MatrixPower.h (23422B)
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@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 /** 18 * \ingroup MatrixFunctions_Module 19 * 20 * \brief Proxy for the matrix power of some matrix. 21 * 22 * \tparam MatrixType type of the base, a matrix. 23 * 24 * This class holds the arguments to the matrix power until it is 25 * assigned or evaluated for some other reason (so the argument 26 * should not be changed in the meantime). It is the return type of 27 * MatrixPower::operator() and related functions and most of the 28 * time this is the only way it is used. 29 */ 30 /* TODO This class is only used by MatrixPower, so it should be nested 31 * into MatrixPower, like MatrixPower::ReturnValue. However, my 32 * compiler complained about unused template parameter in the 33 * following declaration in namespace internal. 34 * 35 * template<typename MatrixType> 36 * struct traits<MatrixPower<MatrixType>::ReturnValue>; 37 */ 38 template<typename MatrixType> 39 class MatrixPowerParenthesesReturnValue : public ReturnByValue< MatrixPowerParenthesesReturnValue<MatrixType> > 40 { 41 public: 42 typedef typename MatrixType::RealScalar RealScalar; 43 44 /** 45 * \brief Constructor. 46 * 47 * \param[in] pow %MatrixPower storing the base. 48 * \param[in] p scalar, the exponent of the matrix power. 49 */ 50 MatrixPowerParenthesesReturnValue(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p) 51 { } 52 53 /** 54 * \brief Compute the matrix power. 55 * 56 * \param[out] result 57 */ 58 template<typename ResultType> 59 inline void evalTo(ResultType& result) const 60 { m_pow.compute(result, m_p); } 61 62 Index rows() const { return m_pow.rows(); } 63 Index cols() const { return m_pow.cols(); } 64 65 private: 66 MatrixPower<MatrixType>& m_pow; 67 const RealScalar m_p; 68 }; 69 70 /** 71 * \ingroup MatrixFunctions_Module 72 * 73 * \brief Class for computing matrix powers. 74 * 75 * \tparam MatrixType type of the base, expected to be an instantiation 76 * of the Matrix class template. 77 * 78 * This class is capable of computing triangular real/complex matrices 79 * raised to a power in the interval \f$ (-1, 1) \f$. 80 * 81 * \note Currently this class is only used by MatrixPower. One may 82 * insist that this be nested into MatrixPower. This class is here to 83 * facilitate future development of triangular matrix functions. 84 */ 85 template<typename MatrixType> 86 class MatrixPowerAtomic : internal::noncopyable 87 { 88 private: 89 enum { 90 RowsAtCompileTime = MatrixType::RowsAtCompileTime, 91 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime 92 }; 93 typedef typename MatrixType::Scalar Scalar; 94 typedef typename MatrixType::RealScalar RealScalar; 95 typedef std::complex<RealScalar> ComplexScalar; 96 typedef Block<MatrixType,Dynamic,Dynamic> ResultType; 97 98 const MatrixType& m_A; 99 RealScalar m_p; 100 101 void computePade(int degree, const MatrixType& IminusT, ResultType& res) const; 102 void compute2x2(ResultType& res, RealScalar p) const; 103 void computeBig(ResultType& res) const; 104 static int getPadeDegree(float normIminusT); 105 static int getPadeDegree(double normIminusT); 106 static int getPadeDegree(long double normIminusT); 107 static ComplexScalar computeSuperDiag(const ComplexScalar&, const ComplexScalar&, RealScalar p); 108 static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p); 109 110 public: 111 /** 112 * \brief Constructor. 113 * 114 * \param[in] T the base of the matrix power. 115 * \param[in] p the exponent of the matrix power, should be in 116 * \f$ (-1, 1) \f$. 117 * 118 * The class stores a reference to T, so it should not be changed 119 * (or destroyed) before evaluation. Only the upper triangular 120 * part of T is read. 121 */ 122 MatrixPowerAtomic(const MatrixType& T, RealScalar p); 123 124 /** 125 * \brief Compute the matrix power. 