MatrixExponential.h (16624B)
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@maths.leeds.ac.uk> 5 // Copyright (C) 2011, 2013 Chen-Pang He <jdh8@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$. 48 */ 49 inline const ComplexScalar operator() (const ComplexScalar& x) const 50 { 51 using std::ldexp; 52 return ComplexScalar(ldexp(x.real(), -m_squarings), ldexp(x.imag(), -m_squarings)); 53 } 54 55 private: 56 int m_squarings; 57 }; 58 59 /** \brief Compute the (3,3)-Padé approximant to the exponential. 60 * 61 * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé 62 * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$. 63 */ 64 template <typename MatA, typename MatU, typename MatV> 65 void matrix_exp_pade3(const MatA& A, MatU& U, MatV& V) 66 { 67 typedef typename MatA::PlainObject MatrixType; 68 typedef typename NumTraits<typename traits<MatA>::Scalar>::Real RealScalar; 69 const RealScalar b[] = {120.L, 60.L, 12.L, 1.L}; 70 const MatrixType A2 = A * A; 71 const MatrixType tmp = b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols()); 72 U.noalias() = A * tmp; 73 V = b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols()); 74 } 75 76 /** \brief Compute the (5,5)-Padé approximant to the exponential. 77 * 78 * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé 79 * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$. 80 */ 81 template <typename MatA, typename MatU, typename MatV> 82 void matrix_exp_pade5(const MatA& A, MatU& U, MatV& V) 83 { 84 typedef typename MatA::PlainObject MatrixType; 85 typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar; 86 const RealScalar b[] = {30240.L, 15120.L, 3360.L, 420.L, 30.L, 1.L}; 87 const MatrixType A2 = A * A; 88 const MatrixType A4 = A2 * A2; 89 const MatrixType tmp = b[5] * A4 + b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols()); 90 U.noalias() = A * tmp; 91 V = b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols()); 92 } 93 94 /** \brief Compute the (7,7)-Padé approximant to the exponential. 95 * 96 * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé 97 * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$. 98 */ 99 template <typename MatA, typename MatU, typename MatV> 100 void matrix_exp_pade7(const MatA& A, MatU& U, MatV& V) 101 { 102 typedef typename MatA::PlainObject MatrixType; 103 typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar; 104 const RealScalar b[] = {17297280.L, 8648640.L, 1995840.L, 277200.L, 25200.L, 1512.L, 56.L, 1.L}; 105 const MatrixType A2 = A * A; 106 const MatrixType A4 = A2 * A2; 107 const MatrixType A6 = A4 * A2; 108 const MatrixType tmp = b[7] * A6 + b[5] * A4 + b[3] * A2 109 + b[1] * MatrixType::Identity(A.rows(), A.cols()); 110 U.noalias() = A * tmp; 111 V = b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols()); 112 113 } 114 115 /** \brief Compute the (9,9)-Padé approximant to the exponential. 116 * 117 * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé 118 * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$. 119 */ 120 template <typename MatA, typename MatU, typename MatV> 121 void matrix_exp_pade9(const MatA& A, MatU& U, MatV& V) 122 { 123 typedef typename MatA::PlainObject MatrixType; 124 typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar; 125 const RealScalar b[] = {17643225600.L, 8821612800.L, 2075673600.L, 302702400.L, 30270240.L, 126 2162160.L, 110880.L, 3960.L, 90.L, 1.L}; 127 const MatrixType A2 = A * A; 128 const MatrixType A4 = A2 * A2; 129 const MatrixType A6 = A4 * A2; 130 const MatrixType A8 = A6 * A2; 131 const MatrixType tmp = b[9] * A8 + b[7] * A6 + b[5] * A4 + b[3] * A2 132 + b[1] * MatrixType::Identity(A.rows(), A.cols()); 133 U.noalias() = A * tmp; 134 V = b[8] * A8 + b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols()); 135 } 136 137 /** \brief Compute the (13,13)-Padé approximant to the exponential. 