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.
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]
A : An N x N matrix, array, sparse matrix, or LinearOperator representing
the operation A * x, where A is a real symmetric matrix 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.
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
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
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:
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
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  for a discussion)
When the requested convergence is not obtained.
The currently converged eigenvalues and eigenvectors can be found as
eigenvectorsattributes of the exception object.
This function is a wrapper to the ARPACK [R371] SSEUPD and DSEUPD functions which use the Implicitly Restarted Lanczos Method to find the eigenvalues and eigenvectors [R372].
[R371] (1, 2) ARPACK Software, http://www.caam.rice.edu/software/ARPACK/ [R372] (1, 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.
>>> import scipy.sparse as sparse >>> id = np.eye(13) >>> vals, vecs = sparse.linalg.eigsh(id, k=6) >>> vals array([ 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j]) >>> vecs.shape (13, 6)