126 * 127 * \param[out] res \f$ A^p \f$ where A and p are specified in the 128 * constructor. 129 */ 130 void compute(ResultType& res) const; 131 }; 132 133 template<typename MatrixType> 134 MatrixPowerAtomic<MatrixType>::MatrixPowerAtomic(const MatrixType& T, RealScalar p) : 135 m_A(T), m_p(p) 136 { 137 eigen_assert(T.rows() == T.cols()); 138 eigen_assert(p > -1 && p < 1); 139 } 140 141 template<typename MatrixType> 142 void MatrixPowerAtomic<MatrixType>::compute(ResultType& res) const 143 { 144 using std::pow; 145 switch (m_A.rows()) { 146 case 0: 147 break; 148 case 1: 149 res(0,0) = pow(m_A(0,0), m_p); 150 break; 151 case 2: 152 compute2x2(res, m_p); 153 break; 154 default: 155 computeBig(res); 156 } 157 } 158 159 template<typename MatrixType> 160 void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, ResultType& res) const 161 { 162 int i = 2*degree; 163 res = (m_p-RealScalar(degree)) / RealScalar(2*i-2) * IminusT; 164 165 for (--i; i; --i) { 166 res = (MatrixType::Identity(IminusT.rows(), IminusT.cols()) + res).template triangularView<Upper>() 167 .solve((i==1 ? -m_p : i&1 ? (-m_p-RealScalar(i/2))/RealScalar(2*i) : (m_p-RealScalar(i/2))/RealScalar(2*i-2)) * IminusT).eval(); 168 } 169 res += MatrixType::Identity(IminusT.rows(), IminusT.cols()); 170 } 171 172 // This function assumes that res has the correct size (see bug 614) 173 template<typename MatrixType> 174 void MatrixPowerAtomic<MatrixType>::compute2x2(ResultType& res, RealScalar p) const 175 { 176 using std::abs; 177 using std::pow; 178 res.coeffRef(0,0) = pow(m_A.coeff(0,0), p); 179 180 for (Index i=1; i < m_A.cols(); ++i) { 181 res.coeffRef(i,i) = pow(m_A.coeff(i,i), p); 182 if (m_A.coeff(i-1,i-1) == m_A.coeff(i,i)) 183 res.coeffRef(i-1,i) = p * pow(m_A.coeff(i,i), p-1); 184 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))) 185 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)); 186 else 187 res.coeffRef(i-1,i) = computeSuperDiag(m_A.coeff(i,i), m_A.coeff(i-1,i-1), p); 188 res.coeffRef(i-1,i) *= m_A.coeff(i-1,i); 189 } 190 } 191 192 template<typename MatrixType> 193 void MatrixPowerAtomic<MatrixType>::computeBig(ResultType& res) const 194 { 195 using std::ldexp; 196 const int digits = std::numeric_limits<RealScalar>::digits; 197 const RealScalar maxNormForPade = RealScalar( 198 digits <= 24? 4.3386528e-1L // single precision 199 : digits <= 53? 2.789358995219730e-1L // double precision 200 : digits <= 64? 2.4471944416607995472e-1L // extended precision 201 : digits <= 106? 1.1016843812851143391275867258512e-1L // double-double 202 : 9.134603732914548552537150753385375e-2L); // quadruple precision 203 MatrixType IminusT, sqrtT, T = m_A.template triangularView<Upper>(); 204 RealScalar normIminusT; 205 int degree, degree2, numberOfSquareRoots = 0; 206 bool hasExtraSquareRoot = false; 207 208 for (Index i=0; i < m_A.cols(); ++i) 209 eigen_assert(m_A(i,i) != RealScalar(0)); 210 211 while (true) { 212 IminusT = MatrixType::Identity(m_A.rows(), m_A.cols()) - T; 213 normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff(); 214 if (normIminusT < maxNormForPade) { 215 degree = getPadeDegree(normIminusT); 216 degree2 = getPadeDegree(normIminusT/2); 217 if (degree - degree2 <= 1 || hasExtraSquareRoot) 218 break; 219 hasExtraSquareRoot = true; 220 } 221 matrix_sqrt_triangular(T, sqrtT); 222 T = sqrtT.template triangularView<Upper>(); 223 ++numberOfSquareRoots; 224 } 225 computePade(degree, IminusT, res); 226 227 for (; numberOfSquareRoots; --numberOfSquareRoots) { 228 compute2x2(res, ldexp(m_p, -numberOfSquareRoots)); 229 res = res.template triangularView<Upper>() * res; 230 } 231 compute2x2(res, m_p); 232 } 233 234 template<typename MatrixType> 235 inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(float