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      1 // This file is part of Eigen, a lightweight C++ template library
      2 // for linear algebra.
      3 //
      4 // Copyright (C) 2009, 2010, 2013 Jitse Niesen <jitse (at) maths.leeds.ac.uk>
      5 // Copyright (C) 2011, 2013 Chen-Pang He <jdh8 (at) ms63.hinet.net>
      6 //
      7 // This Source Code Form is subject to the terms of the Mozilla
      8 // Public License v. 2.0. If a copy of the MPL was not distributed
      9 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
     10 
     11 #ifndef EIGEN_MATRIX_EXPONENTIAL
     12 #define EIGEN_MATRIX_EXPONENTIAL
     13 
     14 #include "StemFunction.h"
     15 
     16 namespace Eigen {
     17 namespace internal {
     18 
     19 /** \brief Scaling operator.
     20  *
     21  * This struct is used by CwiseUnaryOp to scale a matrix by \f$ 2^{-s} \f$.
     22  */
     23 template <typename RealScalar>
     24 struct MatrixExponentialScalingOp
     25 {
     26   /** \brief Constructor.
     27    *
     28    * \param[in] squarings  The integer \f$ s \f$ in this document.
     29    */
     30   MatrixExponentialScalingOp(int squarings) : m_squarings(squarings) { }
     31 
     32 
     33   /** \brief Scale a matrix coefficient.
     34    *
     35    * \param[in,out] x  The scalar to be scaled, becoming \f$ 2^{-s} x \f$.
     36    */
     37   inline const RealScalar operator() (const RealScalar& x) const
     38   {
     39     using std::ldexp;
     40     return ldexp(x, -m_squarings);
     41   }
     42 
     43   typedef std::complex<RealScalar> ComplexScalar;
     44 
     45   /** \brief Scale a matrix coefficient.
     46    *
     47    * \param[in,out] x  The scalar to be scaled, becoming \f$ 2^{-s} x \f$.
     48    */
     49   inline const ComplexScalar operator() (const ComplexScalar& x) const
     50   {
     51     using std::ldexp;
     52     return ComplexScalar(ldexp(x.real(), -m_squarings), ldexp(x.imag(), -m_squarings));
     53   }
     54 
     55   private:
     56     int m_squarings;
     57 };
     58 
     59 /** \brief Compute the (3,3)-Pad&eacute; approximant to the exponential.
     60  *
     61  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
     62  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
     63  */
     64 template <typename MatA, typename MatU, typename MatV>
     65 void matrix_exp_pade3(const MatA& A, MatU& U, MatV& V)
     66 {
     67   typedef typename MatA::PlainObject MatrixType;
     68   typedef typename NumTraits<typename traits<MatA>::Scalar>::Real RealScalar;
     69   const RealScalar b[] = {120.L, 60.L, 12.L, 1.L};
     70   const MatrixType A2 = A * A;
     71   const MatrixType tmp = b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
     72   U.noalias() = A * tmp;
     73   V = b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
     74 }
     75 
     76 /** \brief Compute the (5,5)-Pad&eacute; approximant to the exponential.
     77  *
     78  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
     79  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
     80  */
     81 template <typename MatA, typename MatU, typename MatV>
     82 void matrix_exp_pade5(const MatA& A, MatU& U, MatV& V)
     83 {
     84   typedef typename MatA::PlainObject MatrixType;
     85   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
     86   const RealScalar b[] = {30240.L, 15120.L, 3360.L, 420.L, 30.L, 1.L};
     87   const MatrixType A2 = A * A;
     88   const MatrixType A4 = A2 * A2;
     89   const MatrixType tmp = b[5] * A4 + b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
     90   U.noalias() = A * tmp;
     91   V = b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
     92 }
     93 
     94 /** \brief Compute the (7,7)-Pad&eacute; approximant to the exponential.
     95  *
     96  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
     97  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
     98  */
     99 template <typename MatA, typename MatU, typename MatV>
    100 void matrix_exp_pade7(const MatA& A, MatU& U, MatV& V)
    101 {
    102   typedef typename MatA::PlainObject MatrixType;
    103   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
    104   const RealScalar b[] = {17297280.L, 8648640.L, 1995840.L, 277200.L, 25200.L, 1512.L, 56.L, 1.L};
    105   const MatrixType A2 = A * A;
    106   const MatrixType A4 = A2 * A2;
    107   const MatrixType A6 = A4 * A2;
    108   const MatrixType tmp = b[7] * A6 + b[5] * A4 + b[3] * A2
    109     + b[1] * MatrixType::Identity(A.rows(), A.cols());
    110   U.noalias() = A * tmp;
    111   V = b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
    112 
    113 }
    114 
    115 /** \brief Compute the (9,9)-Pad&eacute; approximant to the exponential.
