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Clean-up of MatrixExponential:
* put internal stuff in the internal namespace * replace member functions by free functions
This commit is contained in:
Jitse Niesen
parent
5879937f58
commit
463343fb37
@ -1,8 +1,8 @@
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// 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, 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
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// Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net>
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// Copyright (C) 2009, 2010, 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
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// Copyright (C) 2011, 2013 Chen-Pang He <jdh8@ms63.hinet.net>
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//
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// This Source Code Form is subject to the terms of the Mozilla
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// Public License v. 2.0. If a copy of the MPL was not distributed
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@ -14,160 +14,21 @@
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#include "StemFunction.h"
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namespace Eigen {
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/** \ingroup MatrixFunctions_Module
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* \brief Class for computing the matrix exponential.
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* \tparam MatrixType type of the argument of the exponential,
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* expected to be an instantiation of the Matrix class template.
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*/
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template <typename MatrixType>
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class MatrixExponential : internal::noncopyable
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{
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public:
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/** \brief Constructor.
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*
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* The class stores a reference to \p M, so it should not be
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* changed (or destroyed) before compute() is called.
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*
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* \param[in] M matrix whose exponential is to be computed.
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*/
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explicit MatrixExponential(const MatrixType &M);
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/** \brief Computes the matrix exponential.
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*
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* \param[out] result the matrix exponential of \p M in the constructor.
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*/
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template <typename ResultType>
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void compute(ResultType &result);
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private:
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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$.
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*
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* \param[in] A Argument of matrix exponential
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*/
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void pade3(const MatrixType &A);
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/** \brief Compute the (5,5)-Padé approximant to the exponential.
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*
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*
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* \param[in] A Argument of matrix exponential
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*/
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void pade5(const MatrixType &A);
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/** \brief Compute the (7,7)-Padé approximant to the exponential.
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*
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*
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* \param[in] A Argument of matrix exponential
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*/
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void pade7(const MatrixType &A);
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/** \brief Compute the (9,9)-Padé approximant to the exponential.
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*
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*
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* \param[in] A Argument of matrix exponential
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*/
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void pade9(const MatrixType &A);
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/** \brief Compute the (13,13)-Padé approximant to the exponential.
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*
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*
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* \param[in] A Argument of matrix exponential
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*/
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void pade13(const MatrixType &A);
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/** \brief Compute the (17,17)-Padé approximant to the exponential.
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*
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*
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* This function activates only if your long double is double-double or quadruple.
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*
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* \param[in] A Argument of matrix exponential
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*/
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void pade17(const MatrixType &A);
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/** \brief Compute Padé approximant to the exponential.
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*
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* Computes \c m_U, \c m_V and \c m_squarings such that
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* \f$ (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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/** \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(long double);
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struct ScalingOp;
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typedef typename internal::traits<MatrixType>::Scalar Scalar;
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typedef typename NumTraits<Scalar>::Real RealScalar;
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typedef typename std::complex<RealScalar> ComplexScalar;
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/** \brief Reference to matrix whose exponential is to be computed. */
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typename internal::nested<MatrixType>::type m_M;
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/** \brief Odd-degree terms in numerator of Padé approximant. */
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MatrixType m_U;
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/** \brief Even-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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RealScalar m_l1norm;
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};
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namespace internal {
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/** \brief Scaling operator.
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*
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* This struct is used by CwiseUnaryOp to scale a matrix by \f$ 2^{-s} \f$.
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*/
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template <typename MatrixType>
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struct MatrixExponential<MatrixType>::ScalingOp
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template <typename RealScalar>
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struct MatrixExponentialScalingOp
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{
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/** \brief Constructor.
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*
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* \param[in] squarings The integer \f$ s \f$ in this document.
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*/
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ScalingOp(int squarings) : m_squarings(squarings) { }
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MatrixExponentialScalingOp(int squarings) : m_squarings(squarings) { }
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/** \brief Scale a matrix coefficient.
