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

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Allo on 23 Jan 2012
Answered: Andrew Knyazev on 15 May 2015
I have been looking at the following set of notes:
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.

Answers (2)

Walter Roberson
Walter Roberson on 23 Jan 2012
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 on 23 Jan 2012
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

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