From 834eb5bfc8deb95179936ea8d854bfee287c1cc9 Mon Sep 17 00:00:00 2001 From: Benoit Jacob Date: Tue, 5 May 2009 20:46:55 +0000 Subject: new unsupported module by Jitse Niesen: matrix exponential --- .../Eigen/src/MatrixFunctions/MatrixExponential.h | 155 +++++++++++++++++++++ 1 file changed, 155 insertions(+) create mode 100644 unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h (limited to 'unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h') diff --git a/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h b/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h new file mode 100644 index 000000000..deb455f47 --- /dev/null +++ b/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h @@ -0,0 +1,155 @@ +// This file is part of Eigen, a lightweight C++ template library +// for linear algebra. Eigen itself is part of the KDE project. +// +// Copyright (C) 2009 Jitse Niesen +// +// Eigen is free software; you can redistribute it and/or +// modify it under the terms of the GNU Lesser General Public +// License as published by the Free Software Foundation; either +// version 3 of the License, or (at your option) any later version. +// +// Alternatively, you can redistribute it and/or +// modify it under the terms of the GNU General Public License as +// published by the Free Software Foundation; either version 2 of +// the License, or (at your option) any later version. +// +// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY +// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS +// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the +// GNU General Public License for more details. +// +// You should have received a copy of the GNU Lesser General Public +// License and a copy of the GNU General Public License along with +// Eigen. If not, see . + +#ifndef EIGEN_MATRIX_EXPONENTIAL +#define EIGEN_MATRIX_EXPONENTIAL + +/** Compute the matrix exponential. + * + * \param M matrix whose exponential is to be computed. + * \param result pointer to the matrix in which to store the result. + * + * The matrix exponential of \f$ M \f$ is defined by + * \f[ \exp(M) = \sum_{k=0}^\infty \frac{M^k}{k!}. \f] + * The matrix exponential can be used to solve linear ordinary + * differential equations: the solution of \f$ y' = My \f$ with the + * initial condition \f$ y(0) = y_0 \f$ is given by + * \f$ y(t) = \exp(M) y_0 \f$. + * + * The cost of the computation is approximately \f$ 20 n^3 \f$ for + * matrices of size \f$ n \f$. The number 20 depends weakly on the + * norm of the matrix. + * + * The matrix exponential is computed using the scaling-and-squaring + * method combined with Padé approximation. The matrix is first + * rescaled, then the exponential of the reduced matrix is computed + * approximant, and then the rescaling is undone by repeated + * squaring. The degree of the Padé approximant is chosen such + * that the approximation error is less than the round-off + * error. However, errors may accumulate during the squaring phase. + * + * Details of the algorithm can be found in: Nicholas J. Higham, "The + * scaling and squaring method for the matrix exponential revisited," + * SIAM J. %Matrix Anal. Applic., 26:1179–1193, + * 2005. + * + * \note Currently, \p M has to be a matrix of \c double . + */ +template +void ei_matrix_exponential(const MatrixBase &M, typename ei_plain_matrix_type::type* result) +{ + typedef typename ei_traits::Scalar Scalar; + typedef typename NumTraits::Real RealScalar; + typedef typename ei_plain_matrix_type::type PlainMatrixType; + + ei_assert(M.rows() == M.cols()); + EIGEN_STATIC_ASSERT(NumTraits::HasFloatingPoint,NUMERIC_TYPE_MUST_BE_FLOATING_POINT) + + PlainMatrixType num, den, U, V; + PlainMatrixType Id = PlainMatrixType::Identity(M.rows(), M.cols()); + RealScalar l1norm = M.cwise().abs().colwise().sum().maxCoeff(); + int squarings = 0; + + // Choose degree of Pade approximant, depending on norm of M + if (l1norm < 1.495585217958292e-002) { + + // Use (3,3)-Pade + const Scalar b[] = {120., 60., 12., 1.}; + PlainMatrixType M2; + M2 = (M * M).lazy(); + num = b[3]*M2 + b[1]*Id; + U = (M * num).lazy(); + V = b[2]*M2 + b[0]*Id; + + } else if (l1norm < 2.539398330063230e-001) { + + // Use (5,5)-Pade + const Scalar b[] = {30240., 15120., 3360., 420., 30., 1.}; + PlainMatrixType M2, M4; + M2 = (M * M).lazy(); + M4 = (M2 * M2).lazy(); + num = b[5]*M4 + b[3]*M2 + b[1]*Id; + U = (M * num).lazy(); + V = b[4]*M4 + b[2]*M2 + b[0]*Id; + + } else if (l1norm < 9.504178996162932e-001) { + + // Use (7,7)-Pade + const Scalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.}; + PlainMatrixType M2, M4, M6; + M2 = (M * M).lazy(); + M4 = (M2 * M2).lazy(); + M6 = (M4 * M2).lazy(); + num = b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id; + U = (M * num).lazy(); + V = b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id; + + } else if (l1norm < 2.097847961257068e+000) { + + // Use (9,9)-Pade + const Scalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240., + 2162160., 110880., 3960., 90., 1.}; + PlainMatrixType M2, M4, M6, M8; + M2 = (M * M).lazy(); + M4 = (M2 * M2).lazy(); + M6 = (M4 * M2).lazy(); + M8 = (M6 * M2).lazy(); + num = b[9]*M8 + b[7]*M6 + b[5]*M4 + b[3]*M2 + b[1]*Id; + U = (M * num).lazy(); + V = b[8]*M8 + b[6]*M6 + b[4]*M4 + b[2]*M2 + b[0]*Id; + + } else { + + // Use (13,13)-Pade; scale matrix by power of 2 so that its norm + // is small enough + + const Scalar maxnorm = 5.371920351148152; + const Scalar b[] = {64764752532480000., 32382376266240000., 7771770303897600., + 1187353796428800., 129060195264000., 10559470521600., 670442572800., + 33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.}; + + squarings = std::max(0, (int)ceil(log2(l1norm / maxnorm))); + PlainMatrixType A, A2, A4, A6; + A = M / pow(2, squarings); + A2 = (A * A).lazy(); + A4 = (A2 * A2).lazy(); + A6 = (A4 * A2).lazy(); + num = b[13]*A6 + b[11]*A4 + b[9]*A2; + V = (A6 * num).lazy(); + num = V + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*Id; + U = (A * num).lazy(); + num = b[12]*A6 + b[10]*A4 + b[8]*A2; + V = (A6 * num).lazy() + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*Id; + } + + num = U + V; // numerator of Pade approximant + den = -U + V; // denominator of Pade approximant + den.lu().solve(num, result); + + // Undo scaling by repeated squaring + for (int i=0; i