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*
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@ -179,6 +40,8 @@ struct MatrixExponential<MatrixType>::ScalingOp
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return ldexp(x, -m_squarings);
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}
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typedef std::complex<RealScalar> ComplexScalar;
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/** \brief Scale a matrix coefficient.
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*
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* \param[in,out] x The scalar to be scaled, becoming \f$ 2^{-s} x \f$.
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@ -193,228 +56,313 @@ struct MatrixExponential<MatrixType>::ScalingOp
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int m_squarings;
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};
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/** \brief Compute the (3,3)-Padé approximant to the exponential.
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*
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* After exit, \f$ (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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template <typename MatrixType>
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MatrixExponential<MatrixType>::MatrixExponential(const MatrixType &M) :
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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(M.cwiseAbs().colwise().sum().maxCoeff())
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void matrix_exp_pade3(const MatrixType &A, MatrixType &U, MatrixType &V)
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{
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/* empty body */
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typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
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const RealScalar b[] = {120., 60., 12., 1.};
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const MatrixType A2 = A * A;
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const MatrixType tmp = b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
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U.noalias() = A * tmp;
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V = b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
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}
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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$.
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*/
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template <typename MatrixType>
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template <typename ResultType>
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void MatrixExponential<MatrixType>::compute(ResultType &result)
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void matrix_exp_pade5(const MatrixType &A, MatrixType &U, MatrixType &V)
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{
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typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
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const RealScalar b[] = {30240., 15120., 3360., 420., 30., 1.};
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const MatrixType A2 = A * A;
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const MatrixType A4 = A2 * A2;
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const MatrixType tmp = b[5] * A4 + b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
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U.noalias() = A * tmp;
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V = b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
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}
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/** \brief Compute the (7,7)-Padé approximant to the exponential.
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*
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*/
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template <typename MatrixType>
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void matrix_exp_pade7(const MatrixType &A, MatrixType &U, MatrixType &V)
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{
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typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
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const RealScalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.};
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const MatrixType A2 = A * A;
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const MatrixType A4 = A2 * A2;
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const MatrixType A6 = A4 * A2;
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const MatrixType tmp = b[7] * A6 + b[5] * A4 + b[3] * A2
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+ b[1] * MatrixType::Identity(A.rows(), A.cols());
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U.noalias() = A * tmp;
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V = b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
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}
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/** \brief Compute the (9,9)-Padé approximant to the exponential.
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*
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* After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé
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* approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
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*/
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template <typename MatrixType>
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void matrix_exp_pade9(const MatrixType &A, MatrixType &U, MatrixType &V)
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{
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typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
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const RealScalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240.,
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2162160., 110880., 3960., 90., 1.};
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const MatrixType A2 = A * A;
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const MatrixType A4 = A2 * A2;
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const MatrixType A6 = A4 * A2;
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const MatrixType A8 = A6 * A2;
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const MatrixType tmp = b[9] * A8 + b[7] * A6 + b[5] * A4 + b[3] * A2
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+ b[1] * MatrixType::Identity(A.rows(), A.cols());
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U.noalias() = A * tmp;
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V = b[8] * A8 + b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
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}
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/** \brief Compute the (13,13)-Padé approximant to the exponential.
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*
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* After exit, \f$ (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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template <typename MatrixType>
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void matrix_exp_pade13(const MatrixType &A, MatrixType &U, MatrixType &V)
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{
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typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
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const RealScalar 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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const MatrixType A2 = A * A;
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const MatrixType A4 = A2 * A2;
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const MatrixType A6 = A4 * A2;
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V = b[13] * A6 + b[11] * A4 + b[9] * A2; // used for temporary storage
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MatrixType tmp = A6 * V;
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tmp += b[7] * A6 + b[5] * A4 + b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
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U.noalias() = A * tmp;
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tmp = b[12] * A6 + b[10] * A4 + b[8] * A2;
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V.noalias() = A6 * tmp;
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V += b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
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}
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/** \brief Compute the (17,17)-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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* This function activates only if your long double is double-double or quadruple.