normIminusT) 236 { 237 const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f }; 238 int degree = 3; 239 for (; degree <= 4; ++degree) 240 if (normIminusT <= maxNormForPade[degree - 3]) 241 break; 242 return degree; 243 } 244 245 template<typename MatrixType> 246 inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(double normIminusT) 247 { 248 const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1, 249 1.999045567181744e-1, 2.789358995219730e-1 }; 250 int degree = 3; 251 for (; degree <= 7; ++degree) 252 if (normIminusT <= maxNormForPade[degree - 3]) 253 break; 254 return degree; 255 } 256 257 template<typename MatrixType> 258 inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(long double normIminusT) 259 { 260 #if LDBL_MANT_DIG == 53 261 const int maxPadeDegree = 7; 262 const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L, 263 1.999045567181744e-1L, 2.789358995219730e-1L }; 264 #elif LDBL_MANT_DIG <= 64 265 const int maxPadeDegree = 8; 266 const long double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L, 267 6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L }; 268 #elif LDBL_MANT_DIG <= 106 269 const int maxPadeDegree = 10; 270 const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ , 271 1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L, 272 2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L, 273 1.1016843812851143391275867258512e-1L }; 274 #else 275 const int maxPadeDegree = 10; 276 const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ , 277 6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L, 278 9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L, 279 3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L, 280 9.134603732914548552537150753385375e-2L }; 281 #endif 282 int degree = 3; 283 for (; degree <= maxPadeDegree; ++degree) 284 if (normIminusT <= maxNormForPade[degree - 3]) 285 break; 286 return degree; 287 } 288 289 template<typename MatrixType> 290 inline typename MatrixPowerAtomic<MatrixType>::ComplexScalar 291 MatrixPowerAtomic<MatrixType>::computeSuperDiag(const ComplexScalar& curr, const ComplexScalar& prev, RealScalar p) 292 { 293 using std::ceil; 294 using std::exp; 295 using std::log; 296 using std::sinh; 297 298 ComplexScalar logCurr = log(curr); 299 ComplexScalar logPrev = log(prev); 300 RealScalar unwindingNumber = ceil((numext::imag(logCurr - logPrev) - RealScalar(EIGEN_PI)) / RealScalar(2*EIGEN_PI)); 301 ComplexScalar w = numext::log1p((curr-prev)/prev)/RealScalar(2) + ComplexScalar(0, RealScalar(EIGEN_PI)*unwindingNumber); 302 return RealScalar(2) * exp(RealScalar(0.5) * p * (logCurr + logPrev)) * sinh(p * w) / (curr - prev); 303 } 304 305 template<typename MatrixType> 306 inline typename MatrixPowerAtomic<MatrixType>::RealScalar 307 MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p) 308 { 309 using std::exp; 310 using std::log; 311 using std::sinh; 312 313 RealScalar w = numext::log1p((curr-prev)/prev)/RealScalar(2); 314 return 2 * exp(p * (log(curr) + log(prev)) / 2) * sinh(p * w) / (curr - prev); 315 } 316 317 /** 318 * \ingroup MatrixFunctions_Module 319 * 320 * \brief Class for computing matrix powers. 321 * 322 * \tparam MatrixType type of the base, expected to be an instantiation 323 * of the Matrix class template. 324 * 325 * This class is capable of computing real/complex matrices raised to 326 * an arbitrary real power. Meanwhile, it saves the result of Schur 327 * decomposition if an non-integral power has even been calculated. 328 * Therefore, if you want to compute multiple (>= 2) matrix powers 329 * for the same matrix, using the class directly is more efficient than 330 * calling MatrixBase::pow(). 