scipy.sparse.linalg.eigsh#
- scipy.sparse.linalg.eigsh(A, k=6, M=None, sigma=None, which='LM', v0=None, ncv=None, maxiter=None, tol=0, return_eigenvectors=True, Minv=None, OPinv=None, mode='normal')[source]#
Find k eigenvalues and eigenvectors of the real symmetric square matrix or complex Hermitian matrix A.
Solves
A @ x[i] = w[i] * x[i]
, the standard eigenvalue problem for w[i] eigenvalues with corresponding eigenvectors x[i].If M is specified, solves
A @ x[i] = w[i] * M @ x[i]
, the generalized eigenvalue problem for w[i] eigenvalues with corresponding eigenvectors x[i].Note that there is no specialized routine for the case when A is a complex Hermitian matrix. In this case,
eigsh()
will calleigs()
and return the real parts of the eigenvalues thus obtained.- Parameters:
- Andarray, sparse matrix or LinearOperator
A square operator representing the operation
A @ x
, whereA
is real symmetric or complex Hermitian. For buckling mode (see below)A
must additionally be positive-definite.- kint, optional
The number of eigenvalues and eigenvectors desired. k must be smaller than N. It is not possible to compute all eigenvectors of a matrix.
- Returns:
- warray
Array of k eigenvalues.
- varray
An array representing the k eigenvectors. The column
v[:, i]
is the eigenvector corresponding to the eigenvaluew[i]
.
- Other Parameters:
- MAn N x N matrix, array, sparse matrix, or linear operator representing
the operation
M @ x
for the generalized eigenvalue problemA @ x = w * M @ x.
M must represent a real symmetric matrix if A is real, and must represent a complex Hermitian matrix if A is complex. For best results, the data type of M should be the same as that of A. Additionally:
If sigma is None, M is symmetric positive definite.
If sigma is specified, M is symmetric positive semi-definite.
In buckling mode, M is symmetric indefinite.
If sigma is None, eigsh requires an operator to compute the solution of the linear equation
M @ x = b
. This is done internally via a (sparse) LU decomposition for an explicit matrix M, or via an iterative solver for a general linear operator. Alternatively, the user can supply the matrix or operator Minv, which givesx = Minv @ b = M^-1 @ b
.- sigmareal
Find eigenvalues near sigma using shift-invert mode. This requires an operator to compute the solution of the linear system
[A - sigma * M] x = b
, where M is the identity matrix if unspecified. This is computed internally via a (sparse) LU decomposition for explicit matrices A & M, or via an iterative solver if either A or M is a general linear operator. Alternatively, the user can supply the matrix or operator OPinv, which givesx = OPinv @ b = [A - sigma * M]^-1 @ b
. Note that when sigma is specified, the keyword ‘which’ refers to the shifted eigenvaluesw'[i]
where:if mode == ‘normal’,
w'[i] = 1 / (w[i] - sigma)
.if mode == ‘cayley’,
w'[i] = (w[i] + sigma) / (w[i] - sigma)
.if mode == ‘buckling’,
w'[i] = w[i] / (w[i] - sigma)
.(see further discussion in ‘mode’ below)
- v0ndarray, optional
Starting vector for iteration. Default: random
- ncvint, optional
The number of Lanczos vectors generated ncv must be greater than k and smaller than n; it is recommended that
ncv > 2*k
. Default:min(n, max(2*k + 1, 20))
- whichstr [‘LM’ | ‘SM’ | ‘LA’ | ‘SA’ | ‘BE’]
If A is a complex Hermitian matrix, ‘BE’ is invalid. Which k eigenvectors and eigenvalues to find:
‘LM’ : Largest (in magnitude) eigenvalues.
‘SM’ : Smallest (in magnitude) eigenvalues.
‘LA’ : Largest (algebraic) eigenvalues.
‘SA’ : Smallest (algebraic) eigenvalues.
‘BE’ : Half (k/2) from each end of the spectrum.
When k is odd, return one more (k/2+1) from the high end. When sigma != None, ‘which’ refers to the shifted eigenvalues
w'[i]
(see discussion in ‘sigma’, above). ARPACK is generally better at finding large values than small values. If small eigenvalues are desired, consider using shift-invert mode for better performance.- maxiterint, optional
Maximum number of Arnoldi update iterations allowed. Default:
n*10
- tolfloat
Relative accuracy for eigenvalues (stopping criterion). The default value of 0 implies machine precision.
- MinvN x N matrix, array, sparse matrix, or LinearOperator
See notes in M, above.
- OPinvN x N matrix, array, sparse matrix, or LinearOperator
See notes in sigma, above.
- return_eigenvectorsbool
Return eigenvectors (True) in addition to eigenvalues. This value determines the order in which eigenvalues are sorted. The sort order is also dependent on the which variable.
- For which = ‘LM’ or ‘SA’:
If return_eigenvectors is True, eigenvalues are sorted by algebraic value.
If return_eigenvectors is False, eigenvalues are sorted by absolute value.
- For which = ‘BE’ or ‘LA’:
eigenvalues are always sorted by algebraic value.
- For which = ‘SM’:
If return_eigenvectors is True, eigenvalues are sorted by algebraic value.
If return_eigenvectors is False, eigenvalues are sorted by decreasing absolute value.
- modestring [‘normal’ | ‘buckling’ | ‘cayley’]
Specify strategy to use for shift-invert mode. This argument applies only for real-valued A and sigma != None. For shift-invert mode, ARPACK internally solves the eigenvalue problem
OP @ x'[i] = w'[i] * B @ x'[i]
and transforms the resulting Ritz vectors x’[i] and Ritz values w’[i] into the desired eigenvectors and eigenvalues of the problemA @ x[i] = w[i] * M @ x[i]
. The modes are as follows:- ‘normal’ :
OP = [A - sigma * M]^-1 @ M, B = M, w’[i] = 1 / (w[i] - sigma)
- ‘buckling’ :
OP = [A - sigma * M]^-1 @ A, B = A, w’[i] = w[i] / (w[i] - sigma)
- ‘cayley’ :
OP = [A - sigma * M]^-1 @ [A + sigma * M], B = M, w’[i] = (w[i] + sigma) / (w[i] - sigma)
The choice of mode will affect which eigenvalues are selected by the keyword ‘which’, and can also impact the stability of convergence (see [2] for a discussion).
- Raises:
- ArpackNoConvergence
When the requested convergence is not obtained.
The currently converged eigenvalues and eigenvectors can be found as
eigenvalues
andeigenvectors
attributes of the exception object.
See also
Notes
This function is a wrapper to the ARPACK [1] SSEUPD and DSEUPD functions which use the Implicitly Restarted Lanczos Method to find the eigenvalues and eigenvectors [2].
References
[1]ARPACK Software, opencollab/arpack-ng
[2]R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK USERS GUIDE: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia, PA, 1998.
Examples
>>> import numpy as np >>> from scipy.sparse.linalg import eigsh >>> identity = np.eye(13) >>> eigenvalues, eigenvectors = eigsh(identity, k=6) >>> eigenvalues array([1., 1., 1., 1., 1., 1.]) >>> eigenvectors.shape (13, 6)