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00001 // This file is part of Eigen, a lightweight C++ template library 00002 // for linear algebra. 00003 // 00004 // Copyright (C) 2012, 2013 Chen-Pang He <jdh8@ms63.hinet.net> 00005 // 00006 // This Source Code Form is subject to the terms of the Mozilla 00007 // Public License v. 2.0. If a copy of the MPL was not distributed 00008 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 00009 00010 #ifndef EIGEN_MATRIX_POWER 00011 #define EIGEN_MATRIX_POWER 00012 00013 namespace Eigen { 00014 00015 template<typename MatrixType> class MatrixPower; 00016 00017 template<typename MatrixType> 00018 class MatrixPowerRetval : public ReturnByValue< MatrixPowerRetval<MatrixType> > 00019 { 00020 public: 00021 typedef typename MatrixType::RealScalar RealScalar; 00022 typedef typename MatrixType::Index Index; 00023 00024 MatrixPowerRetval(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p) 00025 { } 00026 00027 template<typename ResultType> 00028 inline void evalTo(ResultType& res) const 00029 { m_pow.compute(res, m_p); } 00030 00031 Index rows() const { return m_pow.rows(); } 00032 Index cols() const { return m_pow.cols(); } 00033 00034 private: 00035 MatrixPower<MatrixType>& m_pow; 00036 const RealScalar m_p; 00037 MatrixPowerRetval& operator=(const MatrixPowerRetval&); 00038 }; 00039 00040 template<typename MatrixType> 00041 class MatrixPowerAtomic 00042 { 00043 private: 00044 enum { 00045 RowsAtCompileTime = MatrixType::RowsAtCompileTime, 00046 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime 00047 }; 00048 typedef typename MatrixType::Scalar Scalar; 00049 typedef typename MatrixType::RealScalar RealScalar; 00050 typedef std::complex<RealScalar> ComplexScalar; 00051 typedef typename MatrixType::Index Index; 00052 typedef Array<Scalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> ArrayType; 00053 00054 const MatrixType& m_A; 00055 RealScalar m_p; 00056 00057 void computePade(int degree, const MatrixType& IminusT, MatrixType& res) const; 00058 void compute2x2(MatrixType& res, RealScalar p) const; 00059 void computeBig(MatrixType& res) const; 00060 static int getPadeDegree(float normIminusT); 00061 static int getPadeDegree(double normIminusT); 00062 static int getPadeDegree(long double normIminusT); 00063 static ComplexScalar computeSuperDiag(const ComplexScalar&, const ComplexScalar&, RealScalar p); 00064 static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p); 00065 00066 public: 00067 MatrixPowerAtomic(const MatrixType& T, RealScalar p); 00068 void compute(MatrixType& res) const; 00069 }; 00070 00071 template<typename MatrixType> 00072 MatrixPowerAtomic<MatrixType>::MatrixPowerAtomic(const MatrixType& T, RealScalar p) : 00073 m_A(T), m_p(p) 00074 { eigen_assert(T.rows() == T.cols()); } 00075 00076 template<typename MatrixType> 00077 void MatrixPowerAtomic<MatrixType>::compute(MatrixType& res) const 00078 { 00079 res.resizeLike(m_A); 00080 switch (m_A.rows()) { 00081 case 0: 00082 break; 00083 case 1: 00084 res(0,0) = std::pow(m_A(0,0), m_p); 00085 break; 00086 case 2: 00087 compute2x2(res, m_p); 00088 break; 00089 default: 00090 computeBig(res); 00091 } 00092 } 00093 00094 template<typename MatrixType> 00095 void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, MatrixType& res) const 00096 { 00097 int i = degree<<1; 00098 res = (m_p-degree) / ((i-1)<<1) * IminusT; 00099 for (--i; i; --i) { 00100 res = (MatrixType::Identity(IminusT.rows(), IminusT.cols()) + res).template triangularView<Upper>() 00101 .solve((i==1 ? -m_p : i&1 ? (-m_p-(i>>1))/(i<<1) : (m_p-(i>>1))/((i-1)<<1)) * IminusT).eval(); 00102 } 00103 res += MatrixType::Identity(IminusT.rows(), IminusT.cols()); 00104 } 00105 00106 // This function assumes that res has the correct size (see bug 614) 00107 template<typename MatrixType> 00108 void MatrixPowerAtomic<MatrixType>::compute2x2(MatrixType& res, RealScalar p) const 00109 { 00110 using std::abs; 00111 using std::pow; 00112 00113 