 1:n

j = 1:n; % define an array of the coefficients' elements

j(i) = []; % eliminate the unknow's coefficient from the remaining coefficients

Xtemp = X; % copy the unknows to a new variable

Xtemp(i) = []; % eliminate the unknown under question from the set of values

X(i) = (C(i) - sum(A(i,j) * Xtemp)) / A(i,i);

end

Xsolution(:,iteration) = X;

Error_eval = sqrt((X - Z).^2);

end

GaussSeidelTable = [1:iteration;Xsolution]'

MaTrIx = [A X C]

Result