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*/
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#if LDBL_MANT_DIG > 64
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template <typename MatrixType>
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void matrix_exp_pade17(const MatrixType &A, MatrixType &U, MatrixType &V)
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{
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typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
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const RealScalar b[] = {830034394580628357120000.L, 415017197290314178560000.L,
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100610229646136770560000.L, 15720348382208870400000.L,
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1774878043152614400000.L, 153822763739893248000.L, 10608466464820224000.L,
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595373117923584000.L, 27563570274240000.L, 1060137318240000.L,
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33924394183680.L, 899510451840.L, 19554575040.L, 341863200.L, 4651200.L,
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46512.L, 306.L, 1.L};
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const MatrixType A2 = A * A;
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const MatrixType A4 = A2 * A2;
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const MatrixType A6 = A4 * A2;
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const MatrixType A8 = A4 * A4;
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V = b[17] * m_tmp1 + b[15] * A6 + b[13] * A4 + b[11] * A2; // used for temporary storage
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matrixType tmp = A8 * V;
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tmp += b[9] * A8 + b[7] * A6 + b[5] * A4 + b[3] * A2
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+ b[1] * MatrixType::Identity(A.rows(), A.cols());
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U.noalias() = A * tmp;
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tmp = b[16] * A8 + b[14] * A6 + b[12] * A4 + b[10] * A2;
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V.noalias() = tmp * A8;
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V += b[8] * A8 + b[6] * A6 + b[4] * A4 + b[2] * A2
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+ b[0] * MatrixType::Identity(A.rows(), A.cols());
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}
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#endif
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template <typename MatrixType, typename RealScalar = typename NumTraits<typename traits<MatrixType>::Scalar>::Real>
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struct matrix_exp_computeUV
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{
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/** \brief Compute Padé approximant to the exponential.
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*
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* Computes \c U, \c V and \c squarings such that \f$ (V+U)(V-U)^{-1} \f$ is a Padé
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* approximant of \f$ \exp(2^{-\mbox{squarings}}M) \f$ around \f$ M = 0 \f$, where \f$ M \f$
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* denotes the matrix \c arg. The degree of the Padé approximant and the value of squarings
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* are chosen such that the approximation error is no more than the round-off error.
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*/
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static void run(const MatrixType& arg, MatrixType& U, MatrixType& V, int& squarings);
