Issue computing eigenvalues/vectors and orthonormalization

General Usage
1

Hi there,

I am having trouble to compute eigenvectors and orthonormalize them.

I try to solve the generalized eigenvalues problem associated to matrices A and B.

A is not symmetric; B is symmetric but not necessary definit positive.

As A is not symmetric, one has to compute right and left eigenvectors, respectively X and Y.

After some tries, I managed to compute them (eigs from Arpack.jl was a fail; I ended up using ArnoldiMethod.jl with the shift and invert method; KrylovKit.jl does not apply as A is not symmetric nore hermitian).

So, I have now X and Y ans I want to bi-orthonormalize them, i.e., Y^T * B * X = Id (where ^T is the hermitian transpose) and Y^T * A * X = L with L matrix with the eigenvalues on diagonal.

I do this by computing Y^T * B * X = D and by doing for example X <= X * inv(D) I achieve to have Y^T * B * X = Id.

But Y^T * A * X works badly. The eigenvalues are on the diagonal. But extra diagonal terms are very large (1e-2 in norm) when it comes to the last eigenvalues (I compute 10 of them).

So here are my questions :

  • Is there another way to compute eigenvalues/vectors efficiently ?
  • Why is the orthonormalization working so badly ? It works well on matlab.
2

KK does not require this!!

3

I have the following error :

ERROR: LoadError: ArgumentError: Only symmetric or hermitian generalized eigenvalue problems with positive definite `B` matrix are currently supported.
4

for GEV, you need hermitian B.

5

The B matrix I use is hermitian.

issymmetric(B) = true

6

Your scheme works for me with results from ArnoldiMethod.jl on benign matrices, but your remarks suggest that your case is not so easy. Have you used a small tolerance and checked that the residuals satisfy it? Experimenting with different shifts may also help.

7

Maybe the eigenvalues at the edges of the window are badly converged? (I have no experience with ArnoldiMethod.jl but KrylovKit.jl does ensure convergence of the requested number of eigenvectors IIRC).

A quick check is to just request some more vectors and compare the resulting matrices.

Out of curiosity: How large are your matrices?

8

You mean with symmetric B and non symmetric A ?

I did not try on other matrices. I should give it a try.

9

Yes, I could compute more eigenvalues so that the ones I want are more converged.

My matrices are not so large, 30,000 x 30,000, and very sparse.

10

Ok it seems to work. Thanks !

I also tighten a bit the tolerance and it also improves the results.