Finding eigenvectors and eigenvalues of a sparse matrix with ARPACK ( called form PYTHON, MATLAB or as a FORTRAN subroutine)

问题

Few days ago I asked a question how to find the eigenvalues of a large sparse matrix. I got no answers, so I decided to describe a potential solution.

One question remains: 
Can I use the python implementation of ARPACK 
to compute the eigenvalues of a asymmetric sparse matrix.

At the beginning I would like to say that it is not at all necessary to call the subroutines of ARPACK directly using FOTRAN driver program. That is quite difficult and I never got it going. But one can do the following:

OPTION 1: Python

One can install numpy and scipy and run the following code:

import numpy as np
from scipy.linalg import eigh
from scipy.sparse.linalg import eigsh
from scipy.sparse import *
from scipy import * 

# coordinate format storage of the matrix

# rows
ii = array([0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4])
# cols.
jj = array([0, 1, 0, 1, 2, 1, 2, 3, 2, 3, 4])
# and the data
data=array([1.,-1.,-1., 2.,-2.,-2., 1., 1., 1., 1., 1.])
# now put this into sparse storage (CSR-format)
m=csr_matrix( (data,(ii,jj)), shape=(5,5) )
# you can check what you did 
matrix([[ 1, -1,  0,  0,  0],
        [-1,  2, -2,  0,  0],
        [ 0, -2,  1,  1,  0],
        [ 0,  0,  1,  1,  0],
        [ 0,  0,  0,  0,  1]])

# the real part starts here
evals_large, evecs_large = eigsh(m, 4, which='LM')
# print the largest 4 eigenvalues
print evals_all
# and the values are
[-1.04948118  1.          1.48792836  3.90570354]

Well this is all very nice, specially because it spears us the joy of reading the very "well written" manual of ARPACK.

I have a problem with this, I think that it doesn't work with asymmetric matrices. At least comparing the results to matlab was not very convincing.

OPTION 2: MATLAB

       % put your data in a file "matrix.dat"
       % row col. data
       % note that indexing starts at "1"
       1 1 1.
       1 2 -1.
       ......

       load matrix.dat
       M = spconvert(matrix)

       [v,d] = eig(M)

       % v - contains the eigenvectors
       % d - contains the eigenvalues

I think that using matlab is way simpler and works for asymmetric matrices. Well I have a 500000x500000 sparse matrix, so whether this will work in matlab .... Is another cup of tea! I have to note that using python I was able to load matrix of this size and compute it's eigenvalues without too much of a trouble.

Cheers,

来源：https://stackoverflow.com/questions/24613139/finding-eigenvectors-and-eigenvalues-of-a-sparse-matrix-with-arpack-called-for

标签

sparse-matrix

eigenvector

eigenvalue

arpack