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};
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template <typename MatrixType>
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struct matrix_exp_computeUV<MatrixType, float>
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{
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static void run(const MatrixType& arg, MatrixType& U, MatrixType& V, int& squarings)
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{
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using std::frexp;
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using std::pow;
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const float l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
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squarings = 0;
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if (l1norm < 4.258730016922831e-001) {
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matrix_exp_pade3(arg, U, V);
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} else if (l1norm < 1.880152677804762e+000) {
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matrix_exp_pade5(arg, U, V);
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} else {
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const float maxnorm = 3.925724783138660f;
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frexp(l1norm / maxnorm, &squarings);
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if (squarings < 0) squarings = 0;
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MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<float>(squarings));
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matrix_exp_pade7(A, U, V);
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}
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}
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};
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template <typename MatrixType>
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struct matrix_exp_computeUV<MatrixType, double>
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{
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static void run(const MatrixType& arg, MatrixType& U, MatrixType& V, int& squarings)
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{
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using std::frexp;
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using std::pow;
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const double l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
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squarings = 0;
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if (l1norm < 1.495585217958292e-002) {
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matrix_exp_pade3(arg, U, V);
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} else if (l1norm < 2.539398330063230e-001) {
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matrix_exp_pade5(arg, U, V);
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} else if (l1norm < 9.504178996162932e-001) {
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matrix_exp_pade7(arg, U, V);
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} else if (l1norm < 2.097847961257068e+000) {
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matrix_exp_pade9(arg, U, V);
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} else {
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const double maxnorm = 5.371920351148152;
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frexp(l1norm / maxnorm, &squarings);
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if (squarings < 0) squarings = 0;
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MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<double>(squarings));
