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2026ea7ff2
The //@{ ... //@} in unsupported/Eigen/MatrixFunctions for some reason
caused doxygen to list the constructor of the MatrixExponential class
as a separate function in the MatrixFunctions module without any reference
to the class; very confusing.
328 lines
11 KiB
C++
328 lines
11 KiB
C++
// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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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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/** \ingroup MatrixFunctions_Module
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*
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* \brief 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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* Example: The following program checks that
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* \f[ \exp \left[ \begin{array}{ccc}
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* 0 & \frac14\pi & 0 \\
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* -\frac14\pi & 0 & 0 \\
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* 0 & 0 & 0
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* \end{array} \right] = \left[ \begin{array}{ccc}
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* \frac12\sqrt2 & -\frac12\sqrt2 & 0 \\
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* \frac12\sqrt2 & \frac12\sqrt2 & 0 \\
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* 0 & 0 & 1
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* \end{array} \right]. \f]
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* This corresponds to a rotation of \f$ \frac14\pi \f$ radians around
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* the z-axis.
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* \include MatrixExponential.cpp
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* Output: \verbinclude MatrixExponential.out
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*
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* \note \p M has to be a matrix of \c float, \c double,
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* \c complex<float> or \c complex<double> .
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*/
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template <typename Derived>
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EIGEN_STRONG_INLINE void ei_matrix_exponential(const MatrixBase<Derived> &M,
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typename MatrixBase<Derived>::PlainMatrixType* result);
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/** \ingroup MatrixFunctions_Module
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* \brief Class for computing the matrix exponential.
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*/
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template <typename MatrixType>
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class MatrixExponential {
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public:
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/** \brief 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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MatrixExponential(const MatrixType &M, MatrixType *result);
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private:
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// Prevent copying
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MatrixExponential(const MatrixExponential&);
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MatrixExponential& operator=(const MatrixExponential&);
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/** \brief Compute the (3,3)-Padé approximant to the exponential.
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*
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*
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* \param A Argument of matrix exponential
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*/
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void pade3(const MatrixType &A);
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/** \brief Compute the (5,5)-Padé approximant to the exponential.
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*
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*
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* \param A Argument of matrix exponential
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*/
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void pade5(const MatrixType &A);
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/** \brief Compute the (7,7)-Padé approximant to the exponential.
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*
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*
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* \param A Argument of matrix exponential
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*/
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void pade7(const MatrixType &A);
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/** \brief Compute the (9,9)-Padé approximant to the exponential.
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*
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*
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* \param A Argument of matrix exponential
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*/
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void pade9(const MatrixType &A);
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/** \brief Compute the (13,13)-Padé approximant to the exponential.
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*
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*
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* \param A Argument of matrix exponential
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*/
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void pade13(const MatrixType &A);
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/** \brief Compute Padé approximant to the exponential.
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*
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* Computes \c m_U, \c m_V and \c m_squarings such that
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* \f$ (V+U)(V-U)^{-1} \f$ is a Padé of
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* \f$ \exp(2^{-\mbox{squarings}}M) \f$ around \f$ M = 0 \f$. The
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* degree of the Padé approximant and the value of
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* squarings are chosen such that the approximation error is no
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* more than the round-off error.
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*
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* The argument of this function should correspond with the (real
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* part of) the entries of \c m_M. It is used to select the
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* correct implementation using overloading.
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*/
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void computeUV(double);
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/** \brief Compute Padé approximant to the exponential.
