pymor.algorithms.eigs
¶
Module Contents¶
- pymor.algorithms.eigs.eigs(A, E=None, k=3, sigma=None, which='LM', b=None, l=None, maxiter=1000, tol=1e-13, imag_tol=1e-12, complex_pair_tol=1e-12, complex_evp=False, left_evp=False)[source]¶
Approximate a few eigenvalues of a linear
Operator
.Computes
k
eigenvaluesw
with corresponding eigenvectorsv
which solve the eigenvalue problem\[A v_i = w_i v_i\]or the generalized eigenvalue problem
\[A v_i = w_i E v_i\]if
E
is notNone
.The implementation is based on Algorithm 4.2 in [Leh95].
- Parameters:
A – The linear
Operator
for which the eigenvalues are to be computed.E – The linear
Operator
which defines the generalized eigenvalue problem.k – The number of eigenvalues and eigenvectors which are to be computed.
sigma – If not
None
transforms the eigenvalue problem such that the k eigenvalues closest to sigma are computed.which –
A string specifying which
k
eigenvalues and eigenvectors to compute:'LM'
: select eigenvalues with largest magnitude'SM'
: select eigenvalues with smallest magnitude'LR'
: select eigenvalues with largest real part'SR'
: select eigenvalues with smallest real part'LI'
: select eigenvalues with largest imaginary part'SI'
: select eigenvalues with smallest imaginary part
b – Initial vector for Arnoldi iteration. Default is a random vector.
l – The size of the Arnoldi factorization. Default is
min(n - 1, max(2*k + 1, 20))
.maxiter – The maximum number of iterations.
tol – The relative error tolerance for the Ritz estimates.
imag_tol – Relative imaginary parts below this tolerance are set to 0.
complex_pair_tol – Tolerance for detecting pairs of complex conjugate eigenvalues.
complex_evp – Whether to consider an eigenvalue problem with complex operators. When operators are real setting this argument to
False
will increase stability and performance.left_evp – If set to
True
compute left eigenvectors else compute right eigenvectors.
- Returns:
w – A 1D
NumPy array
which contains the computed eigenvalues.v – A
VectorArray
which contains the computed eigenvectors.