问题

I'd like to find the N smallest eigenvalues of a sparse matrix in Python. I've tried using the scipy.sparse.linalg.eigen.arpack package, but it is very slow at computing the smallest eigenvalues. I read somewhere that there is a shift-invert mode, but when I try using it, I receive an error message telling me that the shift-invert mode is not yet supported. Any ideas as to how I should proceed?

回答1:

SciPy Versions

Comparing the documentation of scipy.sparse.linalg.eigs from SciPy v0.9 with the documentation of scipy.sparse.linalg.eigs from SciPy v0.10 it appears that the shift-invert mode is implemented and working since v0.10. Specifically, the explanation of the sigma parameter in the v0.9 documentation states it is not implemented, but the v0.10 documentation does not indicate that is the case.

If you do not have SciPy v0.10, or later, installing the latest should enable you to utilize shift-invert mode with the sparse eigensolver.

Slow Finding Small-magnitude Eigenvalues

As mentioned in the question, it is possible to use the ARPACK interface to find small-magnitude eigenvalues. This is done by passing which='SM' when calling scipy.sparse.linalg.eigs. It is, however, as stated in the question, slow. This is confirmed in the SciPy Tutorial's section on Sparse Eigenvalue Problems with ARPACK, where it states:

Note that ARPACK is generally better at finding extremal eigenvalues: that is, eigenvalues with large magnitudes. In particular, using which = 'SM' may lead to slow execution time and/or anomalous results. A better approach is to use shift-invert mode.

Experiments

Let's look at some code trying to use shift-invert with both v0.9 and v0.10 of SciPy. In both cases, we will use the following code.

from scipy.sparse import identity
from scipy.sparse.linalg import eigs

A = identity(10, format='csc')
A.setdiag(range(1, 11))
eigs(A, 3, sigma=0) # find three eigenvalues near zero using shift-invert mode

SciPy v0.9

Running the code in SciPy v0.9 results in an exception being raised.

NotImplementedError: shifted eigenproblem not supported yet

SciPy v0.10

Running the code in SciPy 0.10 produces expected results.

(array([ 1.+0.j,  2.+0.j,  3.+0.j]),
 array([[ -1.00000000e+00+0.j,   5.96300068e-17+0.j,   9.95488924e-17+0.j],
       [  3.55591776e-17+0.j,   1.00000000e+00+0.j,  -4.88997616e-16+0.j],
       [ -3.79110898e-17+0.j,   1.16635626e-16+0.j,   1.00000000e+00+0.j],
       [ -1.08397454e-17+0.j,   1.23544164e-17+0.j,   1.78854096e-15+0.j],
       [  1.68486368e-17+0.j,  -9.37965967e-18+0.j,   2.05571432e-16+0.j],
       [ -2.97859557e-19+0.j,  -3.43100887e-18+0.j,   3.35947574e-17+0.j],
       [  1.89565432e-17+0.j,  -3.61479402e-17+0.j,  -1.33021453e-17+0.j],
       [ -1.40925577e-18+0.j,   3.16953070e-18+0.j,   7.91193025e-17+0.j],
       [  6.76947854e-19+0.j,  -3.75674631e-19+0.j,   3.61821551e-17+0.j],
       [ -3.07505146e-17+0.j,  -6.52050102e-17+0.j,  -8.57423599e-16+0.j]]))

来源：https://stackoverflow.com/questions/9082782/computing-n-smallest-eigenvalues-of-sparse-matrix-in-python

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python