MatrixExponential.h
00001 // This file is part of Eigen, a lightweight C++ template library
00002 // for linear algebra.
00003 //
00004 // Copyright (C) 2009, 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
00005 // Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net>
00006 //
00007 // Eigen is free software; you can redistribute it and/or
00008 // modify it under the terms of the GNU Lesser General Public
00009 // License as published by the Free Software Foundation; either
00010 // version 3 of the License, or (at your option) any later version.
00011 //
00012 // Alternatively, you can redistribute it and/or
00013 // modify it under the terms of the GNU General Public License as
00014 // published by the Free Software Foundation; either version 2 of
00015 // the License, or (at your option) any later version.
00016 //
00017 // Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
00018 // WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
00019 // FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
00020 // GNU General Public License for more details.
00021 //
00022 // You should have received a copy of the GNU Lesser General Public
00023 // License and a copy of the GNU General Public License along with
00024 // Eigen. If not, see <http://www.gnu.org/licenses/>.
00025 
00026 #ifndef EIGEN_MATRIX_EXPONENTIAL
00027 #define EIGEN_MATRIX_EXPONENTIAL
00028 
00029 #include "StemFunction.h"
00030 
00031 namespace Eigen { 
00032 
00033 #if defined(_MSC_VER) || defined(__FreeBSD__)
00034   template <typename Scalar> Scalar log2(Scalar v) { using std::log; return log(v)/log(Scalar(2)); }
00035 #endif
00036 
00037 
00043 template <typename MatrixType>
00044 class MatrixExponential {
00045 
00046   public:
00047 
00055     MatrixExponential(const MatrixType &M);
00056 
00061     template <typename ResultType> 
00062     void compute(ResultType &result);
00063 
00064   private:
00065 
00066     // Prevent copying
00067     MatrixExponential(const MatrixExponential&);
00068     MatrixExponential& operator=(const MatrixExponential&);
00069 
00077     void pade3(const MatrixType &A);
00078 
00086     void pade5(const MatrixType &A);
00087 
00095     void pade7(const MatrixType &A);
00096 
00104     void pade9(const MatrixType &A);
00105 
00113     void pade13(const MatrixType &A);
00114 
00124     void pade17(const MatrixType &A);
00125 
00139     void computeUV(double);
00140 
00145     void computeUV(float);
00146     
00151     void computeUV(long double);
00152 
00153     typedef typename internal::traits<MatrixType>::Scalar Scalar;
00154     typedef typename NumTraits<Scalar>::Real RealScalar;
00155     typedef typename std::complex<RealScalar> ComplexScalar;
00156 
00158     typename internal::nested<MatrixType>::type m_M;
00159 
00161     MatrixType m_U;
00162 
00164     MatrixType m_V;
00165 
00167     MatrixType m_tmp1;
00168 
00170     MatrixType m_tmp2;
00171 
00173     MatrixType m_Id;
00174 
00176     int m_squarings;
00177 
00179     RealScalar m_l1norm;
00180 };
00181 
00182 template <typename MatrixType>
00183 MatrixExponential<MatrixType>::MatrixExponential(const MatrixType &M) :
00184   m_M(M),
00185   m_U(M.rows(),M.cols()),
00186   m_V(M.rows(),M.cols()),
00187   m_tmp1(M.rows(),M.cols()),
00188   m_tmp2(M.rows(),M.cols()),
00189   m_Id(MatrixType::Identity(M.rows(), M.cols())),
00190   m_squarings(0),
00191   m_l1norm(M.cwiseAbs().colwise().sum().maxCoeff())
00192 {
00193   /* empty body */
00194 }
00195 
00196 template <typename MatrixType>
00197 template <typename ResultType> 
00198 void MatrixExponential<MatrixType>::compute(ResultType &result)
00199 {
00200 #if LDBL_MANT_DIG > 112 // rarely happens
00201   if(sizeof(RealScalar) > 14) {
00202     result = m_M.matrixFunction(StdStemFunctions<ComplexScalar>::exp);
00203     return;
00204   }
00205 #endif
00206   computeUV(RealScalar());
00207   m_tmp1 = m_U + m_V;   // numerator of Pade approximant
00208   m_tmp2 = -m_U + m_V;  // denominator of Pade approximant
00209   result = m_tmp2.partialPivLu().solve(m_tmp1);
00210   for (int i=0; i<m_squarings; i++)
00211     result *= result;   // undo scaling by repeated squaring
00212 }
00213 
00214 template <typename MatrixType>
00215 EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade3(const MatrixType &A)
00216 {
00217   const RealScalar b[] = {120., 60., 12., 1.};
00218   m_tmp1.noalias() = A * A;
00219   m_tmp2 = b[3]*m_tmp1 + b[1]*m_Id;
00220   m_U.noalias() = A * m_tmp2;
00221   m_V = b[2]*m_tmp1 + b[0]*m_Id;
00222 }
00223 
