qriter_test
The approximation to the smallest eigenvalue:
ans =
  -0.059675578871905
   1.564191203698703
   1.987691412628296
   2.349297531449158
   3.409078620563635
  -1.263166795003825
   0.386479838574264
   0.760280531365414
   0.892069805683633
   0.948593844059154
   0.974891322850854
   0.987588914811865
   0.993829600975851
   0.996923497448562
   0.998463910120760
   0.999232493287745
   0.999616380778874
   0.999808223821576
   0.999904120240510
   0.999952062194081
The error in the smallest eigenvalue:
ans =
  1.0597e+000
  5.6419e-001
  9.8769e-001
  1.3493e+000
  2.4091e+000
  2.2632e+000
  6.1352e-001
  2.3972e-001
  1.0793e-001
  5.1406e-002
  2.5109e-002
  1.2411e-002
  6.1704e-003
  3.0765e-003
  1.5361e-003
  7.6751e-004
  3.8362e-004
  1.9178e-004
  9.5880e-005
  4.7938e-005
type qriter_test.m

% Script: qriter_test.m
% Exapmle script for QR iteration to find eigenvalues
%

close all; clear all;
n = 10;
D = diag((1:n)');
V = randn(n);   % Interesting results if orthogonal matrix here instead!
Ak = V*D/V;

maxiter = 20;
approx = [];
for k=1:maxiter
    [Q,R] = qr(Ak); Ak = R*Q; approx = [approx Ak(n,n)];
end
format long
disp('The approximation to the smallest eigenvalue:')
approx'
plot(log(abs(approx)));
xlabel('k'); ylabel('smallest eigenvalue'); 

% now look at error behavior
err = abs( approx-1 );
figure
plot(log(err));
xlabel('k'); ylabel('error in smallest eigenvalue');
format short e
disp('The error in the smallest eigenvalue:')
err'
diary