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matrix_exp_pade13(A, U, V);
|
||||
}
|
||||
}
|
||||
};
|
||||
|
||||
template <typename MatrixType>
|
||||
struct matrix_exp_computeUV<MatrixType, long double>
|
||||
{
|
||||
static void run(const MatrixType& arg, MatrixType& U, MatrixType& V, int& squarings)
|
||||
{
|
||||
#if LDBL_MANT_DIG == 53 // double precision
|
||||
|
||||
matrix_exp_computeUV<MatrixType, double>(arg, U, V, squarings);
|
||||
|
||||
#else
|
||||
|
||||
using std::frexp;
|
||||
using std::pow;
|
||||
const long double l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
|
||||
squarings = 0;
|
||||
|
||||
#if LDBL_MANT_DIG <= 64 // extended precision
|
||||
|
||||
if (l1norm < 4.1968497232266989671e-003L) {
|
||||
matrix_exp_pade3(arg, U, V);
|
||||
} else if (l1norm < 1.1848116734693823091e-001L) {
|
||||
matrix_exp_pade5(arg, U, V);
|
||||
} else if (l1norm < 5.5170388480686700274e-001L) {
|
||||
matrix_exp_pade7(arg, U, V);
|
||||
} else if (l1norm < 1.3759868875587845383e+000L) {
|
||||
matrix_exp_pade9(arg, U, V);
|
||||
} else {
|
||||
const long double maxnorm = 4.0246098906697353063L;
|
||||
frexp(l1norm / maxnorm, &squarings);
|
||||
if (squarings < 0) squarings = 0;
|
||||
MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
|
||||
matrix_exp_pade13(A, U, V);
|
||||
}
|
||||
|
||||
#elif LDBL_MANT_DIG <= 106 // double-double
|
||||
|
||||
if (l1norm < 3.2787892205607026992947488108213e-005L) {
|
||||
matrix_exp_pade3(arg, U, V);
|
||||
} else if (l1norm < 6.4467025060072760084130906076332e-003L) {
|
||||
matrix_exp_pade5(arg, U, V);
|
||||
} else if (l1norm < 6.8988028496595374751374122881143e-002L) {
|
||||
matrix_exp_pade7(arg, U, V);
|
||||
} else if (l1norm < 2.7339737518502231741495857201670e-001L) {
|
||||
matrix_exp_pade9(arg, U, V);
|
||||
} else if (l1norm < 1.3203382096514474905666448850278e+000L) {
|
||||
matrix_exp_pade13(arg, U, V);
|
||||
} else {
|
||||
const long double maxnorm = 3.2579440895405400856599663723517L;
|
||||
frexp(l1norm / maxnorm, &squarings);
|
||||
if (squarings < 0) squarings = 0;
|
||||
MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
|
||||
matrix_exp_pade17(A, U, V);
|
||||
}
|
||||
|
||||
#elif LDBL_MANT_DIG <= 112 // quadruple precison
|
||||
|
||||
if (l1norm < 1.639394610288918690547467954466970e-005L) {
|
||||
matrix_exp_pade3(arg, U, V);
|
||||
} else if (l1norm < 4.253237712165275566025884344433009e-003L) {
|
||||
matrix_exp_pade5(arg, U, V);
|
||||
} else if (l1norm < 5.125804063165764409885122032933142e-002L) {
|
||||
matrix_exp_pade7(arg, U, V);
|
||||
} else if (l1norm < 2.170000765161155195453205651889853e-001L) {
|
||||
matrix_exp_pade9(arg, U, V);
|
||||
} else if (l1norm < 1.125358383453143065081397882891878e+000L) {
|
||||
matrix_exp_pade13(arg, U, V);
|
||||
} else {
|
||||
frexp(l1norm / maxnorm, &squarings);
|
||||
if (squarings < 0) squarings = 0;
|
||||
MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
|
||||
matrix_exp_pade17(A, U, V);
|
||||
}
|
||||
|
||||
#else
|
||||
|
||||
// this case should be handled in compute()
|
||||
eigen_assert(false && "Bug in MatrixExponential");
|
||||
|
||||
#endif
|
||||
#endif // LDBL_MANT_DIG
|
||||
}
|
||||
};
|
||||
|
||||
|
||||
/* Computes the matrix exponential
|
||||
*
|
||||
* \param arg argument of matrix exponential (should be plain object)
|
||||
* \param result variable in which result will be stored
|
||||
*/
|
||||
template <typename MatrixType, typename ResultType>
|
||||
void matrix_exp_compute(const MatrixType& arg, ResultType &result)
|
||||
{
|
||||
#if LDBL_MANT_DIG > 112 // rarely happens
|
||||
if(sizeof(RealScalar) > 14) {
|
||||
result = m_M.matrixFunction(StdStemFunctions<ComplexScalar>::exp);