res.coeffRef(0,0) = pow(m_A.coeff(0,0), p); 00114 00115 for (Index i=1; i < m_A.cols(); ++i) { 00116 res.coeffRef(i,i) = pow(m_A.coeff(i,i), p); 00117 if (m_A.coeff(i-1,i-1) == m_A.coeff(i,i)) 00118 res.coeffRef(i-1,i) = p * pow(m_A.coeff(i,i), p-1); 00119 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))) 00120 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)); 00121 else 00122 res.coeffRef(i-1,i) = computeSuperDiag(m_A.coeff(i,i), m_A.coeff(i-1,i-1), p); 00123 res.coeffRef(i-1,i) *= m_A.coeff(i-1,i); 00124 } 00125 } 00126 00127 template<typename MatrixType> 00128 void MatrixPowerAtomic<MatrixType>::computeBig(MatrixType& res) const 00129 { 00130 const int digits = std::numeric_limits<RealScalar>::digits; 00131 const RealScalar maxNormForPade = digits <= 24? 4.3386528e-1f: // sigle precision 00132 digits <= 53? 2.789358995219730e-1: // double precision 00133 digits <= 64? 2.4471944416607995472e-1L: // extended precision 00134 digits <= 106? 1.1016843812851143391275867258512e-1L: // double-double 00135 9.134603732914548552537150753385375e-2L; // quadruple precision 00136 MatrixType IminusT, sqrtT, T = m_A.template triangularView<Upper>(); 00137 RealScalar normIminusT; 00138 int degree, degree2, numberOfSquareRoots = 0; 00139 bool hasExtraSquareRoot = false; 00140 00141 /* FIXME 00142 * For singular T, norm(I - T) >= 1 but maxNormForPade < 1, leads to infinite 00143 * loop. We should move 0 eigenvalues to bottom right corner. We need not 00144 * worry about tiny values (e.g. 1e-300) because they will reach 1 if 00145 * repetitively sqrt'ed. 00146 * 00147 * If the 0 eigenvalues are semisimple, they can form a 0 matrix at the 00148 * bottom right corner. 00149 * 00150 * [ T A ]^p [ T^p (T^-1 T^p A) ] 00151 * [ ] = [ ] 00152 * [ 0 0 ] [ 0 0 ] 00153 */ 00154 for (Index i=0; i < m_A.cols(); ++i) 00155 eigen_assert(m_A(i,i) != RealScalar(0)); 00156 00157 while (true) { 00158 IminusT = MatrixType::Identity(m_A.rows(), m_A.cols()) - T; 00159 normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff(); 00160 if (normIminusT < maxNormForPade) { 00161 degree = getPadeDegree(normIminusT); 00162 degree2 = getPadeDegree(normIminusT/2); 00163 if (degree - degree2 <= 1 || hasExtraSquareRoot) 00164 break; 00165 hasExtraSquareRoot = true; 00166 } 00167 MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT); 00168 T = sqrtT.template triangularView<Upper>(); 00169 ++numberOfSquareRoots; 00170 } 00171 computePade(degree, IminusT, res); 00172 00173 for (; numberOfSquareRoots; --numberOfSquareRoots) { 00174 compute2x2(res, std::ldexp(m_p, -numberOfSquareRoots)); 00175 res = res.template triangularView<Upper>() * res; 00176 } 00177 compute2x2(res, m_p); 00178 } 00179 00180 template<typename MatrixType> 00181 inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(float normIminusT) 00182 { 00183 const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f }; 00184 int degree = 3; 00185 for (; degree <= 4; ++degree) 00186 if (normIminusT <= maxNormForPade[degree - 3]) 00187 break; 00188 return degree; 00189 } 00190 00191 template<typename MatrixType> 00192 inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(double normIminusT) 00193 { 00194 const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1, 00195 1.999045567181744e-1, 2.789358995219730e-1 }; 00196 int degree = 3; 00197 for (; degree <= 7; ++degree) 00198 if (normIminusT <= maxNormForPade[degree - 3]) 00199 break; 00200 return degree; 00201 } 00202 00203 template<typename MatrixType> 00204 inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(long double normIminusT) 00205 { 00206 #if LDBL_MANT_DIG == 53 00207 const int maxPadeDegree = 7; 00208 const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L, 00209 1.999045567181744e-1L, 2.789358995219730e-1L }; 00210 #elif LDBL_MANT_DIG <= 64 00211 const int maxPadeDegree = 8; 00212 const double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L, 00213 6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L }; 00214 #elif LDBL_MANT_DIG <= 106 00215 const