Python scipy eigh(Arpack) giving wrong eigenvalues for generalized eigenvalue problem

Question

I am trying to solve a generalized eigenvalue problem using Arpack, right now the code is using LAPACK but that's too slow, we only need a few eigenvalues and the matrices are sparse so using Arpack should be the way to go.

Before I start working with the original code I decided to test a simple case using scipy wrapper for Arpack (eigs) but the results that I am getting are wrong and change every time the code runs.

Minimum working example:

    import numpy as np
    from scipy.linalg import eig
    from scipy.sparse.linalg import eigs
    n = 8
    A = np.diag(np.arange(1,n+1,1.0))
    B = np.eye(n) # We want symmetric but a non-diagonal B. eigs gives correct answer for B=np.eye(n)
    B[0][n-1] = 2
    B[n-1][0] = 2


    evals,_ = eigs(A,k=3,M=B,which='LM') 
    print("The eigenvalues obtained by eigs (uses Arpack)")
    print(evals)

    print("Correct eigenvalues using eig (uses Lapack):")
    evals_l,_ = eig(A,b=B)
    print(evals_l)
```

Answer 1

The matrix B (M in the documentation) needs to positive definite according to the documentation: "If sigma is None, M is positive definite", this is in addition to the first requirement "M must represent a real, symmetric matrix if A is real" which your B follows. The eigenvalues of your current matrix B are -1, 1 and 6. So matrix B is not positive definite so it would stand to reason that eigs wouldn't work.

In comparison, the documentation for eig doesn't place any such limitation on the matrix B.

Answer 2

Not really an answer to your question, but the LAPACK implementation of QZ (which solves the GEP) is known to be pretty slow. My PR (https://github.com/Reference-LAPACK/lapack/pull/421) fixes that. You could build my fork from source and use that.

Also watch out when using MKL, they use blocking factor 1 for the HT reduction for some reason, really hampering performance. Depending on the entries, my PR achieves speedups of up to 250, so it might be worth a shot.