331 * 332 * Example: 333 * \include MatrixPower_optimal.cpp 334 * Output: \verbinclude MatrixPower_optimal.out 335 */ 336 template<typename MatrixType> 337 class MatrixPower : internal::noncopyable 338 { 339 private: 340 typedef typename MatrixType::Scalar Scalar; 341 typedef typename MatrixType::RealScalar RealScalar; 342 343 public: 344 /** 345 * \brief Constructor. 346 * 347 * \param[in] A the base of the matrix power. 348 * 349 * The class stores a reference to A, so it should not be changed 350 * (or destroyed) before evaluation. 351 */ 352 explicit MatrixPower(const MatrixType& A) : 353 m_A(A), 354 m_conditionNumber(0), 355 m_rank(A.cols()), 356 m_nulls(0) 357 { eigen_assert(A.rows() == A.cols()); } 358 359 /** 360 * \brief Returns the matrix power. 361 * 362 * \param[in] p exponent, a real scalar. 363 * \return The expression \f$ A^p \f$, where A is specified in the 364 * constructor. 365 */ 366 const MatrixPowerParenthesesReturnValue<MatrixType> operator()(RealScalar p) 367 { return MatrixPowerParenthesesReturnValue<MatrixType>(*this, p); } 368 369 /** 370 * \brief Compute the matrix power. 371 * 372 * \param[in] p exponent, a real scalar. 373 * \param[out] res \f$ A^p \f$ where A is specified in the 374 * constructor. 375 */ 376 template<typename ResultType> 377 void compute(ResultType& res, RealScalar p); 378 379 Index rows() const { return m_A.rows(); } 380 Index cols() const { return m_A.cols(); } 381 382 private: 383 typedef std::complex<RealScalar> ComplexScalar; 384 typedef Matrix<ComplexScalar, Dynamic, Dynamic, 0, 385 MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime> ComplexMatrix; 386 387 /** \brief Reference to the base of matrix power. */ 388 typename MatrixType::Nested m_A; 389 390 /** \brief Temporary storage. */ 391 MatrixType m_tmp; 392 393 /** \brief Store the result of Schur decomposition. */ 394 ComplexMatrix m_T, m_U; 395 396 /** \brief Store fractional power of m_T. */ 397 ComplexMatrix m_fT; 398 399 /** 400 * \brief Condition number of m_A. 401 * 402 * It is initialized as 0 to avoid performing unnecessary Schur 403 * decomposition, which is the bottleneck. 404 */ 405 RealScalar m_conditionNumber; 406 407 /** \brief Rank of m_A. */ 408 Index m_rank; 409 410 /** \brief Rank deficiency of m_A. */ 411 Index m_nulls; 412 413 /** 414 * \brief Split p into integral part and fractional part. 415 * 416 * \param[in] p The exponent. 417 * \param[out] p The fractional part ranging in \f$ (-1, 1) \f$. 418 * \param[out] intpart The integral part. 419 * 420 * Only if the fractional part is nonzero, it calls initialize(). 421 */ 422 void split(RealScalar& p, RealScalar& intpart); 423 424 /** \brief Perform Schur decomposition for fractional power. */ 425 void initialize(); 426 427 template<typename ResultType> 428 void computeIntPower(ResultType& res, RealScalar p); 429 430 template<typename ResultType> 431 void computeFracPower(ResultType& res, RealScalar p); 432 433 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 434 static void revertSchur( 435 Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 436 const ComplexMatrix& T, 437 const ComplexMatrix& U); 438 439 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 440 static void revertSchur( 441 Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 442 const ComplexMatrix& T, 443 const ComplexMatrix& U); 444 }; 445 446 template<typename MatrixType> 447 template<typename ResultType> 448 void MatrixPower<MatrixType>::compute(ResultType& res, RealScalar p) 449 { 450 using std::pow; 451 switch (cols()) { 452 case 0: 453 break; 454 case 1: 455 res(0,0) = pow(m_A.coeff(0,0), p); 456 break; 457 default: 458 RealScalar intpart; 459 split(p, intpart); 460 461 res = MatrixType::Identity(rows(), cols()); 462 computeIntPower(res, intpart); 463 if (p) computeFracPower(res, p); 464 } 465 } 466 467 template<typename MatrixType> 468 void MatrixPower<MatrixType>::split(RealScalar& p, RealScalar& intpart) 469 { 470 using std::floor; 471 using std::pow; 472 473 intpart = floor(p); 474 p -= intpart; 475 476 // Perform Schur decomposition if it is not yet performed and the power is 477 // not an integer. 