int maxPadeDegree = 10; 00216 const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ , 00217 1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L, 00218 2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L, 00219 1.1016843812851143391275867258512e-1L }; 00220 #else 00221 const int maxPadeDegree = 10; 00222 const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ , 00223 6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L, 00224 9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L, 00225 3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L, 00226 9.134603732914548552537150753385375e-2L }; 00227 #endif 00228 int degree = 3; 00229 for (; degree <= maxPadeDegree; ++degree) 00230 if (normIminusT <= maxNormForPade[degree - 3]) 00231 break; 00232 return degree; 00233 } 00234 00235 template<typename MatrixType> 00236 inline typename MatrixPowerAtomic<MatrixType>::ComplexScalar 00237 MatrixPowerAtomic<MatrixType>::computeSuperDiag(const ComplexScalar& curr, const ComplexScalar& prev, RealScalar p) 00238 { 00239 ComplexScalar logCurr = std::log(curr); 00240 ComplexScalar logPrev = std::log(prev); 00241 int unwindingNumber = std::ceil((numext::imag(logCurr - logPrev) - M_PI) / (2*M_PI)); 00242 ComplexScalar w = numext::atanh2(curr - prev, curr + prev) + ComplexScalar(0, M_PI*unwindingNumber); 00243 return RealScalar(2) * std::exp(RealScalar(0.5) * p * (logCurr + logPrev)) * std::sinh(p * w) / (curr - prev); 00244 } 00245 00246 template<typename MatrixType> 00247 inline typename MatrixPowerAtomic<MatrixType>::RealScalar 00248 MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p) 00249 { 00250 RealScalar w = numext::atanh2(curr - prev, curr + prev); 00251 return 2 * std::exp(p * (std::log(curr) + std::log(prev)) / 2) * std::sinh(p * w) / (curr - prev); 00252 } 00253 00273 template<typename MatrixType> 00274 class MatrixPower 00275 { 00276 private: 00277 enum { 00278 RowsAtCompileTime = MatrixType::RowsAtCompileTime, 00279 ColsAtCompileTime = MatrixType::ColsAtCompileTime, 00280 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, 00281 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime 00282 }; 00283 typedef typename MatrixType::Scalar Scalar; 00284 typedef typename MatrixType::RealScalar RealScalar; 00285 typedef typename MatrixType::Index Index; 00286 00287 public: 00296 explicit MatrixPower(const MatrixType& A) : m_A(A), m_conditionNumber(0) 00297 { eigen_assert(A.rows() == A.cols()); } 00298 00306 const MatrixPowerRetval<MatrixType> operator()(RealScalar p) 00307 { return MatrixPowerRetval<MatrixType>(*this, p); } 00308 00316 template<typename ResultType> 00317 void compute(ResultType& res, RealScalar p); 00318 00319 Index rows() const { return m_A.rows(); } 00320 Index cols() const { return m_A.cols(); } 00321 00322 private: 00323 typedef std::complex<RealScalar> ComplexScalar; 00324 typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, MatrixType::Options, 00325 MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrix; 00326 00327 typename MatrixType::Nested m_A; 00328 MatrixType m_tmp; 00329 ComplexMatrix m_T, m_U, m_fT; 00330 RealScalar m_conditionNumber; 00331 00332 RealScalar modfAndInit(RealScalar, RealScalar*); 00333 00334 template<typename ResultType> 00335 void computeIntPower(ResultType&, RealScalar); 00336 00337 template<typename ResultType> 00338 void computeFracPower(ResultType&, RealScalar); 00339 00340 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 00341 static void revertSchur( 00342 Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 00343 const ComplexMatrix& T, 00344 const ComplexMatrix& U); 00345 00346 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 00347 static void revertSchur( 00348 Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 00349 const ComplexMatrix& T, 00350 const ComplexMatrix& U); 00351 }; 00352 00353 template<typename MatrixType> 00354 template<typename ResultType> 00355 void MatrixPower<MatrixType>::compute(ResultType& res, RealScalar p) 00356 { 00357 