Question

Implement the following simple version of QR iteration with shifts for computing the eigenvalues of a general real matrix A Repeat until convergence: 1. o an,n (use corner entry shift) C as 2. Compute QR factorization QR= A- oI 3. A RQ ol (These steps will be casy if you use a package such as MATLAB but more involved if library routine for the QR factorization or write your own.) What convergence test should you use? Test your program on the following you use a matrix 6 2 1 A 2 3 1

Answer 1

MATLAB Script:

close all
clear
clc

A = [6 2 1; 2 3 1; 1 1 1];
l = find_eig_vals(A);
fprintf('EigenValue: %f\n', l)

function out = find_eig_vals(A)
n = size(A,1);
tol = 1e-12; % tolerance
sigma = Inf;
for i = 1:10
sigma_prev = sigma;
sigma = A(n,n);
if abs(sigma - sigma_prev) < tol % convergence criteria
break
else
[Q,R] = qr(A - sigma*eye(n));
A = R*Q + sigma*eye(n);
end
end
out = sigma;
end

Output:

EigenValue: 0.578933