function [E , steps] = myqrmethod (A) % Computes all the eigenvalues of a matrix

Question

function [E , steps] = myqrmethod (A) % Computes all the eigenvalues of a matrix using the QR method. % Input: A-square matrix % Outputs: -vector of eigenvalues steps - the number of iterations it took m n]- size (A); warning('The input matrix is not square.') return end % Set up initial estimate H = hess (A) ; E diag(H); change = 1; steps 0; % loop while estimate changes while change> 0 Eold - E; % apply QR method E = diag(H); % test change change -norm (E Eold); steps = steps +1; end end As you can see the main steps of the program are very simple. The really hard calculations are contained in the built-in commands hess(A) and qr (H) Run this program and compare the results with MATLAB's built in command format long format compact Eqr, steps] myqrmethod (A) Emleig(A) diff norm (Eml - flipud (Eqr ) ) Exercises 17.1 Modify myqrmethod to stop after 1000 iterations. Use the modified program on the matrix A - hilb(n) with n equal to 10, 50, and 200. Use the norm to compare the results to the eigenvalues obtained from MATLAB's built-in program eig. Turn in a printout of your program and a brief report on the experiment

Explanation / Answer

function dummy=myqrmethod1()
clc;
clear all;
format long
format compact
z=[10 50 200];
for s=1:3
A=hilb(z(s));
N=z(s)
[Eqr, steps]=myqrmethod(A);
Eml=eig(A);
diff=norm(Eml-flipud(Eqr))

end

function [E, steps]=myqrmethod(A)
[m n]=size(A);
if (m ~=n)
    warning('the input matrix is not square')
    return
end
Max_step=1000;
H=hess(A);
E=diag(H);
change=1;
steps=0;
while(steps<Max_step)
    Eold=E;
    [Q R]=qr(H);
    H=R*Q;
    E=diag(H);
    change=norm(E-Eold);
    steps=steps+1;
end
end
end

%%% Solution'

N =
    10
diff =
     1.375210822846235e-15
N =
    50
diff =
     4.168496863958721e-15
N =
   200
diff =
     1.589990599203073e-14
>>

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