sparse.linalg.eigs small eigenvalues of sparse.rand matrices: SM vs. shift-invert mode

[denis-bz]

Running sparse.linalg.eigs for small eigenvalues of 100 x 100 sparse.rand matrices,
shift-invert mode which='LM', sigma=0 converges in 10 of 10 cases, seed 0 .. 10;
which='SM' converges in 0 of 10 with e.g.

ARPACK error -1: No convergence (1001 iterations, 0/6 eigenvectors converged) [ARPACK error -14: DNAUPD  did not find any eigenvalues to sufficient accuracy.]

tol=1e-5 doesn't change the picture
nor does OPENBLAS_NUM_THREADS=1 .

It would be easy to always use shift-invert for small eigenvalues --

if which == "SM":
    print( "eigs: using shift-invert mode for small eigenvalues, "
        "because it's more robust than SM on large non-symmetric problems" )
    which = "LM"
    sigma = 0

what do you think ?

(Does anyone use 'SM' for small eigenvalues on real problems ?
Cf. Bai, Day, Demmel, Dongarra, "A test matrix collection for non-Hermitian eigenvalue problems", 1996.)

The testbench for this is 99 % the same as in #11198,
shall I put it up again, here or gist.github ?

Versions: numpy 1.17.4 scipy 1.3.3 python 3.7.3 macos 10.10.5

[pv]

iirc other users of ARPACK typically use LM + sigma=0 instead of SM

[denis-bz]

Yes that's right, but some people who want small eigenvalues may not know that,
and use which='SM' (small) instead of LM (large) -- confusing and wrong.
The above 5 lines are an attempt to help them.
Is the sentence ""eigs: using shift-invert mode ... because ..." clear ?

[pv]

By "users" I mean Octave/Matlab/et al. --- which iiuc don't use the arpack SM mode.

[denis-bz]

Right, don't think we need it either: it's confusing and sometimes plain doesn't work.
The suggestion is to turn SR --> LM, sigma=0 (with a message), then both ways work.

[lobpcg]

For 100 x 100 sparse matrices it's likely faster and surely reliable just to convert them into a dense array and use a dense solver to compute all eigenvalues.