How can the Cholesky decomposition step in eigs() be avoided without passing a matrix to eigs that is a Cholesky decomposition?

Hello,

I have been looking at the following set of notes:

http://people.inf.ethz.ch/arbenz/ewp/Lnotes/chapter10.pdf

and specifically this quote in those notes:

"If SIGMA is a real or complex scalar including 0, EIGS finds the eigenvalues closest to SIGMA. For scalar SIGMA, and when SIGMA = ’SM’, B need only be symmetric (or Hermitian) positive semi-definite since it is not Cholesky factored as in the other cases."

I have a Hermitian positive-semidefinite matrix A, of which I want to find the 3 smallest eigenvalues. The Cholesky-decomposition is too memory intensive for the matrices I am working with. Please, is there a way to use eigs() without having to perform the Cholesky decomposition either in eigs() or outside of it?

Thank you very much.

Answer 1

[Walter Roberson]

Try

eigs(YourArray, 3, 'SM')

However, note that this requires that you be seeking the 3 eigenvalues with smallest absolute magnitude. If you need to find the smallest magnitude (e.g., -11.49 being smaller than -1.149) then you will not be able to use this option.

[Allo]

Hello and thank you for responding.

I tried this

D=eigs(M,6,'SM');

Because M=S*S, where S is my original Hermitian but not positive semi-definite matrix, because I wanted to submit a Hermitian positive semi-definite matrix to the eigs() function for the sake of exploiting the information I found above. It doesn't seem to be yielding the decrease in memory consumption that I expected. I thought the Cholesky decomposition of eigs() was supposed to require far more memory than the rest of eigs(), so I'm a bit confused.

[Allo]

S is a sparse matrix with roughly 10^6 non-zero entries, with a side length of N~160000, if that is useful... u_u

Answer 2

[Andrew Knyazev]

check http://www.mathworks.com/matlabcentral/fileexchange/48-lobpcg-m