1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // Copyright (C) 2012, 2013 Chen-Pang He <jdh8 (at) ms63.hinet.net> 5 // 6 // This Source Code Form is subject to the terms of the Mozilla 7 // Public License v. 2.0. If a copy of the MPL was not distributed 8 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 9 10 #ifndef EIGEN_MATRIX_POWER 11 #define EIGEN_MATRIX_POWER 12 13 namespace Eigen { 14 15 template<typename MatrixType> class MatrixPower; 16 17 template<typename MatrixType> 18 class MatrixPowerRetval : public ReturnByValue< MatrixPowerRetval<MatrixType> > 19 { 20 public: 21 typedef typename MatrixType::RealScalar RealScalar; 22 typedef typename MatrixType::Index Index; 23 24 MatrixPowerRetval(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p) 25 { } 26 27 template<typename ResultType> 28 inline void evalTo(ResultType& res) const 29 { m_pow.compute(res, m_p); } 30 31 Index rows() const { return m_pow.rows(); } 32 Index cols() const { return m_pow.cols(); } 33 34 private: 35 MatrixPower<MatrixType>& m_pow; 36 const RealScalar m_p; 37 MatrixPowerRetval& operator=(const MatrixPowerRetval&); 38 }; 39 40 template<typename MatrixType> 41 class MatrixPowerAtomic 42 { 43 private: 44 enum { 45 RowsAtCompileTime = MatrixType::RowsAtCompileTime, 46 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime 47 }; 48 typedef typename MatrixType::Scalar Scalar; 49 typedef typename MatrixType::RealScalar RealScalar; 50 typedef std::complex<RealScalar> ComplexScalar; 51 typedef typename MatrixType::Index Index; 52 typedef Array<Scalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> ArrayType; 53 54 const MatrixType& m_A; 55 RealScalar m_p; 56 57 void computePade(int degree, const MatrixType& IminusT, MatrixType& res) const; 58 void compute2x2(MatrixType& res, RealScalar p) const; 59 void computeBig(MatrixType& res) const; 60 static int getPadeDegree(float normIminusT); 61 static int getPadeDegree(double normIminusT); 62 static int getPadeDegree(long double normIminusT); 63 static ComplexScalar computeSuperDiag(const ComplexScalar&, const ComplexScalar&, RealScalar p); 64 static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p); 65 66 public: 67 MatrixPowerAtomic(const MatrixType& T, RealScalar p); 68 void compute(MatrixType& res) const; 69 }; 70 71 template<typename MatrixType> 72 MatrixPowerAtomic<MatrixType>::MatrixPowerAtomic(const MatrixType& T, RealScalar p) : 73 m_A(T), m_p(p) 74 { eigen_assert(T.rows() == T.cols()); } 75 76 template<typename MatrixType> 77 void MatrixPowerAtomic<MatrixType>::compute(MatrixType& res) const 78 { 79 res.resizeLike(m_A); 80 switch (m_A.rows()) { 81 case 0: 82 break; 83 case 1: 84 res(0,0) = std::pow(m_A(0,0), m_p); 85 break; 86 case 2: 87 compute2x2(res, m_p); 88 break; 89 default: 90 computeBig(res); 91 } 92 } 93 94 template<typename MatrixType> 95 void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, MatrixType& res) const 96 { 97 int i = degree<<1; 98 res = (m_p-degree) / ((i-1)<<1) * IminusT; 99 for (--i; i; --i) { 100 res = (MatrixType::Identity(IminusT.rows(), IminusT.cols()) + res).template triangularView<Upper>() 101 .solve((i==1 ? -m_p : i&1 ? (-m_p-(i>>1))/(i<<1) : (m_p-(i>>1))/((i-1)<<1)) * IminusT).eval(); 102 } 103 res += MatrixType::Identity(IminusT.rows(), IminusT.cols()); 104 } 105 106 // This function assumes that res has the correct size (see bug 614) 107 template<typename MatrixType> 108 void MatrixPowerAtomic<MatrixType>::compute2x2(MatrixType& res, RealScalar p) const 109 { 110 using std::abs; 111 using std::pow; 112 113 res.coeffRef(0,0) = pow(m_A.coeff(0,0), p); 114 115 for (Index i=1; i < m_A.cols(); ++i) { 116 res.coeffRef(i,i) = pow(m_A.coeff(i,i), p); 117 if (m_A.coeff(i-1,i-1) == m_A.coeff(i,i)) 118 res.coeffRef(i-1,i) = p * pow(m_A.coeff(i,i), p-1); 119 else if (2*abs(m_A.coeff(i-1,i-1)) < abs(m_A.coeff(i,i)) || 2*abs(m_A.coeff(i,i)) < abs(m_A.coeff(i-1,i-1))) 120 res.coeffRef(i-1,i) = (res.coeff(i,i)-res.coeff(i-1,i-1)) / (m_A.coeff(i,i)-m_A.coeff(i-1,i-1)); 121 else 122 res.coeffRef(i-1,i) = computeSuperDiag(m_A.coeff(i,i), m_A.coeff(i-1,i-1), p); 123 res.coeffRef(i-1,i) *= m_A.coeff(i-1,i); 124 } 125 } 126 127 template<typename MatrixType> 128 void MatrixPowerAtomic<MatrixType>::computeBig(MatrixType& res) const 129 { 130 const int digits = std::numeric_limits<RealScalar>::digits; 131 const RealScalar maxNormForPade = digits <= 24? 