    116  *
    117  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
    118  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
    119  */
    120 template <typename MatA, typename MatU, typename MatV>
    121 void matrix_exp_pade9(const MatA& A, MatU& U, MatV& V)
    122 {
    123   typedef typename MatA::PlainObject MatrixType;
    124   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
    125   const RealScalar b[] = {17643225600.L, 8821612800.L, 2075673600.L, 302702400.L, 30270240.L,
    126                           2162160.L, 110880.L, 3960.L, 90.L, 1.L};
    127   const MatrixType A2 = A * A;
    128   const MatrixType A4 = A2 * A2;
    129   const MatrixType A6 = A4 * A2;
    130   const MatrixType A8 = A6 * A2;
    131   const MatrixType tmp = b[9] * A8 + b[7] * A6 + b[5] * A4 + b[3] * A2
    132     + b[1] * MatrixType::Identity(A.rows(), A.cols());
    133   U.noalias() = A * tmp;
    134   V = b[8] * A8 + b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
    135 }
    136 
    137 /** \brief Compute the (13,13)-Pad&eacute; approximant to the exponential.
    138  *
    139  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
    140  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
    141  */
    142 template <typename MatA, typename MatU, typename MatV>
    143 void matrix_exp_pade13(const MatA& A, MatU& U, MatV& V)
    144 {
    145   typedef typename MatA::PlainObject MatrixType;
    146   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
    147   const RealScalar b[] = {64764752532480000.L, 32382376266240000.L, 7771770303897600.L,
    148                           1187353796428800.L, 129060195264000.L, 10559470521600.L, 670442572800.L,
    149                           33522128640.L, 1323241920.L, 40840800.L, 960960.L, 16380.L, 182.L, 1.L};
    150   const MatrixType A2 = A * A;
    151   const MatrixType A4 = A2 * A2;
    152   const MatrixType A6 = A4 * A2;
    153   V = b[13] * A6 + b[11] * A4 + b[9] * A2; // used for temporary storage
    154   MatrixType tmp = A6 * V;
    155   tmp += b[7] * A6 + b[5] * A4 + b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
    156   U.noalias() = A * tmp;
    157   tmp = b[12] * A6 + b[10] * A4 + b[8] * A2;
    158   V.noalias() = A6 * tmp;
    159   V += b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
    160 }
    161 
    162 /** \brief Compute the (17,17)-Pad&eacute; approximant to the exponential.
    163  *
    164  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
    165  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
    166  *
    167  *  This function activates only if your long double is double-double or quadruple.
    168  */
    169 #if LDBL_MANT_DIG > 64
    170 template <typename MatA, typename MatU, typename MatV>
    171 void matrix_exp_pade17(const MatA& A, MatU& U, MatV& V)
    172 {
    173   typedef typename MatA::PlainObject MatrixType;
    174   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
    175   const RealScalar b[] = {830034394580628357120000.L, 415017197290314178560000.L,
    176                           100610229646136770560000.L, 15720348382208870400000.L,
    177                           1774878043152614400000.L, 153822763739893248000.L, 10608466464820224000.L,
    178                           595373117923584000.L, 27563570274240000.L, 1060137318240000.L,
    179                           33924394183680.L, 899510451840.L, 19554575040.L, 341863200.L, 4651200.L,
    180                           46512.L, 306.L, 1.L};
    181   const MatrixType A2 = A * A;
    182   const MatrixType A4 = A2 * A2;
    183   const MatrixType A6 = A4 * A2;
    184   const MatrixType A8 = A4 * A4;
    185   V = b[17] * A8 + b[15] * A6 + b[13] * A4 + b[11] * A2; // used for temporary storage
    186   MatrixType tmp = A8 * V;
    187   tmp += b[9] * A8 + b[7] * A6 + b[5] * A4 + b[3] * A2
    188     + b[1] * MatrixType::Identity(A.rows(), A.cols());
    189   U.noalias() = A * tmp;
    190   tmp = b[16] * A8 + b[14] * A6 + b[12] * A4 + b[10] * A2;
    191   V.noalias() = tmp * A8;
    192   V += b[8] * A8 + b[6] * A6 + b[4] * A4 + b[2] * A2
    193     + b[0] * MatrixType::Identity(A.rows(), A.cols());
    194 }
    195 #endif
    196 
    197 template <typename MatrixType, typename RealScalar = typename NumTraits<typename traits<MatrixType>::Scalar>::Real>
    198 struct matrix_exp_computeUV
    199 {
    200   /** \brief Compute Pad&eacute; approximant to the exponential.
