| // This file is part of Eigen, a lightweight C++ template library |
| // for linear algebra. |
| // |
| // Copyright (C) 2009, 2010 Jitse Niesen <jitse@maths.leeds.ac.uk> |
| // Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net> |
| // |
| // This Source Code Form is subject to the terms of the Mozilla |
| // Public License v. 2.0. If a copy of the MPL was not distributed |
| // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. |
| |
| #ifndef EIGEN_MATRIX_EXPONENTIAL |
| #define EIGEN_MATRIX_EXPONENTIAL |
| |
| #include "StemFunction.h" |
| |
| namespace Eigen { |
| |
| #if defined(_MSC_VER) || defined(__FreeBSD__) |
| template <typename Scalar> Scalar log2(Scalar v) { using std::log; return log(v)/log(Scalar(2)); } |
| #endif |
| |
| |
| /** \ingroup MatrixFunctions_Module |
| * \brief Class for computing the matrix exponential. |
| * \tparam MatrixType type of the argument of the exponential, |
| * expected to be an instantiation of the Matrix class template. |
| */ |
| template <typename MatrixType> |
| class MatrixExponential { |
| |
| public: |
| |
| /** \brief Constructor. |
| * |
| * The class stores a reference to \p M, so it should not be |
| * changed (or destroyed) before compute() is called. |
| * |
| * \param[in] M matrix whose exponential is to be computed. |
| */ |
| MatrixExponential(const MatrixType &M); |
| |
| /** \brief Computes the matrix exponential. |
| * |
| * \param[out] result the matrix exponential of \p M in the constructor. |
| */ |
| template <typename ResultType> |
| void compute(ResultType &result); |
| |
| private: |
| |
| // Prevent copying |
| MatrixExponential(const MatrixExponential&); |
| MatrixExponential& operator=(const MatrixExponential&); |
| |
| /** \brief Compute the (3,3)-Padé approximant to the exponential. |
| * |
| * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé |
| * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$. |
| * |
| * \param[in] A Argument of matrix exponential |
| */ |
| void pade3(const MatrixType &A); |
| |
| /** \brief Compute the (5,5)-Padé approximant to the exponential. |
| * |
| * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé |
| * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$. |
| * |
| * \param[in] A Argument of matrix exponential |
| */ |
| void pade5(const MatrixType &A); |
| |
| /** \brief Compute the (7,7)-Padé approximant to the exponential. |
| * |
| * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé |
| * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$. |
| * |
| * \param[in] A Argument of matrix exponential |
| */ |
| void pade7(const MatrixType &A); |
| |
| /** \brief Compute the (9,9)-Padé approximant to the exponential. |
| * |
| * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé |
| * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$. |
| * |
| * \param[in] A Argument of matrix exponential |
| */ |
| void pade9(const MatrixType &A); |
| |
| /** \brief Compute the (13,13)-Padé approximant to the exponential. |
| * |
| * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé |
| * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$. |
| * |
| * \param[in] A Argument of matrix exponential |
| */ |
| void pade13(const MatrixType &A); |
| |
| /** \brief Compute the (17,17)-Padé approximant to the exponential. |
| * |
| * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé |
| * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$. |
| * |
| * This function activates only if your long double is double-double or quadruple. |
| * |
| * \param[in] A Argument of matrix exponential |
| */ |
| void pade17(const MatrixType &A); |
| |
| /** \brief Compute Padé approximant to the exponential. |
| * |
| * Computes \c m_U, \c m_V and \c m_squarings such that |
| * \f$ (V+U)(V-U)^{-1} \f$ is a Padé of |
| * \f$ \exp(2^{-\mbox{squarings}}M) \f$ around \f$ M = 0 \f$. The |
| * degree of the Padé approximant and the value of |
| * squarings are chosen such that the approximation error is no |
| * more than the round-off error. |
| * |
| * The argument of this function should correspond with the (real |
| * part of) the entries of \c m_M. It is used to select the |
| * correct implementation using overloading. |
| */ |
| void computeUV(double); |
| |
| /** \brief Compute Padé approximant to the exponential. |
| * |
| * \sa computeUV(double); |
| */ |
| void computeUV(float); |
| |
| /** \brief Compute Padé approximant to the exponential. |
| * |
| * \sa computeUV(double); |
| */ |
| void computeUV(long double); |
| |
