to develop a MATLAB program for eigenvalue solving using MATRIX ITERATION method

I am trying to create a program which will compute eigenvalue using matrix iteration method. I just know some basic commands and still learning. I have created the following function but I know it has a lot of errors and is not complete. Can anyone plz help

function l = ww(m,k)

for i=1:1000

for j=1:1000

m = [i 0 0;0 2*i 0;0 0 i];

k = [2*j -j 0;-j 7*j -4*j;0 -4*j 5*j];

end

end

n = length(m);

y = [];

x = [];

for i = 1:n % starting vector

x(i) = m(i,1);

end;

l = 0;

blad = k; % starting value of error

while blad>=k

for i = 1:n % A*x

y(i) = 0;

for j = 1:n

y(i) = y(i) + m(i,j)*x(j);

end;

end;

blad = l;

l = 0; % Rayleigh

t = 0;

for i = 1:n

l = l + x(i)*y(i);

t = t + t(i)*x(i);

end;

l = l/t; % eigenvalue

blad = abs(l - blad); % error

x = y;

end;

end

Answer 1

[Vinay]

Hello Muhammad

Matrix iteration method calculates the eigen vector of a square matrix by starting with an initial guess for the eigen vector, repeatedly multiplying it by the matrix, and normalizing the result until convergence is achieved.

MATLAB function “rand” can be used for generating initial eigen vector and “norm” function for normalizing the eigen vector.

Kindly refer to the following documentations for “rand” and “norm” functions:

• https://www.mathworks.com/help/matlab/ref/rand.html
• https://www.mathworks.com/help/matlab/ref/norm.html?searchHighlight=norm&s_tid=srchtitle_support_results_1_norm

function [eigenvalue, eigenvector] = power_iteration(A, tol, max_iter)
    % A: Square matrix
    % tol: Tolerance 
    % max_iter: Maximum number of iterations

    [~,n] = size(A);
    % Initial guess for the eigenvector
    b_k = rand(n, 1);

    % Normalize the initial vector
    b_k = b_k / norm(b_k);

    % Iteration
    for k = 1:max_iter
        % Multiply by matrix A
        b_k1 = A * b_k;

        % Normalize the eigen vector
        b_k1 = b_k1 / norm(b_k1);

        % Check whether convergence is achieved 
        if norm(b_k1 - b_k) < tol
            break;
        end

        % Update the eigen vector
        b_k = b_k1;
    end

    % The dominant eigenvalue
    eigenvalue = b_k' * A * b_k;

    % The corresponding eigenvector
    eigenvector = b_k;
end

% Example
A = [4, 1; 5, 3];
tol = 1e-6;
max_iter = 1000;

[eigenvalue, eigenvector] = power_iteration(A, tol, max_iter);

disp(['Dominant Eigenvalue: ', num2str(lambda)]);
disp('Corresponding Eigenvector:');
disp(eigenvector);

I hope this helps