whats the fastest way to find eigenvalues/vectors in python?

• **if your matrix is sparse, then instantiate your matrix using a constructor from scipy.sparse then use the analogous eigenvector/eigenvalue methods in spicy.sparse.linalg. From a performance point of view, this has two advantages:

  • your matrix, built from the spicy.sparse constructor, will be smaller in proportion to how sparse it is.

  • the eigenvalue/eigenvector methods for sparse matrices (eigs, eigsh) accept an optional argument, k which is the number of eigenvector/eigenvalue pairs you want returned. Nearly always the number required to account for the >99% of the variance is far less then the number of columns, which you can verify ex post; in other words, you can tell method not to calculate and return all of the eigenvectors/eigenvalue pairs–beyond the (usually) small subset required to account for the variance, it’s unlikely you need the rest.

• use the linear algebra library in SciPy, scipy.linalg, instead
  of the NumPy library of the same name. These two libraries have
  the same name and use the same method names. Yet there’s a difference in performance.
  This difference is caused by the fact that numpy.linalg is a
  less faithful wrapper on the analogous LAPACK routines which
  sacrifice some performance for portability and convenience (i.e.,
  to comply with the NumPy design goal that the entire NumPy library
  should be built without a Fortran compiler). linalg in SciPy on
  the other hand is a much more complete wrapper on LAPACK and which
  uses f2py.

• select the function appropriate for your use case; in other words, don’t use a function does more than you need. In scipy.linalg
  there are several functions to calculate eigenvalues; the
  differences are not large, though by careful choice of the function
  to calculate eigenvalues, you should see a performance boost. For
  instance:

  • scipy.linalg.eig returns both the eigenvalues and
    eigenvectors
  • scipy.linalg.eigvals, returns only the eigenvalues. So if you only need the eigenvalues of a matrix then do not use linalg.eig, use linalg.eigvals instead.
  • if you have a real-valued square symmetric matrices (equal to its transpose) then use scipy.linalg.eigsh

• optimize your Scipy build Preparing your SciPy build environement
  is done largely in SciPy’s setup.py script. Perhaps the
  most significant option performance-wise is identifying any optimized
  LAPACK libraries such as ATLAS or Accelerate/vecLib framework (OS X
  only?) so that SciPy can detect them and build against them.
  Depending on the rig you have at the moment, optimizing your SciPy
  build then re-installing can give you a substantial performance
  increase. Additional notes from the SciPy core team are here.

Will these functions work for large matrices?

I should think so. These are industrial strength matrix decomposition methods, and which are just thin wrappers over the analogous Fortran LAPACK routines.

I have used most of the methods in the linalg library to decompose matrices in which the number of columns is usually between about 5 and 50, and in which the number of rows usually exceeds 500,000. Neither the SVD nor the eigenvalue methods seem to have any problem handling matrices of this size.

Using the SciPy library linalg you can calculate eigenvectors and eigenvalues, with a single call, using any of several methods from this library, eig, eigvalsh, and eigh.

>>> import numpy as NP
>>> from scipy import linalg as LA

>>> A = NP.random.randint(0, 10, 25).reshape(5, 5)
>>> A
    array([[9, 5, 4, 3, 7],
           [3, 3, 2, 9, 7],
           [6, 5, 3, 4, 0],
           [7, 3, 5, 5, 5],
           [2, 5, 4, 7, 8]])

>>> e_vals, e_vecs = LA.eig(A)