switch (cols()) { 00358 case 0: 00359 break; 00360 case 1: 00361 res(0,0) = std::pow(m_A.coeff(0,0), p); 00362 break; 00363 default: 00364 RealScalar intpart, x = modfAndInit(p, &intpart); 00365 computeIntPower(res, intpart); 00366 computeFracPower(res, x); 00367 } 00368 } 00369 00370 template<typename MatrixType> 00371 typename MatrixPower<MatrixType>::RealScalar 00372 MatrixPower<MatrixType>::modfAndInit(RealScalar x, RealScalar* intpart) 00373 { 00374 typedef Array<RealScalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> RealArray; 00375 00376 *intpart = std::floor(x); 00377 RealScalar res = x - *intpart; 00378 00379 if (!m_conditionNumber && res) { 00380 const ComplexSchur<MatrixType> schurOfA(m_A); 00381 m_T = schurOfA.matrixT(); 00382 m_U = schurOfA.matrixU(); 00383 00384 const RealArray absTdiag = m_T.diagonal().array().abs(); 00385 m_conditionNumber = absTdiag.maxCoeff() / absTdiag.minCoeff(); 00386 } 00387 00388 if (res>RealScalar(0.5) && res>(1-res)*std::pow(m_conditionNumber, res)) { 00389 --res; 00390 ++*intpart; 00391 } 00392 return res; 00393 } 00394 00395 template<typename MatrixType> 00396 template<typename ResultType> 00397 void MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p) 00398 { 00399 RealScalar pp = std::abs(p); 00400 00401 if (p<0) m_tmp = m_A.inverse(); 00402 else m_tmp = m_A; 00403 00404 res = MatrixType::Identity(rows(), cols()); 00405 while (pp >= 1) { 00406 if (std::fmod(pp, 2) >= 1) 00407 res = m_tmp * res; 00408 m_tmp *= m_tmp; 00409 pp /= 2; 00410 } 00411 } 00412 00413 template<typename MatrixType> 00414 template<typename ResultType> 00415 void MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p) 00416 { 00417 if (p) { 00418 eigen_assert(m_conditionNumber); 00419 MatrixPowerAtomic<ComplexMatrix>(m_T, p).compute(m_fT); 00420 revertSchur(m_tmp, m_fT, m_U); 00421 res = m_tmp * res; 00422 } 00423 } 00424 00425 template<typename MatrixType> 00426 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 00427 inline void MatrixPower<MatrixType>::revertSchur( 00428 Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 00429 const ComplexMatrix& T, 00430 const ComplexMatrix& U) 00431 { res.noalias() = U * (T.template triangularView<Upper>() * U.adjoint()); } 00432 00433 template<typename MatrixType> 00434 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 00435 inline void MatrixPower<MatrixType>::revertSchur( 00436 Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 00437 const ComplexMatrix& T, 00438 const ComplexMatrix& U) 00439 { res.noalias() = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); } 00440 00454 template<typename Derived> 00455 class MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Derived> > 00456 { 00457 public: 00458 typedef typename Derived::PlainObject PlainObject; 00459 typedef typename Derived::RealScalar RealScalar; 00460 typedef typename Derived::Index Index; 00461 00468 MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p) 00469 { } 00470 00477 template<typename ResultType> 00478 inline void evalTo(ResultType& res) const 00479 { MatrixPower<PlainObject>(m_A.eval()).compute(res, m_p); } 00480 00481 Index rows() const { return m_A.rows(); } 00482 Index cols() const { return m_A.cols(); } 00483 00484 private: 00485 const Derived& m_A; 00486 const RealScalar m_p; 00487 MatrixPowerReturnValue& operator=(const MatrixPowerReturnValue&); 00488 }; 00489 00490 namespace internal { 00491 00492 template<typename MatrixPowerType> 00493 struct traits< MatrixPowerRetval<MatrixPowerType> > 00494 { typedef typename MatrixPowerType::PlainObject ReturnType; }; 00495 00496 template<typename Derived> 00497 struct traits< MatrixPowerReturnValue<Derived> > 00498 { typedef typename Derived::PlainObject ReturnType; }; 00499 00500 } 00501 00502 template<typename Derived> 00503 const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const 00504 { return MatrixPowerReturnValue<Derived>(derived(), p); } 00505 00506 } // namespace Eigen 00507 00508 #endif // EIGEN_MATRIX_POWER