|
||||
typedef typename traits<MatrixType>::Scalar Scalar;
|
||||
typedef typename NumTraits<Scalar>::Real RealScalar;
|
||||
typedef typename std::complex<RealScalar> ComplexScalar;
|
||||
if (sizeof(RealScalar) > 14) {
|
||||
result = arg.matrixFunction(StdStemFunctions<ComplexScalar>::exp);
|
||||
return;
|
||||
}
|
||||
#endif
|
||||
computeUV(RealScalar());
|
||||
m_tmp1 = m_U + m_V; // numerator of Pade approximant
|
||||
m_tmp2 = -m_U + m_V; // denominator of Pade approximant
|
||||
result = m_tmp2.partialPivLu().solve(m_tmp1);
|
||||
for (int i=0; i<m_squarings; i++)
|
||||
MatrixType U, V;
|
||||
int squarings;
|
||||
matrix_exp_computeUV<MatrixType>::run(arg, U, V, squarings); // Pade approximant is (U+V) / (-U+V)
|
||||
MatrixType numer = U + V;
|
||||
MatrixType denom = -U + V;
|
||||
result = denom.partialPivLu().solve(numer);
|
||||
for (int i=0; i<squarings; i++)
|
||||
result *= result; // undo scaling by repeated squaring
|
||||
}
|
||||
|
||||
template <typename MatrixType>
|
||||
EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade3(const MatrixType &A)
|
||||
{
|
||||
const RealScalar b[] = {120., 60., 12., 1.};
|
||||
m_tmp1.noalias() = A * A;
|
||||
m_tmp2 = b[3]*m_tmp1 + b[1]*m_Id;
|
||||
m_U.noalias() = A * m_tmp2;
|
||||
m_V = b[2]*m_tmp1 + b[0]*m_Id;
|
||||
}
|
||||
|
||||
template <typename MatrixType>
|
||||
EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade5(const MatrixType &A)
|
||||
{
|
||||
const RealScalar b[] = {30240., 15120., 3360., 420., 30., 1.};
|
||||
MatrixType A2 = A * A;
|
||||
m_tmp1.noalias() = A2 * A2;
|
||||
m_tmp2 = b[5]*m_tmp1 + b[3]*A2 + b[1]*m_Id;
|
||||
m_U.noalias() = A * m_tmp2;
|
||||
m_V = b[4]*m_tmp1 + b[2]*A2 + b[0]*m_Id;
|
||||
}
|
||||
|
||||
template <typename MatrixType>
|
||||
EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade7(const MatrixType &A)
|
||||
{
|
||||
const RealScalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.};
|
||||
MatrixType A2 = A * A;
|
||||
MatrixType A4 = A2 * A2;
|
||||
m_tmp1.noalias() = A4 * A2;
|
||||
m_tmp2 = b[7]*m_tmp1 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
|
||||
m_U.noalias() = A * m_tmp2;
|
||||
m_V = b[6]*m_tmp1 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
|
||||
}
|
||||
|
||||
template <typename MatrixType>
|
||||
EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade9(const MatrixType &A)
|
||||
{
|
||||
const RealScalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240.,
|
||||
2162160., 110880., 3960., 90., 1.};
|
||||
MatrixType A2 = A * A;
|
||||
MatrixType A4 = A2 * A2;
|
||||
MatrixType A6 = A4 * A2;
|
||||
m_tmp1.noalias() = A6 * A2;
|
||||
m_tmp2 = b[9]*m_tmp1 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
|
||||
m_U.noalias() = A * m_tmp2;
|
||||
m_V = b[8]*m_tmp1 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
|
||||
}
|
||||
|
||||
template <typename MatrixType>
|
||||
EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade13(const MatrixType &A)
|
||||
{
|
||||
const RealScalar b[] = {64764752532480000., 32382376266240000., 7771770303897600.,
|
||||
1187353796428800., 129060195264000., 10559470521600., 670442572800.,
|
||||
33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.};
|
||||
MatrixType A2 = A * A;
|
||||
MatrixType A4 = A2 * A2;
|
||||
m_tmp1.noalias() = A4 * A2;
|
||||
m_V = b[13]*m_tmp1 + b[11]*A4 + b[9]*A2; // used for temporary storage
|
||||
m_tmp2.noalias() = m_tmp1 * m_V;
|
||||
m_tmp2 += b[7]*m_tmp1 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
|
||||
m_U.noalias() = A * m_tmp2;
|
||||
m_tmp2 = b[12]*m_tmp1 + b[10]*A4 + b[8]*A2;
|
||||
m_V.noalias() = m_tmp1 * m_tmp2;
|
||||
m_V += b[6]*m_tmp1 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
|
||||
}
|
||||
|
||||
#if LDBL_MANT_DIG > 64