138 * 139 * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé 140 * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$. 141 */ 142 template <typename MatA, typename MatU, typename MatV> 143 void matrix_exp_pade13(const MatA& A, MatU& U, MatV& V) 144 { 145 typedef typename MatA::PlainObject MatrixType; 146 typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar; 147 const RealScalar b[] = {64764752532480000.L, 32382376266240000.L, 7771770303897600.L, 148 1187353796428800.L, 129060195264000.L, 10559470521600.L, 670442572800.L, 149 33522128640.L, 1323241920.L, 40840800.L, 960960.L, 16380.L, 182.L, 1.L}; 150 const MatrixType A2 = A * A; 151 const MatrixType A4 = A2 * A2; 152 const MatrixType A6 = A4 * A2; 153 V = b[13] * A6 + b[11] * A4 + b[9] * A2; // used for temporary storage 154 MatrixType tmp = A6 * V; 155 tmp += b[7] * A6 + b[5] * A4 + b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols()); 156 U.noalias() = A * tmp; 157 tmp = b[12] * A6 + b[10] * A4 + b[8] * A2; 158 V.noalias() = A6 * tmp; 159 V += b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols()); 160 } 161 162 /** \brief Compute the (17,17)-Padé approximant to the exponential. 163 * 164 * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Padé 165 * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$. 166 * 167 * This function activates only if your long double is double-double or quadruple. 168 */ 169 #if LDBL_MANT_DIG > 64 170 template <typename MatA, typename MatU, typename MatV> 171 void matrix_exp_pade17(const MatA& A, MatU& U, MatV& V) 172 { 173 typedef typename MatA::PlainObject MatrixType; 174 typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar; 175 const RealScalar b[] = {830034394580628357120000.L, 415017197290314178560000.L, 176 100610229646136770560000.L, 15720348382208870400000.L, 177 1774878043152614400000.L, 153822763739893248000.L, 10608466464820224000.L, 178 595373117923584000.L, 27563570274240000.L, 1060137318240000.L, 179 33924394183680.L, 899510451840.L, 19554575040.L, 341863200.L, 4651200.L, 180 46512.L, 306.L, 1.L}; 181 const MatrixType A2 = A * A; 182 const MatrixType A4 = A2 * A2; 183 const MatrixType A6 = A4 * A2; 184 const MatrixType A8 = A4 * A4; 185 V = b[17] * A8 + b[15] * A6 + b[13] * A4 + b[11] * A2; // used for temporary storage 186 MatrixType tmp = A8 * V; 187 tmp += b[9] * A8 + b[7] * A6 + b[5] * A4 + b[3] * A2 188 + b[1] * MatrixType::Identity(A.rows(), A.cols()); 189 U.noalias() = A * tmp; 190 tmp = b[16] * A8 + b[14] * A6 + b[12] * A4 + b[10] * A2; 191 V.noalias() = tmp * A8; 192 V += b[8] * A8 + b[6] * A6 + b[4] * A4 + b[2] * A2 193 + b[0] * MatrixType::Identity(A.rows(), A.cols()); 194 } 195 #endif 196 197 template <typename MatrixType, typename RealScalar = typename NumTraits<typename traits<MatrixType>::Scalar>::Real> 198 struct matrix_exp_computeUV 199 { 200 /** \brief Compute Padé approximant to the exponential. 