00224 template <typename MatrixType>
00225 EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade5(const MatrixType &A)
00226 {
00227   const RealScalar b[] = {30240., 15120., 3360., 420., 30., 1.};
00228   MatrixType A2 = A * A;
00229   m_tmp1.noalias() = A2 * A2;
00230   m_tmp2 = b[5]*m_tmp1 + b[3]*A2 + b[1]*m_Id;
00231   m_U.noalias() = A * m_tmp2;
00232   m_V = b[4]*m_tmp1 + b[2]*A2 + b[0]*m_Id;
00233 }
00234 
00235 template <typename MatrixType>
00236 EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade7(const MatrixType &A)
00237 {
00238   const RealScalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.};
00239   MatrixType A2 = A * A;
00240   MatrixType A4 = A2 * A2;
00241   m_tmp1.noalias() = A4 * A2;
00242   m_tmp2 = b[7]*m_tmp1 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
00243   m_U.noalias() = A * m_tmp2;
00244   m_V = b[6]*m_tmp1 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
00245 }
00246 
00247 template <typename MatrixType>
00248 EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade9(const MatrixType &A)
00249 {
00250   const RealScalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240.,
00251                       2162160., 110880., 3960., 90., 1.};
00252   MatrixType A2 = A * A;
00253   MatrixType A4 = A2 * A2;
00254   MatrixType A6 = A4 * A2;
00255   m_tmp1.noalias() = A6 * A2;
00256   m_tmp2 = b[9]*m_tmp1 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
00257   m_U.noalias() = A * m_tmp2;
00258   m_V = b[8]*m_tmp1 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
00259 }
00260 
00261 template <typename MatrixType>
00262 EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade13(const MatrixType &A)
00263 {
00264   const RealScalar b[] = {64764752532480000., 32382376266240000., 7771770303897600.,
00265                       1187353796428800., 129060195264000., 10559470521600., 670442572800.,
00266                       33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.};
00267   MatrixType A2 = A * A;
00268   MatrixType A4 = A2 * A2;
00269   m_tmp1.noalias() = A4 * A2;
00270   m_V = b[13]*m_tmp1 + b[11]*A4 + b[9]*A2; // used for temporary storage
00271   m_tmp2.noalias() = m_tmp1 * m_V;
00272   m_tmp2 += b[7]*m_tmp1 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
00273   m_U.noalias() = A * m_tmp2;
00274   m_tmp2 = b[12]*m_tmp1 + b[10]*A4 + b[8]*A2;
00275   m_V.noalias() = m_tmp1 * m_tmp2;
00276   m_V += b[6]*m_tmp1 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
00277 }
00278 
00279 #if LDBL_MANT_DIG > 64
00280 template <typename MatrixType>
00281 EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade17(const MatrixType &A)
00282 {
00283   const RealScalar b[] = {830034394580628357120000.L, 415017197290314178560000.L,
00284             100610229646136770560000.L, 15720348382208870400000.L,
00285             1774878043152614400000.L, 153822763739893248000.L, 10608466464820224000.L,
00286             595373117923584000.L, 27563570274240000.L, 1060137318240000.L,
00287             33924394183680.L, 899510451840.L, 19554575040.L, 341863200.L, 4651200.L,
00288             46512.L, 306.L, 1.L};
00289   MatrixType A2 = A * A;
00290   MatrixType A4 = A2 * A2;
00291   MatrixType A6 = A4 * A2;
00292   m_tmp1.noalias() = A4 * A4;
00293   m_V = b[17]*m_tmp1 + b[15]*A6 + b[13]*A4 + b[11]*A2; // used for temporary storage
00294   m_tmp2.noalias() = m_tmp1 * m_V;
00295   m_tmp2 += b[9]*m_tmp1 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
00296   m_U.noalias() = A * m_tmp2;
00297   m_tmp2 = b[16]*m_tmp1 + b[14]*A6 + b[12]*A4 + b[10]*A2;
00298   m_V.noalias() = m_tmp1 * m_tmp2;
00299   m_V += b[8]*m_tmp1 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
00300 }
00301 #endif
00302 
00303 template <typename MatrixType>
00304 void MatrixExponential<MatrixType>::computeUV(float)
00305 {
00306   using std::max;
00307   using std::pow;
00308   using std::ceil;
00309   if (m_l1norm < 4.258730016922831e-001) {
00310     pade3(m_M);
00311   } else if (m_l1norm < 1.880152677804762e+000) {
00312     pade5(m_M);
00313   } else {
00314     const float maxnorm = 3.925724783138660f;
00315     m_squarings = (max)(0, (int)ceil(log2(m_l1norm / maxnorm)));
00316     MatrixType A = m_M / pow(Scalar(2), m_squarings);
00317     pade7(A);
00318   }
00319 }
00320 
00321 template <typename MatrixType>
00322 void MatrixExponential<MatrixType>::computeUV(double)
00323 {
00324   using std::max;
00325   using std::pow;
00326   using std::ceil;
00327   if (m_l1norm < 1.495585217958292e-002) {