|
||||
template <typename MatrixType>
|
||||
EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade17(const MatrixType &A)
|
||||
{
|
||||
const RealScalar b[] = {830034394580628357120000.L, 415017197290314178560000.L,
|
||||
100610229646136770560000.L, 15720348382208870400000.L,
|
||||
1774878043152614400000.L, 153822763739893248000.L, 10608466464820224000.L,
|
||||
595373117923584000.L, 27563570274240000.L, 1060137318240000.L,
|
||||
33924394183680.L, 899510451840.L, 19554575040.L, 341863200.L, 4651200.L,
|
||||
46512.L, 306.L, 1.L};
|
||||
MatrixType A2 = A * A;
|
||||
MatrixType A4 = A2 * A2;
|
||||
MatrixType A6 = A4 * A2;
|
||||
m_tmp1.noalias() = A4 * A4;
|
||||
m_V = b[17]*m_tmp1 + b[15]*A6 + b[13]*A4 + b[11]*A2; // used for temporary storage
|
||||
m_tmp2.noalias() = m_tmp1 * m_V;
|
||||
m_tmp2 += b[9]*m_tmp1 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
|
||||
m_U.noalias() = A * m_tmp2;
|
||||
m_tmp2 = b[16]*m_tmp1 + b[14]*A6 + b[12]*A4 + b[10]*A2;
|
||||
m_V.noalias() = m_tmp1 * m_tmp2;
|
||||
m_V += b[8]*m_tmp1 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
|
||||
}
|
||||
#endif
|
||||
|
||||
template <typename MatrixType>
|
||||
void MatrixExponential<MatrixType>::computeUV(float)
|
||||
{
|
||||
using std::frexp;
|
||||
if (m_l1norm < 4.258730016922831e-001) {
|
||||
pade3(m_M);
|
||||
} else if (m_l1norm < 1.880152677804762e+000) {
|
||||
pade5(m_M);
|
||||
} else {
|
||||
const float maxnorm = 3.925724783138660f;
|
||||
frexp(m_l1norm / maxnorm, &m_squarings);
|
||||
if (m_squarings < 0) m_squarings = 0;
|
||||
MatrixType A = CwiseUnaryOp<ScalingOp, const MatrixType>(m_M, m_squarings);
|
||||
pade7(A);
|
||||
}
|
||||
}
|
||||
|
||||
template <typename MatrixType>
|
||||
void MatrixExponential<MatrixType>::computeUV(double)
|
||||
{
|
||||
using std::frexp;
|
||||
if (m_l1norm < 1.495585217958292e-002) {
|
||||
pade3(m_M);
|
||||
} else if (m_l1norm < 2.539398330063230e-001) {
|
||||
pade5(m_M);
|
||||
} else if (m_l1norm < 9.504178996162932e-001) {
|
||||
pade7(m_M);
|
||||
} else if (m_l1norm < 2.097847961257068e+000) {
|
||||
pade9(m_M);
|
||||
} else {
|
||||
const double maxnorm = 5.371920351148152;
|
||||
frexp(m_l1norm / maxnorm, &m_squarings);
|
||||
if (m_squarings < 0) m_squarings = 0;
|
||||
MatrixType A = CwiseUnaryOp<ScalingOp, const MatrixType>(m_M, m_squarings);
|
||||
pade13(A);
|
||||
}
|
||||
}
|
||||
|
||||
template <typename MatrixType>
|
||||
void MatrixExponential<MatrixType>::computeUV(long double)
|
||||
{
|
||||
using std::frexp;
|
||||
#if LDBL_MANT_DIG == 53 // double precision
|
||||
computeUV(double());
|
||||
#elif LDBL_MANT_DIG <= 64 // extended precision
|
||||
if (m_l1norm < 4.1968497232266989671e-003L) {
|
||||
pade3(m_M);
|
||||
} else if (m_l1norm < 1.1848116734693823091e-001L) {
|
||||
pade5(m_M);
|
||||
} else if (m_l1norm < 5.5170388480686700274e-001L) {
|
||||
pade7(m_M);
|
||||
} else if (m_l1norm < 1.3759868875587845383e+000L) {
|
||||
pade9(m_M);
|
||||
} else {
|
||||
const long double maxnorm = 4.0246098906697353063L;
|
||||
frexp(m_l1norm / maxnorm, &m_squarings);
|
||||
if (m_squarings < 0) m_squarings = 0;
|
||||
MatrixType A = CwiseUnaryOp<ScalingOp, const MatrixType>(m_M, m_squarings);
|
||||
pade13(A);
|
||||
}
|
||||
#elif LDBL_MANT_DIG <= 106 // double-double
|
||||
if (m_l1norm < 3.2787892205607026992947488108213e-005L) {
|
||||
pade3(m_M);
|
||||
} else if (m_l1norm < 6.4467025060072760084130906076332e-003L) {
|
||||
pade5(m_M);
|
||||
} else if (m_l1norm < 6.8988028496595374751374122881143e-002L) {
|
||||
pade7(m_M);
|
||||
} else if (m_l1norm < 2.7339737518502231741495857201670e-001L) {
|
||||
pade9(m_M);
|
||||
} else if (m_l1norm < 1.3203382096514474905666448850278e+000L) {
|
||||
pade13(m_M);
|
||||