201 * 202 * Computes \c U, \c V and \c squarings such that \f$ (V+U)(V-U)^{-1} \f$ is a Padé 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 Padé 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 typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar; 238 template <typename ArgType> 239 static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings) 240 { 241 using std::frexp; 242 using std::pow; 243 const RealScalar l1norm = arg.cwiseAbs().colwise().sum().maxCoeff(); 244 squarings = 0; 245 if (l1norm < 1.495585217958292e-002) { 246 matrix_exp_pade3(arg, U, V); 247 } else if (l1norm < 2.539398330063230e-001) { 248 matrix_exp_pade5(arg, U, V); 249 } else if (l1norm < 9.504178996162932e-001) { 250 matrix_exp_pade7(arg, U, V); 251 } else if (l1norm < 2.097847961257068e+000) { 252 matrix_exp_pade9(arg, U, V); 253 } else { 254 const RealScalar maxnorm = 5.371920351148152; 255 frexp(l1norm / maxnorm, &squarings); 256 if (squarings < 0) squarings = 0; 257 MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<RealScalar>(squarings)); 258 matrix_exp_pade13(A, U, V); 259 } 260 } 261 }; 262 263 template <typename MatrixType> 264 struct matrix_exp_computeUV<MatrixType, long double> 265 { 266 template <typename ArgType> 267 static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings) 268 { 269 #if LDBL_MANT_DIG == 53 // double precision 270 matrix_exp_computeUV<MatrixType, double>::run(arg, U, V, squarings); 271 272 #else 273 274 using std::frexp; 275 using std::pow; 276 const long double l1norm = arg.cwiseAbs().colwise().sum().maxCoeff(); 277 squarings = 0; 278 279 #if LDBL_MANT_DIG <= 64 // extended precision 280 281 if (l1norm < 4.1968497232266989671e-003L) { 282 matrix_exp_pade3(arg, U, V); 283 } else if (l1norm < 1.1848116734693823091e-001L) { 284 matrix_exp_pade5(arg, U, V); 285 } else if (l1norm < 5.5170388480686700274e-001L) { 286 matrix_exp_pade7(arg, U, V); 287 } else if (l1norm < 1.3759868875587845383e+000L) { 288 matrix_exp_pade9(arg, U, V); 289 } else { 290 const long double maxnorm = 4.0246098906697353063L; 291 frexp(l1norm / maxnorm, &squarings); 292 if (squarings < 0) squarings = 0; 293 MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings)); 294 matrix_exp_pade13(A, U, V); 295 } 296 297 #elif LDBL_MANT_DIG <= 106 // double-double 298 299 if (l1norm < 3.2787892205607026992947488108213e-005L) { 300 matrix_exp_pade3(arg, U, V); 301 } else if (l1norm < 6.4467025060072760084130906076332e-003L) { 302 matrix_exp_pade5(arg, U, V); 303 } else if (l1norm < 6.8988028496595374751374122881143e-002L) { 304 matrix_exp_pade7(arg, U, V); 305 } else if (l1norm < 2.7339737518502231741495857201670e-001L) { 306 matrix_exp_pade9(arg, U, V); 307 } else if (l1norm < 1.3203382096514474905666448850278e+000L) { 308 matrix_exp_pade13(arg, U, V); 309 } else { 310 const long double maxnorm = 3.2579440895405400856599663723517L; 311 frexp(l1norm / maxnorm, &squarings); 312 if (squarings < 0) squarings = 0; 313 MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings)); 314 matrix_exp_pade17(A, U, V); 315 } 316 317 #elif LDBL_MANT_DIG <= 113 // quadruple precision 318 319 if (l1norm < 1.639394610288918690547467954466970e-005L) { 320 matrix_exp_pade3(arg, U, V); 321 } else if (l1norm < 4.253237712165275566025884344433009e-003L) { 322 matrix_exp_pade5(arg, U, V); 323 } else if (l1norm < 5.125804063165764409885122032933142e-002L) { 324 matrix_exp_pade7(arg, U, V); 325 } else if (l1norm < 2.170000765161155195453205651889853e-001L) { 326 matrix_exp_pade9(arg, U, V); 327 } else if (l1norm < 1.125358383453143065081397882891878e+000L) { 328 matrix_exp_pade13(arg, U, V); 329 } else { 330 const long double maxnorm = 2.884233277829519311757165057717815L; 331 frexp(l1norm / maxnorm, &squarings); 332 if (squarings < 0) squarings = 0; 333 MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings)); 334 matrix_exp_pade17(A, U, V); 335 } 336 337 #else 338 339 // this case should be handled in compute() 340 eigen_assert(false && "Bug in MatrixExponential"); 