    201     *
    202     * Computes \c U, \c V and \c squarings such that \f$ (V+U)(V-U)^{-1} \f$ is a Pad&eacute;
    203     * approximant of \f$ \exp(2^{-\mbox{squarings}}M) \f$ around \f$ M = 0 \f$, where \f$ M \f$
    204     * denotes the matrix \c arg. The degree of the Pad&eacute; approximant and the value of squarings
    205     * are chosen such that the approximation error is no more than the round-off error.
    206     */
    207   static void run(const MatrixType& arg, MatrixType& U, MatrixType& V, int& squarings);
    208 };
    209 
    210 template <typename MatrixType>
    211 struct matrix_exp_computeUV<MatrixType, float>
    212 {
    213   template <typename ArgType>
    214   static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
    215   {
    216     using std::frexp;
    217     using std::pow;
    218     const float l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
    219     squarings = 0;
    220     if (l1norm < 4.258730016922831e-001f) {
    221       matrix_exp_pade3(arg, U, V);
    222     } else if (l1norm < 1.880152677804762e+000f) {
    223       matrix_exp_pade5(arg, U, V);
    224     } else {
    225       const float maxnorm = 3.925724783138660f;
    226       frexp(l1norm / maxnorm, &squarings);
    227       if (squarings < 0) squarings = 0;
    228       MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<float>(squarings));
    229       matrix_exp_pade7(A, U, V);
    230     }
    231   }
    232 };
    233 
    234 template <typename MatrixType>
    235 struct matrix_exp_computeUV<MatrixType, double>
    236 {
    237   template <typename ArgType>
    238   static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
    239   {
    240     using std::frexp;
    241     using std::pow;
    242     const double l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
    243     squarings = 0;
    244     if (l1norm < 1.495585217958292e-002) {
    245       matrix_exp_pade3(arg, U, V);
    246     } else if (l1norm < 2.539398330063230e-001) {
    247       matrix_exp_pade5(arg, U, V);
    248     } else if (l1norm < 9.504178996162932e-001) {
    249       matrix_exp_pade7(arg, U, V);
    250     } else if (l1norm < 2.097847961257068e+000) {
    251       matrix_exp_pade9(arg, U, V);
    252     } else {
    253       const double maxnorm = 5.371920351148152;
    254       frexp(l1norm / maxnorm, &squarings);
    255       if (squarings < 0) squarings = 0;
    256       MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<double>(squarings));
    257       matrix_exp_pade13(A, U, V);
    258     }
    259   }
    260 };
    261 
    262 template <typename MatrixType>
    263 struct matrix_exp_computeUV<MatrixType, long double>
    264 {
    265   template <typename ArgType>
    266   static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
    267   {
    268 #if   LDBL_MANT_DIG == 53   // double precision
    269     matrix_exp_computeUV<MatrixType, double>::run(arg, U, V, squarings);
    270 
    271 #else
    272 
    273     using std::frexp;
    274     using std::pow;
    275     const long double l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
    276     squarings = 0;
    277 
    278 #if LDBL_MANT_DIG <= 64   // extended precision
    279 
    280     if (l1norm < 4.1968497232266989671e-003L) {
    281       matrix_exp_pade3(arg, U, V);
    282     } else if (l1norm < 1.1848116734693823091e-001L) {
    283       matrix_exp_pade5(arg, U, V);
    284     } else if (l1norm < 5.5170388480686700274e-001L) {
    285       matrix_exp_pade7(arg, U, V);
    286     } else if (l1norm < 1.3759868875587845383e+000L) {
    287       matrix_exp_pade9(arg, U, V);
    288     } else {
    289       const long double maxnorm = 4.0246098906697353063L;
    290       frexp(l1norm / maxnorm, &squarings);
    291       if (squarings < 0) squarings = 0;
    292       MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
    293       matrix_exp_pade13(A, U, V);
    294     }
    295 
    296 #elif LDBL_MANT_DIG <= 106  // double-double
    297 
    298     if (l1norm < 3.2787892205607026992947488108213e-005L) {
    299       matrix_exp_pade3(arg, U, V);
    300     } else if (l1norm < 6.4467025060072760084130906076332e-003L) {
    301       matrix_exp_pade5(arg, U, V);
    302     } else if (l1norm < 6.8988028496595374751374122881143e-002L) {
    303       matrix_exp_pade7(arg, U, V);
    304     } else if (l1norm < 2.7339737518502231741495857201670e-001L) {
    305       matrix_exp_pade9(arg, U, V);
    306     } else if (l1norm < 1.3203382096514474905666448850278e+000L) {
    307       matrix_exp_pade13(arg, U, V);
    308     } else {
    309       const long double maxnorm = 3.2579440895405400856599663723517L;
    310       frexp(l1norm / maxnorm, &squarings);
    311       if (squarings < 0) squarings = 0;
