Source code for pymor.algorithms.arnoldi

# This file is part of the pyMOR project (http://www.pymor.org).
# Copyright 2013-2018 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