478 if (!m_conditionNumber && p) 479 initialize(); 480 481 // Choose the more stable of intpart = floor(p) and intpart = ceil(p). 482 if (p > RealScalar(0.5) && p > (1-p) * pow(m_conditionNumber, p)) { 483 --p; 484 ++intpart; 485 } 486 } 487 488 template<typename MatrixType> 489 void MatrixPower<MatrixType>::initialize() 490 { 491 const ComplexSchur<MatrixType> schurOfA(m_A); 492 JacobiRotation<ComplexScalar> rot; 493 ComplexScalar eigenvalue; 494 495 m_fT.resizeLike(m_A); 496 m_T = schurOfA.matrixT(); 497 m_U = schurOfA.matrixU(); 498 m_conditionNumber = m_T.diagonal().array().abs().maxCoeff() / m_T.diagonal().array().abs().minCoeff(); 499 500 // Move zero eigenvalues to the bottom right corner. 501 for (Index i = cols()-1; i>=0; --i) { 502 if (m_rank <= 2) 503 return; 504 if (m_T.coeff(i,i) == RealScalar(0)) { 505 for (Index j=i+1; j < m_rank; ++j) { 506 eigenvalue = m_T.coeff(j,j); 507 rot.makeGivens(m_T.coeff(j-1,j), eigenvalue); 508 m_T.applyOnTheRight(j-1, j, rot); 509 m_T.applyOnTheLeft(j-1, j, rot.adjoint()); 510 m_T.coeffRef(j-1,j-1) = eigenvalue; 511 m_T.coeffRef(j,j) = RealScalar(0); 512 m_U.applyOnTheRight(j-1, j, rot); 513 } 514 --m_rank; 515 } 516 } 517 518 m_nulls = rows() - m_rank; 519 if (m_nulls) { 520 eigen_assert(m_T.bottomRightCorner(m_nulls, m_nulls).isZero() 521 && "Base of matrix power should be invertible or with a semisimple zero eigenvalue."); 522 m_fT.bottomRows(m_nulls).fill(RealScalar(0)); 523 } 524 } 525 526 template<typename MatrixType> 527 template<typename ResultType> 528 void MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p) 529 { 530 using std::abs; 531 using std::fmod; 532 RealScalar pp = abs(p); 533 534 if (p<0) 535 m_tmp = m_A.inverse(); 536 else 537 m_tmp = m_A; 538 539 while (true) { 540 if (fmod(pp, 2) >= 1) 541 res = m_tmp * res; 542 pp /= 2; 543 if (pp < 1) 544 break; 545 m_tmp *= m_tmp; 546 } 547 } 548 549 template<typename MatrixType> 550 template<typename ResultType> 551 void MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p) 552 { 553 Block<ComplexMatrix,Dynamic,Dynamic> blockTp(m_fT, 0, 0, m_rank, m_rank); 554 eigen_assert(m_conditionNumber); 555 eigen_assert(m_rank + m_nulls == rows()); 556 557 MatrixPowerAtomic<ComplexMatrix>(m_T.topLeftCorner(m_rank, m_rank), p).compute(blockTp); 558 if (m_nulls) { 559 m_fT.topRightCorner(m_rank, m_nulls) = m_T.topLeftCorner(m_rank, m_rank).template triangularView<Upper>() 560 .solve(blockTp * m_T.topRightCorner(m_rank, m_nulls)); 561 } 562 revertSchur(m_tmp, m_fT, m_U); 563 res = m_tmp * res; 564 } 565 566 template<typename MatrixType> 567 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 568 inline void MatrixPower<MatrixType>::revertSchur( 569 Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 570 const ComplexMatrix& T, 571 const ComplexMatrix& U) 572 { res.noalias() = U * (T.template triangularView<Upper>() * U.adjoint()); } 573 574 template<typename MatrixType> 575 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 576 inline void MatrixPower<MatrixType>::revertSchur( 577 Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 578 const ComplexMatrix& T, 579 const ComplexMatrix& U) 580 { res.noalias() = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); } 581 582 /** 583 * \ingroup MatrixFunctions_Module 584 * 585 * \brief Proxy for the matrix power of some matrix (expression). 586 * 587 * \tparam Derived type of the base, a matrix (expression). 