How to compute interior eigenvectors that exclude certain eigenvalues?

I have a FEM matrix equation of the form:

(K - T)*x = T*b

Where T is a mass matrix and K is a stiffness matrix. I am using matlab's eigs function to compute the eigenvalues and eigenvectors of this system in a generalized eigenvalue problem where A = K-T and B = T.

The expected eigenspectrum is a flat line at and then a linearly increasing slope for . It seems as if avoiding the computation of eigenvectors siginificantly increases the speed of the eigs function. I currently try to avoid the computation by using the sigma option for eigs. Is there a better way to exclude certain eigenvalues from the eigs computation?

[Matt J]

What sigma option setting are you using? And what is 'b' in your equation?

[Lucas Banting]

I was using sigma=30, I figure if I know the slope of the eigenplot I can extrapolate a line and center point from lambda=-1.

[Matt J]

Seems like a good idea. I assume you're using eigs(A<B,k,30) with k>1. What isn't working well with that approach?

[Lucas Banting]

The issue is the computation speed is extremely slow when the lambda=-1 eigenvectors are computed, but if I shift the spectrum enough to avoid them it is much faster to compute eigenvectors.

[Matt J]

But once you've done your piecewise linear fit to the spectrum, you should be able to avoid processing lambda=-1. Just set sigma and k to include only lambda>-1. Isn't that what you are already doing, and if so what's wrong with it?

[Lucas Banting]

I was basically wondering if there was an eigenvalue algorithm where I could just specify as inputs (a, b) to compute all eigen values within the range (a, b).

Answer 1

[Matt J]

I was basically wondering if there was an eigenvalue algorithm where I could just specify as inputs (a, b) to compute all eigen values within the range (a, b).

It doesn't appear that there is, however, a faster way to compute the lambda=-1 eigenvectors might be to recognize that they are the null vectors of K, and so you can do,

[~,S,nullVectors]=svds(K,800,'smallest');

Not only should this find you the lambda=-1 eigenvectors, but also inspection of diag(S) should also tell you were the up-slope in your attached figure begins.

Together with the maximum eigenvectors,

eigmax=eigs(A,B,10,'largestabs')

you should be able to fit the slope more accurately than with sigma=30.

[Lucas Banting]

Thanks for your help. The solution I ended up using was limiting the MaxIterations and SubspaceDimension parameters, the eigenvectors associated with the nullspace of K converge last so they are just not computed in this case.

Answer 2

[Matt J]

If you'll be computing the majority of the eigenvalues anyway, it would be faster to use eig() than eigs().

[Lucas Banting]

The image I shared was only a portion of the eigen spectrum, the size of the matrices are practically one to two orders of magnitude larger than the number of needed eigenvectors.