4.3386528e-1f: // sigle precision 132 digits <= 53? 2.789358995219730e-1: // double precision 133 digits <= 64? 2.4471944416607995472e-1L: // extended precision 134 digits <= 106? 1.1016843812851143391275867258512e-1L: // double-double 135 9.134603732914548552537150753385375e-2L; // quadruple precision 136 MatrixType IminusT, sqrtT, T = m_A.template triangularView<Upper>(); 137 RealScalar normIminusT; 138 int degree, degree2, numberOfSquareRoots = 0; 139 bool hasExtraSquareRoot = false; 140 141 /* FIXME 142 * For singular T, norm(I - T) >= 1 but maxNormForPade < 1, leads to infinite 143 * loop. We should move 0 eigenvalues to bottom right corner. We need not 144 * worry about tiny values (e.g. 1e-300) because they will reach 1 if 145 * repetitively sqrt'ed. 146 * 147 * If the 0 eigenvalues are semisimple, they can form a 0 matrix at the 148 * bottom right corner. 149 * 150 * [ T A ]^p [ T^p (T^-1 T^p A) ] 151 * [ ] = [ ] 152 * [ 0 0 ] [ 0 0 ] 153 */ 154 for (Index i=0; i < m_A.cols(); ++i) 155 eigen_assert(m_A(i,i) != RealScalar(0)); 156 157 while (true) { 158 IminusT = MatrixType::Identity(m_A.rows(), m_A.cols()) - T; 159 normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff(); 160 if (normIminusT < maxNormForPade) { 161 degree = getPadeDegree(normIminusT); 162 degree2 = getPadeDegree(normIminusT/2); 163 if (degree - degree2 <= 1 || hasExtraSquareRoot) 164 break; 165 hasExtraSquareRoot = true; 166 } 167 MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT); 168 T = sqrtT.template triangularView<Upper>(); 169 ++numberOfSquareRoots; 170 } 171 computePade(degree, IminusT, res); 172 173 for (; numberOfSquareRoots; --numberOfSquareRoots) { 174 compute2x2(res, std::ldexp(m_p, -numberOfSquareRoots)); 175 res = res.template triangularView<Upper>() * res; 176 } 177 compute2x2(res, m_p); 178 } 179 180 template<typename MatrixType> 181 inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(float normIminusT) 182 { 183 const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f }; 184 int degree = 3; 185 for (; degree <= 4; ++degree) 186 if (normIminusT <= maxNormForPade[degree - 3]) 187 break; 188 return degree; 189 } 190 191 template<typename MatrixType> 192 inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(double normIminusT) 193 { 194 const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1, 195 1.999045567181744e-1, 2.789358995219730e-1 }; 196 int degree = 3; 197 for (; degree <= 7; ++degree) 198 if (normIminusT <= maxNormForPade[degree - 3]) 199 break; 200 return degree; 201 } 202 203 template<typename MatrixType> 204 inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(long double normIminusT) 205 { 206 #if LDBL_MANT_DIG == 53 207 const int maxPadeDegree = 7; 208 const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L, 209 1.999045567181744e-1L, 2.789358995219730e-1L }; 210 #elif LDBL_MANT_DIG <= 64 211 const int maxPadeDegree = 8; 212 const double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L, 213 6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L }; 214 #elif LDBL_MANT_DIG <= 106 215 const int maxPadeDegree = 10; 216 const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ , 217 1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L, 218 2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L, 219 1.1016843812851143391275867258512e-1L }; 220 #else 221 const int maxPadeDegree = 10; 222 const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ , 223 6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L, 224 9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L, 225 3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L, 226 9.134603732914548552537150753385375e-2L }; 227 #endif 228 int degree = 3; 229 for (; degree <= maxPadeDegree; ++degree) 230 if (normIminusT <= maxNormForPade[degree - 3]) 231 break; 232 return degree; 233 } 234 235 template<typename MatrixType> 236 inline typename MatrixPowerAtomic<MatrixType>::ComplexScalar 237 MatrixPowerAtomic<MatrixType>::computeSuperDiag(const ComplexScalar& curr, const ComplexScalar& prev, RealScalar p) 238 { 239 ComplexScalar logCurr = std::log(curr); 240 ComplexScalar logPrev = std::log(prev); 241 int unwindingNumber = std::ceil((numext::imag(logCurr - logPrev) - M_PI) / (2*M_PI)); 242 ComplexScalar w = numext::atanh2(curr - prev, curr + prev) + ComplexScalar(0, M_PI*unwindingNumber); 243 return RealScalar(2) * std::exp(RealScalar(0.5) * p * (logCurr + logPrev)) * std::sinh(p * w) / (curr - prev); 244 } 245 246 template<typename MatrixType> 247 inline typename MatrixPowerAtomic<MatrixType>::RealScalar 248 MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p) 249 { 250 RealScalar w = numext::atanh2(curr - prev, curr + prev); 251 return 2 * std::exp(p * (std::log(curr) + std::log(prev)) / 2) * std::sinh(p * w) / (curr - prev); 252 } 253 254 /** 255 * \ingroup MatrixFunctions_Module 256 * 257 * \brief Class for computing matrix powers. 258 * 259 * \tparam MatrixType type of the base, expected to be an instantiation 260 * of the Matrix class template. 261 * 262 * This class is capable of computing real/complex matrices raised to 263 * an arbitrary real power. Meanwhile, it saves the result of Schur 264 * decomposition if an non-integral power has even been calculated. 265 * Therefore, if you want to compute multiple (>= 2) matrix powers 266 * for the same matrix, using the class directly is more efficient than 267 * calling MatrixBase::pow(). 268 * 269 * Example: 270 * \include MatrixPower_optimal.cpp 271 * Output: \verbinclude MatrixPower_optimal.out 272 */ 273 template<typename MatrixType> 274 class MatrixPower 275 { 276 private: 277 enum { 278 RowsAtCompileTime = MatrixType::RowsAtCompileTime, 279 ColsAtCompileTime = MatrixType::ColsAtCompileTime, 280 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, 281 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime 282 }; 283 typedef typename MatrixType::Scalar Scalar; 284 typedef typename MatrixType::RealScalar RealScalar; 285 typedef typename MatrixType::Index Index; 286 287 public: 288 /** 289 * \brief Constructor. 290 * 291 * \param[in] A the base of the matrix power. 292 * 293 * The class stores a reference to A, so it should not be changed 294 * (or destroyed) before evaluation. 295 */ 296 explicit MatrixPower(const MatrixType& A) : m_A(A), m_conditionNumber(0) 297 { eigen_assert(A.rows() == A.cols()); } 298 299 /** 300 * \brief Returns the matrix power. 301 * 302 * \param[in] p exponent, a real scalar. 303 * \return The expression \f$ A^p \f$, where A is specified in the 304 * constructor. 305 */ 306 const MatrixPowerRetval<MatrixType> operator()(RealScalar p) 307 { return MatrixPowerRetval<MatrixType>(*this, p); } 308 309 /** 310 * \brief Compute the matrix power. 311 * 312 * \param[in] p exponent, a real scalar. 313 * \param[out] res \f$ A^p \f$ where A is specified in the 314 * constructor. 315 */ 316 template<typename ResultType> 317 void compute(ResultType& res, RealScalar p); 318 319 Index rows() const { return m_A.rows(); } 320 Index cols() const { return m_A.cols(); } 321 322 private: 323 typedef std::complex<RealScalar> ComplexScalar; 324 typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, MatrixType::Options, 325 MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrix; 326 327 typename MatrixType::Nested m_A; 328 MatrixType m_tmp; 329 ComplexMatrix m_T, m_U, m_fT; 330 RealScalar m_conditionNumber; 331 332 RealScalar modfAndInit(RealScalar, RealScalar*); 333 334 template<typename ResultType> 335 void computeIntPower(ResultType&, RealScalar); 336 337 template<typename ResultType> 338 void computeFracPower(ResultType&, RealScalar); 339 340 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 341 static void revertSchur( 342 Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 343 const ComplexMatrix& T, 344 const ComplexMatrix& U); 345 346 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 347 static void revertSchur( 348 Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 349 const ComplexMatrix& T, 350 const ComplexMatrix& U); 351 }; 352 353 template<typename MatrixType> 354 template<typename ResultType> 355 void MatrixPower<MatrixType>::compute(ResultType& res, RealScalar p) 356 { 357 switch (cols()) { 358 case 0: 359 break; 360 case 1: 361 res(0,0) = std::pow(m_A.coeff(0,0), p); 362 break; 363 default: 364 RealScalar intpart, x = modfAndInit(p, &intpart); 365 computeIntPower(res, intpart); 366 computeFracPower(res, x); 367 } 368 } 369 370 template<typename MatrixType> 371 typename