Bi7740: Scientific computing
Eigenvalues and eigenvectors - Exercises
Vlad Popovici
popovici@iba.muni.cz
Institute of Biostatistics and Analyses
Masaryk University, Brno
Vlad Bi7740: Scientific computing
Power method
function [v, lambda] = eig_power(A, x0, max_iter, ∆)
...
x0 = x0(:); % make it a column vector
lambda = 0;
for k = 1:max_iter
lambda1 = lambda;
x1 = A*x0/norm(x0,inf);
[xm,m] = max(abs(x1));
lambda = x1(m); % new eigenvalue: the ...
largest in new x
if (norm(x1−x0) < ∆) && (abs(lambda1−lambda) < ∆)
break
end
x0 = x1; % new approximation is x0
end
...
v = x0/norm(x0); % ensure unit norm
Vlad Bi7740: Scientific computing
Try it:
>> A = [2 0 1;0 −2 0;1 0 2];
>> [v,l] = eig_power(A)
v =
3
l =
0.7071
−0.0000
0.7071
>> [v,l] = eig(A)
v =
0 0.7071 0.7071
−1.000 0 0
0 −0.7071 0.7071
l =
−2 0 0
0 1 0
0 0 3
Vlad Bi7740: Scientific computing