341 342 #endif 343 #endif // LDBL_MANT_DIG 344 } 345 }; 346 347 template<typename T> struct is_exp_known_type : false_type {}; 348 template<> struct is_exp_known_type<float> : true_type {}; 349 template<> struct is_exp_known_type<double> : true_type {}; 350 #if LDBL_MANT_DIG <= 113 351 template<> struct is_exp_known_type<long double> : true_type {}; 352 #endif 353 354 template <typename ArgType, typename ResultType> 355 void matrix_exp_compute(const ArgType& arg, ResultType &result, true_type) // natively supported scalar type 356 { 357 typedef typename ArgType::PlainObject MatrixType; 358 MatrixType U, V; 359 int squarings; 360 matrix_exp_computeUV<MatrixType>::run(arg, U, V, squarings); // Pade approximant is (U+V) / (-U+V) 361 MatrixType numer = U + V; 362 MatrixType denom = -U + V; 363 result = denom.partialPivLu().solve(numer); 364 for (int i=0; i<squarings; i++) 365 result *= result; // undo scaling by repeated squaring 366 } 367 368 369 /* Computes the matrix exponential 370 * 371 * \param arg argument of matrix exponential (should be plain object) 372 * \param result variable in which result will be stored 373 */ 374 template <typename ArgType, typename ResultType> 375 void matrix_exp_compute(const ArgType& arg, ResultType &result, false_type) // default 376 { 377 typedef typename ArgType::PlainObject MatrixType; 378 typedef typename traits<MatrixType>::Scalar Scalar; 379 typedef typename NumTraits<Scalar>::Real RealScalar; 380 typedef typename std::complex<RealScalar> ComplexScalar; 381 result = arg.matrixFunction(internal::stem_function_exp<ComplexScalar>); 382 } 383 384 } // end namespace Eigen::internal 385 386 /** \ingroup MatrixFunctions_Module 387 * 388 * \brief Proxy for the matrix exponential of some matrix (expression). 389 * 390 * \tparam Derived Type of the argument to the matrix exponential. 391 * 392 * This class holds the argument to the matrix exponential until it is assigned or evaluated for 393 * some other reason (so the argument should not be changed in the meantime). It is the return type 394 * of MatrixBase::exp() and most of the time this is the only way it is used. 395 */ 396 template<typename Derived> struct MatrixExponentialReturnValue 397 : public ReturnByValue<MatrixExponentialReturnValue<Derived> > 398 { 399 public: 400 /** \brief Constructor. 401 * 402 * \param src %Matrix (expression) forming the argument of the matrix exponential. 403 */ 404 MatrixExponentialReturnValue(const Derived& src) : m_src(src) { } 405 406 /** \brief Compute the matrix exponential. 407 * 408 * \param result the matrix exponential of \p src in the constructor. 409 */ 410 template <typename ResultType> 411 inline void evalTo(ResultType& result) const 412 { 413 const typename internal::nested_eval<Derived, 10>::type tmp(m_src); 414 internal::matrix_exp_compute(tmp, result, internal::is_exp_known_type<typename Derived::RealScalar>()); 415 } 416 417 Index rows() const { return m_src.rows(); } 418 Index cols() const { return m_src.cols(); } 419 420 protected: 421 const typename internal::ref_selector<Derived>::type m_src; 422 }; 423 424 namespace internal { 425 template<typename Derived> 426 struct traits<MatrixExponentialReturnValue<Derived> > 427 { 428 typedef typename Derived::PlainObject ReturnType; 429 }; 430 } 431 432 template <typename Derived> 433 const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp() const 434 { 435 eigen_assert(rows() == cols()); 436 return MatrixExponentialReturnValue<Derived>(derived()); 437 } 438 439 } // end namespace Eigen 440 441 #endif // EIGEN_MATRIX_EXPONENTIAL