Why do eigh and eigsh from scipy.sparse.linalg as used below give incorrect results when solving the generalized eigenvalue problem A * x = lambda * M * x , if M is non-diagonal?
import mkl
import numpy as np
from scipy import linalg as LA
from scipy.sparse import linalg as LAsp
from scipy.sparse import csr_matrix
A = np.diag(np.arange(1.0,7.0))
M = np.array([[ 25.1, 0. , 0. , 17.3, 0. , 0. ],
[ 0. , 33.6, 16.8, 8.4, 4.2, 2.1],
[ 0. , 16.8, 3.6, 0. , 11. , 0. ],
[ 17.3, 8.4, 0. , 4.2, 0. , 9.5],
[ 0. , 4.2, 11. , 0. , 2.7, 8.3],
[ 0. , 2.1, 0. , 9.5, 8.3, 4.4]])
Asp = csr_matrix(np.matrix(A,dtype=float))
Msp = csr_matrix(np.matrix(M,dtype=float))
D, V = LA.eig(A, b=M)
eigno = 4
Dsp0, Vsp0 = LAsp.eigs(csr_matrix(np.matrix(np.dot(np.linalg.inv(M),A))),
k=eigno,which='LM',return_eigenvectors=True)
Dsp1, Vsp1 = LAsp.eigs(Asp,k=eigno,M=Msp,which='LM',return_eigenvectors=True)
Dsp2, Vsp2 = LAsp.eigsh(Asp,k=eigno,M=Msp,which='LA',return_eigenvectors=True,
maxiter=1000)
From LA.eig and checking with MatLab the eigenvalues for this small generalized eigenvalue problem with test matrices A and M should be:
D = [ 0.7208+0.j, 0.3979+0.j, -0.3011+0.j, -0.3251+0.j, 0.0357+0.j, 0.0502+0.j]
I want to use sparse matrices because the actual A and M matrices involved are around 30,000 x 30,000. A is always square, real and diagonal, M is always square, real and symmetric. When M is diagonal I get the correct results. However, both eigs and eigsh give incorrect results when solving the generalized eigenvalue problem for a non-diagonal M matrix.
Dsp1 = [-1.6526+2.3357j, -1.6526-2.3357j, -0.6243+2.7334j, -0.6243-2.7334j]
Dsp2 = [ 2.01019097, 3.09248265, 4.06799498, 7.01216316]
When I convert the problem to the standard eigenvalue form M^-1 * A * x = lambda * x, eigs gives the correct result (Dsp0). For large matrices this is not an option because it takes too long to compute the inverse of M.
I noticed that using mkl or not yields different Dsp1 and Dsp2 eigenvalues as well. Could this eigenvalue problem be caused by an issue with my Python installation? I am running Python 2.7.8 anaconda with SciPy 0.15.1 - np19py27_p0 [mkl] on Mac OS 10.10.2.
Both eigs and eigsh require that M be positive definite (see the descriptions of M in the docstrings for more details).
Your matrix M is not positive definite. Note the negative eigenvalues:
In [212]: M
Out[212]:
array([[ 25.1, 0. , 0. , 17.3, 0. , 0. ],
[ 0. , 33.6, 16.8, 8.4, 4.2, 2.1],
[ 0. , 16.8, 3.6, 0. , 11. , 0. ],
[ 17.3, 8.4, 0. , 4.2, 0. , 9.5],
[ 0. , 4.2, 11. , 0. , 2.7, 8.3],
[ 0. , 2.1, 0. , 9.5, 8.3, 4.4]])
In [213]: np.linalg.eigvals(M)
Out[213]:
array([ 45.92443169, 33.92113421, -13.12639751, -10.6991868 ,
5.34183619, 12.23818222])