Spectral radius of matrix?

In order to find the spectral radius of a matrix, I commonly use
max(abs(eig(A)))
that computes all the eigenvalues. Is there a faster way? I'm interested only on the spectral radius, and don't need anything else..

Answers (1)

Ask the eigs function to return the largest magnitude eigenvalue.
% Generate a vector of eigenvalues suitable for the 'randcorr' option of the gallery function
x = rand(1, 100);
x = 100*x./sum(x);
% Generate a matrix whose eigenvalues are the elements of x
A = gallery('randcorr', x);
% Compute the largest magnitude eigenvalue
largest = eigs(A, 1, 'lm')
% Check: this should be small
largest - max(x)
If you're using release R2017b or later, you might want to use 'largestabs' instead of 'lm' in the eigs call. The old options are still accepted, but 'largestabs' is more descriptive of what it's doing than 'lm'.

5 Comments

X = rand(1000, 1000);
tic; e1 = max(abs(eig(X))); toc
tic; e2 = eigs(A, 1, 'lm'); toc
e1, e2
Elapsed time is 0.733548 seconds.
Elapsed time is 0.069295 seconds.
e1 =
499.5490
e2 =
499.5490
+1.
Since we are just using rand for a test case,
X = rand(1000, 1000);
tic; e1 = max(abs(eig(X))); toc
tic; e2 = eigs(X, 1, 'lm'); toc
tic;
n = rand(1000,1); % line added per comment below
for k = 1:4
n = X*n/max(abs(n));
end
e3 = mean(X*n./n);
toc
e1, e2, e3
Elapsed time is 1.426244 seconds.
Elapsed time is 0.021678 seconds.
Elapsed time is 0.004378 seconds.
e1 = 499.6231
e2 = 499.6231
e3 = 499.6231
:)
@David: What is n initially?
David Goodmanson
David Goodmanson on 12 May 2018
Edited: David Goodmanson on 12 May 2018
Hi Jan,
I forgot that line. The initial n is rand(1000,1), so I addended the comment.
The code was not a serious suggestion since it does not contain a tolerance check to stop the iteration, which might have slowed it down too much. But I found that for the special case of the random matrix it takes surprisingly few iterations to get a pretty accurate result.
eigs(X, 1, 'lm') will give negative answers

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