Question

(a) Write the following function in Matlab [eval, evec] -power method (A, x, tol) The inputs are a matrix A, an initial starting vector r, and tolerance tol. The return value is an approximation to the largest eigenvalue (eval) of A, and the corresponding normalized eigenvector (evec) Your power method implementation should halt (i.e., converge) when this cri- teria is met: where (k) is the current approximate (normalized) eigenvector after k itera- tions. (Note that the "sign" of r() and ( must be compatible. For example, con- sider the vectors x = Į1: 0; 0]T and y = [-1; 0; 0]T, and note that if is an eigenvector of A, then y is also. That is, r and y denote the same eigenvector, but the norm of their difference is V2, not 0. To account for such cases, engineer your power method code to appropriately measure y. That is, in some (but not all) cases, you will need to multiply x(k-1) by-1, so that llx㈦-x(k-1)112 accurately measures the distance between two potential eigenvectors

Answer 1

MATLAB Script (Run it as a script, not from command window):

close all
clear
clc

A = [1 2 3; 4 5 6; 1 0 1];
x = [1;0;0];
tol = 1e-6;

[eval, evec] = power_method(A,x,tol);
fprintf('Using Power Method:\n')
fprintf('Approximate Dominant Eigen Value: %.8f\n', eval);
disp('Approximate Eigen Vector:')
disp(evec)

function [eval, evec] = power_method(A, x, tol)
evec = x;
eval = zeros(size(A,1), 1);
while true
evec_h = evec;
eval = (evec' * A * evec) / (evec' * evec);
evec = (A*evec) / norm(A*evec);
if ~isempty(find(evec_h.*evec < 0, 1))
% If the signs of the element of eigen vectors in
% consecutive iterations is not same, i.e. to make the signs
% compatible
evec_h = (-1)*evec_h;
end
if norm(evec - evec_h, 2) < tol
break
end
end
end

Output:

Using Power Method:
Approximate Dominant Eigen Value: 6.87298407
Approximate Eigen Vector:
0.3488
0.9353
0.0594