How expensive is it to compute the eigenvalues of a matrix?

Question

How expensive is it to compute the eigenvalues of a matrix?

What is the complexity of the best algorithms?

How long might it take in practice if I have a 1

Answer 1

With big matrices you usually don't want all the eigenvalues. You just want the top few to do (say) a dimension reduction.

The canonical algorithm is the Arnoldi-Lanczos iterative algorithm implemented in ARPACK:

www.caam.rice.edu/software/ARPACK/

There is a matlab interface in eigs:

http://www.mathworks.com/access/helpdesk/help/techdoc/index.html?/access/helpdesk/help/techdoc/ref/eigs.html

eigs(A,k) and eigs(A,B,k) return the k largest magnitude eigenvalues.

And there is now an R interface as well:

http://igraph.sourceforge.net/doc-0.5/R/arpack.html