# Source code for pymor.algorithms.krylov

# This file is part of the pyMOR project (http://www.pymor.org).

"""Module for computing (rational) Krylov subspaces' bases."""

from pymor.algorithms.gram_schmidt import gram_schmidt

[docs]def rational_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)^{-1} E (\sigma_i E - A)^{-1} b,
\ldots,
\left((\sigma_i E - A)^{-1} E\right)^{m - 1} (\sigma_i E - A)^{-1} 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
Sequence of interpolation points (closed under conjugation).
trans
Boolean, see above.

Returns
-------
V
Orthonormal basis for the Krylov subspace |VectorArray|.
"""
assert A.source == A.range
assert E.source == A.source
assert E.range == A.source
assert (b.range if not trans else b.source) == A.source
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:
continue
if sigma[i].imag == 0:
sEmA = sigma[i].real * E - A
else:
sEmA = sigma[i] * E - A
if not trans:
v = sEmA.apply_inverse(v if len(V) == 0 else E.apply(v))
else:
if sigma[i].imag == 0:
V.append(v)
gram_schmidt(V, atol=0, rtol=0, offset=len(V) - 1, copy=False)
else:
V.append(v.real)
V.append(v.imag)
gram_schmidt(V, atol=0, rtol=0, offset=len(V) - 2, copy=False)
v = V[-1]

return V

[docs]def tangential_rational_krylov(A, E, B, b, sigma, trans=False, orth=True):
r"""Tangential Rational Krylov subspace.

If trans == False, computes a real basis for the rational Krylov
subspace

.. math::
\mathrm{span}\{
(\sigma_1 E - A)^{-1} B b_1,
(\sigma_2 E - A)^{-1} B b_2,
\ldots,
(\sigma_r E - A)^{-1} B b_r
\},

otherwise, computes the same for

.. math::
\mathrm{span}\{
(\sigma_1 E - A)^{-T} B^T b_1,
(\sigma_2 E - A)^{-T} B^T b_2,
\ldots,
(\sigma_r E - A)^{-T} B^T b_r
\}.

Interpolation points in sigma are assumed to be pairwise distinct.

Parameters
----------
A
Real |Operator| A.
E
Real |Operator| E.
B
Real |Operator| B.
b
|VectorArray| from B.source, if trans == False, or
B.range, if trans == True.
sigma
Sequence of interpolation points (closed under conjugation), of
the same length as b.
trans
Boolean, see above.
orth
If True, orthonormalizes the basis using
:meth:pymor.algorithms.gram_schmidt.gram_schmidt.

Returns
-------
V
Optionally orthonormal basis for the Krylov subspace |VectorArray|.
"""
assert A.source == A.range
assert E.source == A.source
assert E.range == A.source
assert (B.range if not trans else B.source) == A.source
assert b in (B.source if not trans else B.range)
assert len(b) == len(sigma)

r = len(sigma)
V = A.source.empty(reserve=r)
for i in range(r):
if sigma[i].imag == 0:
sEmA = sigma[i].real * E - A
if not trans:
Bb = B.apply(b.real[i])
V.append(sEmA.apply_inverse(Bb))
else:
elif sigma[i].imag > 0:
sEmA = sigma[i] * E - A
if not trans:
Bb = B.apply(b[i])
v = sEmA.apply_inverse(Bb)
else: