%Lecture 22
% Illustrating the power algorithm
A=rand(2)
% get the real eigevalues and eigenvectors
[V,D]=eig(A)
% illustrate the power iteration 
x=[1 1]' % initialize
x=A*x % repeat usig the uparrow till the vector appears to converge (in direction)
% remark on the fact x either is blowing up or going to zero in magnitude
% go to lecture slides and illustrate solutions to this problem via
% normalization

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% illustrating the QR algorithm without shifts
A=rand(3)
[V,D]=eig(A);
[Q,R]=qr(A);A=R*Q; % repeat using uparrow
% remark on slow convergence to triangular form
% remark on 
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%
% illusrate QR algorithm with shifts
A=rand(3);
n = size(A,1);
I = eye(n,n);
s = A(n,n); [Q,R] = qr(A-s*I); A = R*Q+s*I  % repeat with uparrow
% remark on faster convergence