MatrixPower<MatrixType>::RealScalar 372 MatrixPower<MatrixType>::modfAndInit(RealScalar x, RealScalar* intpart) 373 { 374 typedef Array<RealScalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> RealArray; 375 376 *intpart = std::floor(x); 377 RealScalar res = x - *intpart; 378 379 if (!m_conditionNumber && res) { 380 const ComplexSchur<MatrixType> schurOfA(m_A); 381 m_T = schurOfA.matrixT(); 382 m_U = schurOfA.matrixU(); 383 384 const RealArray absTdiag = m_T.diagonal().array().abs(); 385 m_conditionNumber = absTdiag.maxCoeff() / absTdiag.minCoeff(); 386 } 387 388 if (res>RealScalar(0.5) && res>(1-res)*std::pow(m_conditionNumber, res)) { 389 --res; 390 ++*intpart; 391 } 392 return res; 393 } 394 395 template<typename MatrixType> 396 template<typename ResultType> 397 void MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p) 398 { 399 RealScalar pp = std::abs(p); 400 401 if (p<0) m_tmp = m_A.inverse(); 402 else m_tmp = m_A; 403 404 res = MatrixType::Identity(rows(), cols()); 405 while (pp >= 1) { 406 if (std::fmod(pp, 2) >= 1) 407 res = m_tmp * res; 408 m_tmp *= m_tmp; 409 pp /= 2; 410 } 411 } 412 413 template<typename MatrixType> 414 template<typename ResultType> 415 void MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p) 416 { 417 if (p) { 418 eigen_assert(m_conditionNumber); 419 MatrixPowerAtomic<ComplexMatrix>(m_T, p).compute(m_fT); 420 revertSchur(m_tmp, m_fT, m_U); 421 res = m_tmp * res; 422 } 423 } 424 425 template<typename MatrixType> 426 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 427 inline void MatrixPower<MatrixType>::revertSchur( 428 Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 429 const ComplexMatrix& T, 430 const ComplexMatrix& U) 431 { res.noalias() = U * (T.template triangularView<Upper>() * U.adjoint()); } 432 433 template<typename MatrixType> 434 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 435 inline void MatrixPower<MatrixType>::revertSchur( 436 Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 437 const ComplexMatrix& T, 438 const ComplexMatrix& U) 439 { res.noalias() = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); } 440 441 /** 442 * \ingroup MatrixFunctions_Module 443 * 444 * \brief Proxy for the matrix power of some matrix (expression). 445 * 446 * \tparam Derived type of the base, a matrix (expression). 447 * 448 * This class holds the arguments to the matrix power until it is 449 * assigned or evaluated for some other reason (so the argument 450 * should not be changed in the meantime). It is the return type of 451 * MatrixBase::pow() and related functions and most of the 452 * time this is the only way it is used. 453 */ 454 template<typename Derived> 455 class MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Derived> > 456 { 457 public: 458 typedef typename Derived::PlainObject PlainObject; 459 typedef typename Derived::RealScalar RealScalar; 460 typedef typename Derived::Index Index; 461 462 /** 463 * \brief Constructor. 464 * 465 * \param[in] A %Matrix (expression), the base of the matrix power. 466 * \param[in] p scalar, the exponent of the matrix power. 467 */ 468 MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p) 469 { } 470 471 /** 472 * \brief Compute the matrix power. 473 * 474 * \param[out] result \f$ A^p \f$ where \p A and \p p are as in the 475 * constructor. 476 */ 477 template<typename ResultType> 478 inline void evalTo(ResultType& res) const 479 { MatrixPower<PlainObject>(m_A.eval()).compute(res, m_p); } 480 481 Index rows() const { return m_A.rows(); } 482 Index cols() const { return m_A.cols(); } 483 484 private: 485 const Derived& m_A; 486 const RealScalar m_p; 487 MatrixPowerReturnValue& operator=(const MatrixPowerReturnValue&); 488 }; 489 490 namespace internal { 491 492 template<typename MatrixPowerType> 493 struct traits< MatrixPowerRetval<MatrixPowerType> > 494 { typedef typename MatrixPowerType::PlainObject ReturnType; }; 495 496 template<typename Derived> 497 struct traits< MatrixPowerReturnValue<Derived> > 498 { typedef typename Derived::PlainObject ReturnType; }; 499 500 } 501 502 template<typename Derived> 503 const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const 504 { return MatrixPowerReturnValue<Derived>(derived(), p); } 505 506 } // namespace Eigen 507 508 #endif // EIGEN_MATRIX_POWER 509
