Sparse Eigenvalue Problems with ARPACK
======================================
.. sectionauthor:: Jake Vanderplas <vanderplas@astro.washington.edu>
.. currentmodule:: scipy.sparse.linalg
Introduction
------------
ARPACK is a Fortran package which provides routines for quickly finding a few
eigenvalues/eigenvectors of large sparse matrices. In order to find these
solutions, it requires only left-multiplication by the matrix in question.
This operation is performed through a *reverse-communication* interface. The
result of this structure is that ARPACK is able to find eigenvalues and
eigenvectors of any linear function mapping a vector to a vector.
All of the functionality provided in ARPACK is contained within the two
high-level interfaces :func:`scipy.sparse.linalg.eigs` and
:func:`scipy.sparse.linalg.eigsh`. :func:`eigs`
provides interfaces to find the
eigenvalues/vectors of real or complex nonsymmetric square matrices, while
:func:`eigsh` provides interfaces for real-symmetric or complex-hermitian
matrices.
Basic Functionality
-------------------
ARPACK can solve either standard eigenvalue problems of the form
.. math::
A \mathbf{x} = \lambda \mathbf{x}
or general eigenvalue problems of the form
.. math::
A \mathbf{x} = \lambda M \mathbf{x}
The power of ARPACK is that it can compute only a specified subset of
eigenvalue/eigenvector pairs. This is accomplished through the keyword
``which``. The following values of ``which`` are available:
* ``which = 'LM'`` : Eigenvalues with largest magnitude (``eigs``, ``eigsh``),
that is, largest eigenvalues in the euclidean norm of complex numbers.
* ``which = 'SM'`` : Eigenvalues with smallest magnitude (``eigs``, ``eigsh``),
that is, smallest eigenvalues in the euclidean norm of complex numbers.
* ``which = 'LR'`` : Eigenvalues with largest real part (``eigs``)
* ``which = 'SR'`` : Eigenvalues with smallest real part (``eigs``)
* ``which = 'LI'`` : Eigenvalues with largest imaginary part (``eigs``)
* ``which = 'SI'`` : Eigenvalues with smallest imaginary part (``eigs``)
* ``which = 'LA'`` : Eigenvalues with largest algebraic value (``eigsh``),
that is, largest eigenvalues inclusive of any negative sign.
* ``which = 'SA'`` : Eigenvalues with smallest algebraic value (``eigsh``),
that is, smallest eigenvalues inclusive of any negative sign.
* ``which = 'BE'`` : Eigenvalues from both ends of the spectrum (``eigsh``)
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*.
Shift-Invert Mode
-----------------
Shift invert mode relies on the following observation. For the generalized
eigenvalue problem
.. math::
A \mathbf{x} = \lambda M \mathbf{x}
it can be shown that
.. math::
(A - \sigma M)^{-1} M \mathbf{x} = \nu \mathbf{x}
where
.. math::
\nu = \frac{1}{\lambda - \sigma}
Examples
--------
Imagine you'd like to find the smallest and largest eigenvalues and the
corresponding eigenvectors for a large matrix. ARPACK can handle many
forms of input: dense matrices such as :func:`numpy.ndarray` instances, sparse
matrices such as :func:`scipy.sparse.csr_matrix`, or a general linear operator
derived from :func:`scipy.sparse.linalg.LinearOperator`. For this example, for
simplicity, we'll construct a symmetric, positive-definite matrix.
>>> import numpy as np
>>> from scipy.linalg import eigh
>>> from scipy.sparse.linalg import eigsh
>>> np.set_printoptions(suppress=True)
>>>
>>> np.random.seed(0)
>>> X = np.random.random((100,100)) - 0.5
>>> X = np.dot(X, X.T) #create a symmetric matrix
We now have a symmetric matrix ``X`` with which to test the routines. First
compute a standard eigenvalue decomposition using ``eigh``:
>>> evals_all, evecs_all = eigh(X)
As the dimension of ``X`` grows, this routine becomes very slow. Especially
if only a few eigenvectors and eigenvalues are needed, ``ARPACK`` can be a
better option. First let's compute the largest eigenvalues (``which = 'LM'``)
of ``X`` and compare them to the known results:
>>> evals_large, evecs_large = eigsh(X, 3, which='LM')
>>> print evals_all[-3:]
[ 29.1446102 30.05821805 31.19467646]
>>> print evals_large
[ 29.1446102 30.05821805 31.19467646]
>>> print np.dot(evecs_large.T, evecs_all[:,-3:])
array([[-1. 0. 0.], # may vary (signs)
[ 0. 1. 0.],
[-0. 0. -1.]])
