Question

QUESTION:

USING MATLAB, Carry out three iterations of the Gauss-Seidel method, starting from the initial vector Use the  ,  and  norm to calculate the residual error after each iteration, until all errors are below 0.0001.

[Use 5 decimal place accuracy in you calculations]

Answer 1

%Output:

% S = the solution( M x 1 matrix ; jacobi approximation)

% j = the number of iterations it took to

% converge to the user inputed value

% R = Residual Values

%establishes the variables needed

%B is an M x 1 matrix

%A is an M x M matrix

%P is the initial M x 1 matrix

%Z = remembering matrix

%Ask the user for each input statement required

Imax = input('What do you want the maximum iteration to be? ');

N = input('How many equations do you want? ');

C_n = input('What number do you want to converge to? ');

%Assigns the values inputed by the user into the matrices

for x=1:1:N

for y=1:1:N

strA = ['What do you desire your numbers in the matrix to be? ' num2str(x) 'Row: ' num2str(y) 'Column: '];

A = input(strA);

end

end

for l=1:1:N

strB = ('What do you desire the Solution matrix to be? ');

B = input(strB);

end

n = length(B);

X = zeros(n,1);

e = ones(n,1);

%%Check if the matrix A is diagonally dominant

for i = 1:n

j = 1:n;

j(i) = [];

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

Check(i,1) = 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(e) > C_n%check error

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,1) = (B(i,1) - sum(A(i,j) * Xtemp)) / A(i,i);

end

Xs = X;

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

end