    312       MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
    313       matrix_exp_pade17(A, U, V);
    314     }
    315 
    316 #elif LDBL_MANT_DIG <= 112  // quadruple precison
    317 
    318     if (l1norm < 1.639394610288918690547467954466970e-005L) {
    319       matrix_exp_pade3(arg, U, V);
    320     } else if (l1norm < 4.253237712165275566025884344433009e-003L) {
    321       matrix_exp_pade5(arg, U, V);
    322     } else if (l1norm < 5.125804063165764409885122032933142e-002L) {
    323       matrix_exp_pade7(arg, U, V);
    324     } else if (l1norm < 2.170000765161155195453205651889853e-001L) {
    325       matrix_exp_pade9(arg, U, V);
    326     } else if (l1norm < 1.125358383453143065081397882891878e+000L) {
    327       matrix_exp_pade13(arg, U, V);
    328     } else {
    329       frexp(l1norm / maxnorm, &squarings);
    330       if (squarings < 0) squarings = 0;
    331       MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
    332       matrix_exp_pade17(A, U, V);
    333     }
    334 
    335 #else
    336 
    337     // this case should be handled in compute()
    338     eigen_assert(false && "Bug in MatrixExponential");
    339 
    340 #endif
    341 #endif  // LDBL_MANT_DIG
    342   }
    343 };
    344 
    345 
    346 /* Computes the matrix exponential
    347  *
    348  * \param arg    argument of matrix exponential (should be plain object)
    349  * \param result variable in which result will be stored
    350  */
    351 template <typename ArgType, typename ResultType>
    352 void matrix_exp_compute(const ArgType& arg, ResultType &result)
    353 {
    354   typedef typename ArgType::PlainObject MatrixType;
    355 #if LDBL_MANT_DIG > 112 // rarely happens
    356   typedef typename traits<MatrixType>::Scalar Scalar;
    357   typedef typename NumTraits<Scalar>::Real RealScalar;
    358   typedef typename std::complex<RealScalar> ComplexScalar;
    359   if (sizeof(RealScalar) > 14) {
    360     result = arg.matrixFunction(internal::stem_function_exp<ComplexScalar>);
    361     return;
    362   }
    363 #endif
    364   MatrixType U, V;
    365   int squarings;
    366   matrix_exp_computeUV<MatrixType>::run(arg, U, V, squarings); // Pade approximant is (U+V) / (-U+V)
    367   MatrixType numer = U + V;
    368   MatrixType denom = -U + V;
    369   result = denom.partialPivLu().solve(numer);
    370   for (int i=0; i<squarings; i++)
    371     result *= result;   // undo scaling by repeated squaring
    372 }
    373 
    374 } // end namespace Eigen::internal
    375 
    376 /** \ingroup MatrixFunctions_Module
    377   *
    378   * \brief Proxy for the matrix exponential of some matrix (expression).
    379   *
    380   * \tparam Derived  Type of the argument to the matrix exponential.
    381   *
    382   * This class holds the argument to the matrix exponential until it is assigned or evaluated for
    383   * some other reason (so the argument should not be changed in the meantime). It is the return type
    384   * of MatrixBase::exp() and most of the time this is the only way it is used.
    385   */
    386 template<typename Derived> struct MatrixExponentialReturnValue
    387 : public ReturnByValue<MatrixExponentialReturnValue<Derived> >
    388 {
    389     typedef typename Derived::Index Index;
    390   public:
    391     /** \brief Constructor.
    392       *
    393       * \param src %Matrix (expression) forming the argument of the matrix exponential.
    394       */
    395     MatrixExponentialReturnValue(const Derived& src) : m_src(src) { }
    396 
    397     /** \brief Compute the matrix exponential.
    398       *
    399       * \param result the matrix exponential of \p src in the constructor.
    400       */
    401     template <typename ResultType>
    402     inline void evalTo(ResultType& result) const
    403     {
    404       const typename internal::nested_eval<Derived, 10>::type tmp(m_src);
    405       internal::matrix_exp_compute(tmp, result);
    406     }
    407 
    408     Index rows() const { return m_src.rows(); }
    409     Index cols() const { return m_src.cols(); }
    410 
    411   protected:
    412     const typename internal::ref_selector<Derived>::type m_src;
    413 };
    414 
    415 namespace internal {
    416 template<typename Derived>
    417 struct traits<MatrixExponentialReturnValue<Derived> >
    418 {
    419   typedef typename Derived::PlainObject ReturnType;
    420 };
    421 }
    422 
    423 template <typename Derived>
    424 const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp() const
    425 {
    426   eigen_assert(rows() == cols());
    427   return MatrixExponentialReturnValue<Derived>(derived());
    428 }
    429 
    430 } // end namespace Eigen
    431 
    432 #endif // EIGEN_MATRIX_EXPONENTIAL
    433