588 * 589 * This class holds the arguments to the matrix power until it is 590 * assigned or evaluated for some other reason (so the argument 591 * should not be changed in the meantime). It is the return type of 592 * MatrixBase::pow() and related functions and most of the 593 * time this is the only way it is used. 594 */ 595 template<typename Derived> 596 class MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Derived> > 597 { 598 public: 599 typedef typename Derived::PlainObject PlainObject; 600 typedef typename Derived::RealScalar RealScalar; 601 602 /** 603 * \brief Constructor. 604 * 605 * \param[in] A %Matrix (expression), the base of the matrix power. 606 * \param[in] p real scalar, the exponent of the matrix power. 607 */ 608 MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p) 609 { } 610 611 /** 612 * \brief Compute the matrix power. 613 * 614 * \param[out] result \f$ A^p \f$ where \p A and \p p are as in the 615 * constructor. 616 */ 617 template<typename ResultType> 618 inline void evalTo(ResultType& result) const 619 { MatrixPower<PlainObject>(m_A.eval()).compute(result, m_p); } 620 621 Index rows() const { return m_A.rows(); } 622 Index cols() const { return m_A.cols(); } 623 624 private: 625 const Derived& m_A; 626 const RealScalar m_p; 627 }; 628 629 /** 630 * \ingroup MatrixFunctions_Module 631 * 632 * \brief Proxy for the matrix power of some matrix (expression). 633 * 634 * \tparam Derived type of the base, a matrix (expression). 635 * 636 * This class holds the arguments to the matrix power until it is 637 * assigned or evaluated for some other reason (so the argument 638 * should not be changed in the meantime). It is the return type of 639 * MatrixBase::pow() and related functions and most of the 640 * time this is the only way it is used. 641 */ 642 template<typename Derived> 643 class MatrixComplexPowerReturnValue : public ReturnByValue< MatrixComplexPowerReturnValue<Derived> > 644 { 645 public: 646 typedef typename Derived::PlainObject PlainObject; 647 typedef typename std::complex<typename Derived::RealScalar> ComplexScalar; 648 649 /** 650 * \brief Constructor. 651 * 652 * \param[in] A %Matrix (expression), the base of the matrix power. 653 * \param[in] p complex scalar, the exponent of the matrix power. 654 */ 655 MatrixComplexPowerReturnValue(const Derived& A, const ComplexScalar& p) : m_A(A), m_p(p) 656 { } 657 658 /** 659 * \brief Compute the matrix power. 660 * 661 * Because \p p is complex, \f$ A^p \f$ is simply evaluated as \f$ 662 * \exp(p \log(A)) \f$. 663 * 664 * \param[out] result \f$ A^p \f$ where \p A and \p p are as in the 665 * constructor. 666 */ 667 template<typename ResultType> 668 inline void evalTo(ResultType& result) const 669 { result = (m_p * m_A.log()).exp(); } 670 671 Index rows() const { return m_A.rows(); } 672 Index cols() const { return m_A.cols(); } 673 674 private: 675 const Derived& m_A; 676 const ComplexScalar m_p; 677 }; 678 679 namespace internal { 680 681 template<typename MatrixPowerType> 682 struct traits< MatrixPowerParenthesesReturnValue<MatrixPowerType> > 683 { typedef typename MatrixPowerType::PlainObject ReturnType; }; 684 685 template<typename Derived> 686 struct traits< MatrixPowerReturnValue<Derived> > 687 { typedef typename Derived::PlainObject ReturnType; }; 688 689 template<typename Derived> 690 struct traits< MatrixComplexPowerReturnValue<Derived> > 691 { typedef typename Derived::PlainObject ReturnType; }; 692 693 } 694 695 template<typename Derived> 696 const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const 697 { return MatrixPowerReturnValue<Derived>(derived(), p); } 698 699 template<typename Derived> 700 const MatrixComplexPowerReturnValue<Derived> MatrixBase<Derived>::pow(const std::complex<RealScalar>& p) const 701 { return MatrixComplexPowerReturnValue<Derived>(derived(), p); } 702 703 } // namespace Eigen 704 705 #endif // EIGEN_MATRIX_POWER