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