polyeig calculating few polynomial eigenvalue

The eigs routine calculate only few eigenvalues compared to eig. How can few polynomial eigenvalue be calculated rather than all using polyeig?

[Raghava S N]

Could you share more details about the code you are working with? That will help in diagnosing the issue better.

[Fabre François]

A solution would be to linearize the eigenvalue problem and use eigs.

Answer 1

[Bruno Luong]

I'm affraid there is no equivalent of EIGS to POLYEIG (vs EIG).

Answer 2

[Christine Tobler]

While there is no equivalent to eigs for polyeig, if you just need some eigenvalues, you could step through polyeig.m and get what you need.

In the first step, polyeig takes the input matrices and computes a generalized eigenvalue problem of size n*p (where the inputs to polyeig were p+1 matrices of size n). Then, it simply calls eig on this generalized eigenvalue problem. At this point, you could instead call eigs with any number of eigenvalues you want to compute.

If you just need the eigenvalues, you're done at this point - the eigenvalues returned by eig are the eigenvalues of the polynomial eigenvalue problem. If you also need the eigenvectors, you would have to adapt the contents of the "if nargin > 2" branch in polyeig.m to only loop over as many eigenvectors as have been computed, instead of over all eigenvectors like is done in polyeig.

[Christine Tobler]

You would want to use sparse matrices instead of dense for computing A and B (speye instead of eye to initialize A, and sparse(nB, nB) instead of zeros(nB).

[Christine Tobler]

Since this has come up once or twice before, I've written a quick modification of the polyeig.m shipped with MATLAB so that it calls eigs to compute just a few eigenvalues and eigenvectors. This takes a cell array of the matrices passed to polyeig, followed by k, and optionally any additional inputs for eigs.

This is just a quick-and-dirty prototype, no guarantees here. But it should hopefully be easier to try this out than follow along my explanation above.