} else {
|
||||
const long double maxnorm = 3.2579440895405400856599663723517L;
|
||||
frexp(m_l1norm / maxnorm, &m_squarings);
|
||||
if (m_squarings < 0) m_squarings = 0;
|
||||
MatrixType A = CwiseUnaryOp<ScalingOp, const MatrixType>(m_M, m_squarings);
|
||||
pade17(A);
|
||||
}
|
||||
#elif LDBL_MANT_DIG <= 112 // quadruple precison
|
||||
if (m_l1norm < 1.639394610288918690547467954466970e-005L) {
|
||||
pade3(m_M);
|
||||
} else if (m_l1norm < 4.253237712165275566025884344433009e-003L) {
|
||||
pade5(m_M);
|
||||
} else if (m_l1norm < 5.125804063165764409885122032933142e-002L) {
|
||||
pade7(m_M);
|
||||
} else if (m_l1norm < 2.170000765161155195453205651889853e-001L) {
|
||||
pade9(m_M);
|
||||
} else if (m_l1norm < 1.125358383453143065081397882891878e+000L) {
|
||||
pade13(m_M);
|
||||
} else {
|
||||
const long double maxnorm = 2.884233277829519311757165057717815L;
|
||||
frexp(m_l1norm / maxnorm, &m_squarings);
|
||||
if (m_squarings < 0) m_squarings = 0;
|
||||
MatrixType A = CwiseUnaryOp<ScalingOp, const MatrixType>(m_M, m_squarings);
|
||||
pade17(A);
|
||||
}
|
||||
#else
|
||||
// this case should be handled in compute()
|
||||
eigen_assert(false && "Bug in MatrixExponential");
|
||||
#endif // LDBL_MANT_DIG
|
||||
}
|
||||
} // end namespace Eigen::internal
|
||||
|
||||
/** \ingroup MatrixFunctions_Module
|
||||
*
|
||||
@ -422,11 +370,9 @@ void MatrixExponential<MatrixType>::computeUV(long double)
|
||||
*
|
||||
* \tparam Derived Type of the argument to the matrix exponential.
|
||||
*
|
||||
* This class holds the argument to the matrix exponential until it
|
||||
* is assigned or evaluated for some other reason (so the argument
|
||||
* should not be changed in the meantime). It is the return type of
|
||||
* MatrixBase::exp() and most of the time this is the only way it is
|
||||
* used.
|
||||
* This class holds the argument to the matrix exponential until it is assigned or evaluated for
|
||||
* some other reason (so the argument should not be changed in the meantime). It is the return type
|
||||
* of MatrixBase::exp() and most of the time this is the only way it is used.
|
||||
*/
|
||||
template<typename Derived> struct MatrixExponentialReturnValue
|
||||
: public ReturnByValue<MatrixExponentialReturnValue<Derived> >
|
||||
@ -435,22 +381,19 @@ template<typename Derived> struct MatrixExponentialReturnValue
|
||||
public:
|
||||
/** \brief Constructor.
|
||||
*
|
||||
* \param[in] src %Matrix (expression) forming the argument of the
|
||||
* matrix exponential.
|
||||
* \param src %Matrix (expression) forming the argument of the matrix exponential.
|
||||
*/
|
||||
explicit MatrixExponentialReturnValue(const Derived& src) : m_src(src) { }
|
||||
MatrixExponentialReturnValue(const Derived& src) : m_src(src) { }
|
||||
|
||||
/** \brief Compute the matrix exponential.
|
||||
*
|
||||
* \param[out] result the matrix exponential of \p src in the
|
||||
* constructor.
|
||||
* \param result the matrix exponential of \p src in the constructor.
|
||||
*/
|
||||
template <typename ResultType>
|
||||
inline void evalTo(ResultType& result) const
|
||||
{
|
||||
const typename Derived::PlainObject srcEvaluated = m_src.eval();
|
||||
MatrixExponential<typename Derived::PlainObject> me(srcEvaluated);
|
||||
me.compute(result);
|
||||
internal::matrix_exp_compute(srcEvaluated, result);
|
||||
}
|
||||
|
||||
Index rows() const { return m_src.rows(); }
|
||||
@ -458,6 +401,8 @@ template<typename Derived> struct MatrixExponentialReturnValue
|
||||
|
||||
protected:
|
||||
const Derived& m_src;
|
||||
private:
|
||||
MatrixExponentialReturnValue& operator=(const MatrixExponentialReturnValue&);
|
||||
};
|
||||
|
||||
namespace internal {
|
||||
|
||||
Loading…
Reference in New Issue
Block a user