| typedef typename internal::traits<MatrixType>::Scalar Scalar; |
| typedef typename NumTraits<Scalar>::Real RealScalar; |
| typedef typename std::complex<RealScalar> ComplexScalar; |
| |
| /** \brief Reference to matrix whose exponential is to be computed. */ |
| typename internal::nested<MatrixType>::type m_M; |
| |
| /** \brief Odd-degree terms in numerator of Padé approximant. */ |
| MatrixType m_U; |
| |
| /** \brief Even-degree terms in numerator of Padé approximant. */ |
| MatrixType m_V; |
| |
| /** \brief Used for temporary storage. */ |
| MatrixType m_tmp1; |
| |
| /** \brief Used for temporary storage. */ |
| MatrixType m_tmp2; |
| |
| /** \brief Identity matrix of the same size as \c m_M. */ |
| MatrixType m_Id; |
| |
| /** \brief Number of squarings required in the last step. */ |
| int m_squarings; |
| |
| /** \brief L1 norm of m_M. */ |
| RealScalar m_l1norm; |
| }; |
| |
| template <typename MatrixType> |
| MatrixExponential<MatrixType>::MatrixExponential(const MatrixType &M) : |
| m_M(M), |
| m_U(M.rows(),M.cols()), |
| m_V(M.rows(),M.cols()), |
| m_tmp1(M.rows(),M.cols()), |
| m_tmp2(M.rows(),M.cols()), |
| m_Id(MatrixType::Identity(M.rows(), M.cols())), |
| m_squarings(0), |
| m_l1norm(M.cwiseAbs().colwise().sum().maxCoeff()) |
| { |
| /* empty body */ |
| } |
| |
| template <typename MatrixType> |
| template <typename ResultType> |
| void MatrixExponential<MatrixType>::compute(ResultType &result) |
| { |
| #if LDBL_MANT_DIG > 112 // rarely happens |
| if(sizeof(RealScalar) > 14) { |
| result = m_M.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++) |
| 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::max; |
| using std::pow; |
| using std::ceil; |
| 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; |
| m_squarings = (max)(0, (int)ceil(log2(m_l1norm / maxnorm))); |
| MatrixType A = m_M / pow(Scalar(2), m_squarings); |
| pade7(A); |
| } |
| } |
| |
| template <typename MatrixType> |
| void MatrixExponential<MatrixType>::computeUV(double) |
| { |
| using std::max; |
| using std::pow; |
| using std::ceil; |
| 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; |
| m_squarings = (max)(0, (int)ceil(log2(m_l1norm / maxnorm))); |
| MatrixType A = m_M / pow(Scalar(2), m_squarings); |
| pade13(A); |
| } |
| } |
| |
| template <typename MatrixType> |
| void MatrixExponential<MatrixType>::computeUV(long double) |
| { |
| using std::max; |
| using std::pow; |
| using std::ceil; |
| #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; |
| m_squarings = (max)(0, (int)ceil(log2(m_l1norm / maxnorm))); |
| MatrixType A = m_M / pow(Scalar(2), 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; |
| m_squarings = (max)(0, (int)ceil(log2(m_l1norm / maxnorm))); |
| MatrixType A = m_M / pow(Scalar(2), 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; |
| m_squarings = (max)(0, (int)ceil(log2(m_l1norm / maxnorm))); |
| MatrixType A = m_M / pow(Scalar(2), m_squarings); |
| pade17(A); |
| } |
| #else |
| // this case should be handled in compute() |
| eigen_assert(false && "Bug in MatrixExponential"); |
| #endif // LDBL_MANT_DIG |
| } |
| |
| /** \ingroup MatrixFunctions_Module |
| * |
| * \brief Proxy for the matrix exponential of some matrix (expression). |
| * |
| * \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. |
| */ |
| template<typename Derived> struct MatrixExponentialReturnValue |
| : public ReturnByValue<MatrixExponentialReturnValue<Derived> > |
| { |
| typedef typename Derived::Index Index; |
| public: |
| /** \brief Constructor. |
| * |
| * \param[in] src %Matrix (expression) forming the argument of the |
| * matrix exponential. |
| */ |
| MatrixExponentialReturnValue(const Derived& src) : m_src(src) { } |
| |
| /** \brief Compute the matrix exponential. |
| * |
| * \param[out] 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); |
| } |
| |
| Index rows() const { return m_src.rows(); } |
| Index cols() const { return m_src.cols(); } |
| |
| protected: |
| const Derived& m_src; |
| private: |
| MatrixExponentialReturnValue& operator=(const MatrixExponentialReturnValue&); |
| }; |
| |
| namespace internal { |
| template<typename Derived> |
| struct traits<MatrixExponentialReturnValue<Derived> > |
| { |
| typedef typename Derived::PlainObject ReturnType; |
| }; |
| } |
| |
| template <typename Derived> |
| const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp() const |
| { |
| eigen_assert(rows() == cols()); |
| return MatrixExponentialReturnValue<Derived>(derived()); |
| } |
| |
| } // end namespace Eigen |
| |
| #endif // EIGEN_MATRIX_EXPONENTIAL |
