Source code for pymor.algorithms.arnoldi
# This file is part of the pyMOR project (http://www.pymor.org).
# Copyright 2013-2019 pyMOR developers and contributors. All rights reserved.
# License: BSD 2-Clause License (http://opensource.org/licenses/BSD-2-Clause)
from pymor.algorithms.gram_schmidt import gram_schmidt
[docs]def arnoldi(A, E, b, sigma, trans=False):
r"""Rational Arnoldi algorithm.
If `trans == False`, using Arnoldi process, computes a real
orthonormal basis for the rational Krylov subspace
.. math::
\mathrm{span}\{(\sigma_1 E - A)^{-1} b, (\sigma_2 E - A)^{-1} b, \ldots,
(\sigma_r E - A)^{-1} b\},
otherwise, computes the same for
.. math::
\mathrm{span}\{(\sigma_1 E - A)^{-T} b^T, (\sigma_2 E - A)^{-T} b^T,
\ldots, (\sigma_r E - A)^{-T} b^T\}.
Interpolation points in `sigma` are allowed to repeat (in any
order). Then, in the above expression,
.. math::
\underbrace{(\sigma_i E - A)^{-1} b, \ldots,
(\sigma_i E - A)^{-1} b}_{m \text{ times}}
is replaced by
.. math::
(\sigma_i E - A)^{-1} b, (\sigma_i E - A)^{-2} b, \ldots,
(\sigma_i E - A)^{-m} b.
Analogously for the `trans == True` case.
Parameters
----------
A
Real |Operator| A.
E
Real |Operator| E.
b
Real vector-like operator (if trans is False) or functional (if
trans is True).
sigma
Interpolation points (closed under conjugation).
trans
Boolean, see above.
Returns
-------
V
Projection matrix.
"""
assert not trans and b.source.dim == 1 or trans and b.range.dim == 1
r = len(sigma)
V = A.source.empty(reserve=r)
v = b.as_vector()
v.scal(1 / v.l2_norm()[0])
for i in range(r):
if sigma[i].imag == 0:
sEmA = sigma[i].real * E - A
if not trans:
v = sEmA.apply_inverse(v if len(V) == 0 else E.apply(v))
else:
v = sEmA.apply_inverse_adjoint(v if len(V) == 0 else E.apply_adjoint(v))
V.append(v)
V = gram_schmidt(V, atol=0, rtol=0, offset=len(V) - 1, copy=False)
v = V[-1]
elif sigma[i].imag > 0:
sEmA = sigma[i] * E - A
if not trans:
v = sEmA.apply_inverse(v if len(V) == 0 else E.apply(v))
else:
v = sEmA.apply_inverse_adjoint(v if len(V) == 0 else E.apply_adjoint(v))
V.append(v.real)
V.append(v.imag)
V = gram_schmidt(V, atol=0, rtol=0, offset=len(V) - 2, copy=False)
v = V[-1]
return V