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.
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
Starting vector for iteration.
ncv : int
The number of Lanczos vectors generated ncv must be greater than k and
smaller than n; it is recommended that ncv > 2*k.
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
Maximum number of Arnoldi update iterations allowed
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
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)