The results are as expected. ARPACK recovers the desired eigenvalues, and they
match the previously known results. Furthermore, the eigenvectors are
orthogonal, as we'd expect. Now let's attempt to solve for the eigenvalues
with smallest magnitude:
>>> evals_small, evecs_small = eigsh(X, 3, which='SM')
Traceback (most recent call last): # may vary (convergence)
...
scipy.sparse.linalg.eigen.arpack.arpack.ArpackNoConvergence:
ARPACK error -1: No convergence (1001 iterations, 0/3 eigenvectors converged)
Oops. We see that as mentioned above, ``ARPACK`` is not quite as adept at
finding small eigenvalues. There are a few ways this problem can be
addressed. We could increase the tolerance (``tol``) to lead to faster
convergence:
>>> evals_small, evecs_small = eigsh(X, 3, which='SM', tol=1E-2)
>>> evals_all[:3]
array([0.0003783, 0.00122714, 0.00715878])
>>> evals_small
array([0.00037831, 0.00122714, 0.00715881])
>>> np.dot(evecs_small.T, evecs_all[:,:3])
array([[ 0.99999999 0.00000024 -0.00000049], # may vary (signs)
[-0.00000023 0.99999999 0.00000056],
[ 0.00000031 -0.00000037 0.99999852]])
This works, but we lose the precision in the results. Another option is
to increase the maximum number of iterations (``maxiter``) from 1000 to 5000:
>>> evals_small, evecs_small = eigsh(X, 3, which='SM', maxiter=5000)
>>> evals_all[:3]
array([0.0003783, 0.00122714, 0.00715878])
>>> evals_small
array([0.0003783, 0.00122714, 0.00715878])
>>> np.dot(evecs_small.T, evecs_all[:,:3])
array([[ 1. 0. 0.], # may vary (signs)
[-0. 1. 0.],
[ 0. 0. -1.]])
We get the results we'd hoped for, but the computation time is much longer.
Fortunately, ``ARPACK`` contains a mode that allows quick determination of
non-external eigenvalues: *shift-invert mode*. As mentioned above, this
mode involves transforming the eigenvalue problem to an equivalent problem
with different eigenvalues. In this case, we hope to find eigenvalues near
zero, so we'll choose ``sigma = 0``. The transformed eigenvalues will
then satisfy :math:`\nu = 1/(\sigma - \lambda) = 1/\lambda`, so our
small eigenvalues :math:`\lambda` become large eigenvalues :math:`\nu`.
>>> evals_small, evecs_small = eigsh(X, 3, sigma=0, which='LM')
>>> evals_all[:3]
array([0.0003783, 0.00122714, 0.00715878])
>>> evals_small
array([0.0003783, 0.00122714, 0.00715878])
>>> np.dot(evecs_small.T, evecs_all[:,:3])
array([[ 1. 0. 0.], # may vary (signs)
[ 0. -1. -0.],
[-0. -0. 1.]])
We get the results we were hoping for, with much less computational time.
Note that the transformation from :math:`\nu \to \lambda` takes place
entirely in the background. The user need not worry about the details.
The shift-invert mode provides more than just a fast way to obtain a few
small eigenvalues. Say you
desire to find internal eigenvalues and eigenvectors, e.g. those nearest to
:math:`\lambda = 1`. Simply set ``sigma = 1`` and ARPACK takes care of
the rest:
>>> evals_mid, evecs_mid = eigsh(X, 3, sigma=1, which='LM')
>>> i_sort = np.argsort(abs(1. / (1 - evals_all)))[-3:]
>>> evals_all[i_sort]
array([1.16577199, 0.85081388, 1.06642272])
>>> evals_mid
array([0.85081388, 1.06642272, 1.16577199])
>>> print np.dot(evecs_mid.T, evecs_all[:,i_sort])
array([[-0. 1. 0.], # may vary (signs)
[-0. -0. 1.],
[ 1. 0. 0.]]
The eigenvalues come out in a different order, but they're all there.
Note that the shift-invert mode requires the internal solution of a matrix
inverse. This is taken care of automatically by ``eigsh`` and `eigs`,
but the operation can also be specified by the user. See the docstring of
:func:`scipy.sparse.linalg.eigsh` and
:func:`scipy.sparse.linalg.eigs` for details.
References
----------
.. [1] http://www.caam.rice.edu/software/ARPACK/