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def 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')
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 call ``eigs()`` and return the
real parts of the eigenvalues thus obtained.
Parameters
----------
A : ndarray, sparse matrix or LinearOperator
A square operator representing the operation ``A @ x``, where ``A`` is
real symmetric or complex Hermitian. For buckling mode (see below)
``A`` must additionally be positive-definite.
k : int, 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
-------
w : array
Array of k eigenvalues.
v : array
An array representing the `k` eigenvectors. The column ``v[:, i]`` is
the eigenvector corresponding to the eigenvalue ``w[i]``.
Other Parameters
----------------
M : An N x N matrix, array, sparse matrix, or linear operator representing
the operation ``M @ x`` for the generalized eigenvalue problem
A @ 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 gives
``x = Minv @ b = M^-1 @ b``.
sigma : real
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 gives ``x = OPinv @ b = [A - sigma * M]^-1 @ b``.
Note that when sigma is specified, the keyword 'which' refers to
the shifted eigenvalues ``w'[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)
v0 : ndarray, optional
Starting vector for iteration.
Default: random
ncv : int, 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))``
which : str ['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.
maxiter : int, optional
Maximum number of Arnoldi update iterations allowed.
Default: ``n*10``
tol : float
Relative accuracy for eigenvalues (stopping criterion).
The default value of 0 implies machine precision.
Minv : N x N matrix, array, sparse matrix, or LinearOperator
See notes in M, above.
OPinv : N x N matrix, array, sparse matrix, or LinearOperator
See notes in sigma, above.
return_eigenvectors : bool
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.
mode : string ['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 problem
``A @ 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`` and ``eigenvectors`` attributes of the exception
object.
See Also
--------
eigs : eigenvalues and eigenvectors for a general (nonsymmetric) matrix A
svds : singular value decomposition for a matrix A
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, https://github.com/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)
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