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*
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* \sa computeUV(double);
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*/
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void computeUV(float);
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typedef typename ei_traits<MatrixType>::Scalar Scalar;
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typedef typename NumTraits<typename ei_traits<MatrixType>::Scalar>::Real RealScalar;
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/** \brief Pointer to matrix whose exponential is to be computed. */
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const MatrixType* m_M;
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/** \brief Even-degree terms in numerator of Padé approximant. */
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MatrixType m_U;
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/** \brief Odd-degree terms in numerator of Padé approximant. */
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MatrixType m_V;
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/** \brief Used for temporary storage. */
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MatrixType m_tmp1;
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/** \brief Used for temporary storage. */
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MatrixType m_tmp2;
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/** \brief Identity matrix of the same size as \c m_M. */
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MatrixType m_Id;
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/** \brief Number of squarings required in the last step. */
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int m_squarings;
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/** \brief L1 norm of m_M. */
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float m_l1norm;
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};
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template <typename MatrixType>
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MatrixExponential<MatrixType>::MatrixExponential(const MatrixType &M, MatrixType *result) :
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m_M(&M),
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m_U(M.rows(),M.cols()),
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m_V(M.rows(),M.cols()),
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m_tmp1(M.rows(),M.cols()),
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m_tmp2(M.rows(),M.cols()),
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m_Id(MatrixType::Identity(M.rows(), M.cols())),
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m_squarings(0),
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m_l1norm(static_cast<float>(M.cwise().abs().colwise().sum().maxCoeff()))
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{
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computeUV(RealScalar());
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m_tmp1 = m_U + m_V; // numerator of Pade approximant
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m_tmp2 = -m_U + m_V; // denominator of Pade approximant
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*result = m_tmp2.partialPivLu().solve(m_tmp1);
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for (int i=0; i<m_squarings; i++)
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*result *= *result; // undo scaling by repeated squaring
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}
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template <typename MatrixType>
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EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade3(const MatrixType &A)
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{
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const Scalar b[] = {120., 60., 12., 1.};
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m_tmp1.noalias() = A * A;
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m_tmp2 = b[3]*m_tmp1 + b[1]*m_Id;
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m_U.noalias() = A * m_tmp2;
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m_V = b[2]*m_tmp1 + b[0]*m_Id;
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}
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template <typename MatrixType>
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EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade5(const MatrixType &A)
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{
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const Scalar b[] = {30240., 15120., 3360., 420., 30., 1.};
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MatrixType A2 = A * A;
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m_tmp1.noalias() = A2 * A2;
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m_tmp2 = b[5]*m_tmp1 + b[3]*A2 + b[1]*m_Id;
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m_U.noalias() = A * m_tmp2;
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m_V = b[4]*m_tmp1 + b[2]*A2 + b[0]*m_Id;
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}
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template <typename MatrixType>
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EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade7(const MatrixType &A)
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{
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const Scalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.};
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MatrixType A2 = A * A;
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MatrixType A4 = A2 * A2;
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m_tmp1.noalias() = A4 * A2;
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m_tmp2 = b[7]*m_tmp1 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
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m_U.noalias() = A * m_tmp2;
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m_V = b[6]*m_tmp1 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
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}
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template <typename MatrixType>
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EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade9(const MatrixType &A)
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{
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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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MatrixType A2 = A * A;
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MatrixType A4 = A2 * A2;
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MatrixType A6 = A4 * A2;
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m_tmp1.noalias() = A6 * A2;
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m_tmp2 = b[9]*m_tmp1 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
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m_U.noalias() = A * m_tmp2;
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m_V = b[8]*m_tmp1 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
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}
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template <typename MatrixType>
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EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade13(const MatrixType &A)
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{
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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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MatrixType A2 = A * A;
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MatrixType A4 = A2 * A2;
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m_tmp1.noalias() = A4 * A2;
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m_V = b[13]*m_tmp1 + b[11]*A4 + b[9]*A2; // used for temporary storage
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m_tmp2.noalias() = m_tmp1 * m_V;
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m_tmp2 += b[7]*m_tmp1 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
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m_U.noalias() = A * m_tmp2;
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m_tmp2 = b[12]*m_tmp1 + b[10]*A4 + b[8]*A2;
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m_V.noalias() = m_tmp1 * m_tmp2;
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m_V += b[6]*m_tmp1 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
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}
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template <typename MatrixType>
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void MatrixExponential<MatrixType>::computeUV(float)
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{
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if (m_l1norm < 4.258730016922831e-001) {
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pade3(*m_M);
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} else if (m_l1norm < 1.880152677804762e+000) {
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pade5(*m_M);
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} else {
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const float maxnorm = 3.925724783138660f;
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m_squarings = std::max(0, (int)ceil(log2(m_l1norm / maxnorm)));
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MatrixType A = *m_M / std::pow(Scalar(2), Scalar(m_squarings));
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pade7(A);
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}
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}
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template <typename MatrixType>
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void MatrixExponential<MatrixType>::computeUV(double)
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{
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if (m_l1norm < 1.495585217958292e-002) {
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pade3(*m_M);
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} else if (m_l1norm < 2.539398330063230e-001) {
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pade5(*m_M);
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} else if (m_l1norm < 9.504178996162932e-001) {
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pade7(*m_M);
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} else if (m_l1norm < 2.097847961257068e+000) {
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pade9(*m_M);
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} else {
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const double maxnorm = 5.371920351148152;
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m_squarings = std::max(0, (int)ceil(log2(m_l1norm / maxnorm)));
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MatrixType A = *m_M / std::pow(Scalar(2), Scalar(m_squarings));
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pade13(A);
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}
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}
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template <typename Derived>
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EIGEN_STRONG_INLINE void ei_matrix_exponential(const MatrixBase<Derived> &M,
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typename MatrixBase<Derived>::PlainMatrixType* result)
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{
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ei_assert(M.rows() == M.cols());
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MatrixExponential<typename MatrixBase<Derived>::PlainMatrixType>(M, result);
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}
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#endif // EIGEN_MATRIX_EXPONENTIAL
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