00328     pade3(m_M);
00329   } else if (m_l1norm < 2.539398330063230e-001) {
00330     pade5(m_M);
00331   } else if (m_l1norm < 9.504178996162932e-001) {
00332     pade7(m_M);
00333   } else if (m_l1norm < 2.097847961257068e+000) {
00334     pade9(m_M);
00335   } else {
00336     const double maxnorm = 5.371920351148152;
00337     m_squarings = (max)(0, (int)ceil(log2(m_l1norm / maxnorm)));
00338     MatrixType A = m_M / pow(Scalar(2), m_squarings);
00339     pade13(A);
00340   }
00341 }
00342 
00343 template <typename MatrixType>
00344 void MatrixExponential<MatrixType>::computeUV(long double)
00345 {
00346   using std::max;
00347   using std::pow;
00348   using std::ceil;
00349 #if   LDBL_MANT_DIG == 53   // double precision
00350   computeUV(double());
00351 #elif LDBL_MANT_DIG <= 64   // extended precision
00352   if (m_l1norm < 4.1968497232266989671e-003L) {
00353     pade3(m_M);
00354   } else if (m_l1norm < 1.1848116734693823091e-001L) {
00355     pade5(m_M);
00356   } else if (m_l1norm < 5.5170388480686700274e-001L) {
00357     pade7(m_M);
00358   } else if (m_l1norm < 1.3759868875587845383e+000L) {
00359     pade9(m_M);
00360   } else {
00361     const long double maxnorm = 4.0246098906697353063L;
00362     m_squarings = (max)(0, (int)ceil(log2(m_l1norm / maxnorm)));
00363     MatrixType A = m_M / pow(Scalar(2), m_squarings);
00364     pade13(A);
00365   }
00366 #elif LDBL_MANT_DIG <= 106  // double-double
00367   if (m_l1norm < 3.2787892205607026992947488108213e-005L) {
00368     pade3(m_M);
00369   } else if (m_l1norm < 6.4467025060072760084130906076332e-003L) {
00370     pade5(m_M);
00371   } else if (m_l1norm < 6.8988028496595374751374122881143e-002L) {
00372     pade7(m_M);
00373   } else if (m_l1norm < 2.7339737518502231741495857201670e-001L) {
00374     pade9(m_M);
00375   } else if (m_l1norm < 1.3203382096514474905666448850278e+000L) {
00376     pade13(m_M);
00377   } else {
00378     const long double maxnorm = 3.2579440895405400856599663723517L;
00379     m_squarings = (max)(0, (int)ceil(log2(m_l1norm / maxnorm)));
00380     MatrixType A = m_M / pow(Scalar(2), m_squarings);
00381     pade17(A);
00382   }
00383 #elif LDBL_MANT_DIG <= 112  // quadruple precison
00384   if (m_l1norm < 1.639394610288918690547467954466970e-005L) {
00385     pade3(m_M);
00386   } else if (m_l1norm < 4.253237712165275566025884344433009e-003L) {
00387     pade5(m_M);
00388   } else if (m_l1norm < 5.125804063165764409885122032933142e-002L) {
00389     pade7(m_M);
00390   } else if (m_l1norm < 2.170000765161155195453205651889853e-001L) {
00391     pade9(m_M);
00392   } else if (m_l1norm < 1.125358383453143065081397882891878e+000L) {
00393     pade13(m_M);
00394   } else {
00395     const long double maxnorm = 2.884233277829519311757165057717815L;
00396     m_squarings = (max)(0, (int)ceil(log2(m_l1norm / maxnorm)));
00397     MatrixType A = m_M / pow(Scalar(2), m_squarings);
00398     pade17(A);
00399   }
00400 #else
00401   // this case should be handled in compute()
00402   eigen_assert(false && "Bug in MatrixExponential"); 
00403 #endif  // LDBL_MANT_DIG
00404 }
00405 
00418 template<typename Derived> struct MatrixExponentialReturnValue
00419 : public ReturnByValue<MatrixExponentialReturnValue<Derived> >
00420 {
00421     typedef typename Derived::Index Index;
00422   public:
00428     MatrixExponentialReturnValue(const Derived& src) : m_src(src) { }
00429 
00435     template <typename ResultType>
00436     inline void evalTo(ResultType& result) const
00437     {
00438       const typename Derived::PlainObject srcEvaluated = m_src.eval();
00439       MatrixExponential<typename Derived::PlainObject> me(srcEvaluated);
00440       me.compute(result);
00441     }
00442 
00443     Index rows() const { return m_src.rows(); }
00444     Index cols() const { return m_src.cols(); }
00445 
00446   protected:
00447     const Derived& m_src;
00448   private:
00449     MatrixExponentialReturnValue& operator=(const MatrixExponentialReturnValue&);
00450 };
00451 
00452 namespace internal {
00453 template<typename Derived>
00454 struct traits<MatrixExponentialReturnValue<Derived> >
00455 {
00456   typedef typename Derived::PlainObject ReturnType;
00457 };
00458 }
00459 
00460 template <typename Derived>
00461 const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp() const
00462 {
00463   eigen_assert(rows() == cols());
00464   return MatrixExponentialReturnValue<Derived>(derived());
00465 }
00466 
00467 } // end namespace Eigen
00468 
00469 #endif // EIGEN_MATRIX_EXPONENTIAL