src/MatrixFunctions/MatrixPower.h

0001 // This file is part of Eigen, a lightweight C++ template library
0002 // for linear algebra.
0003 //
0004 // Copyright (C) 2012, 2013 Chen-Pang He <jdh8@ms63.hinet.net>
0005 //
0006 // This Source Code Form is subject to the terms of the Mozilla
0007 // Public License v. 2.0. If a copy of the MPL was not distributed
0008 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
0009
0010 #ifndef EIGEN_MATRIX_POWER
0011 #define EIGEN_MATRIX_POWER
0012
0013 namespace Eigen {
0014
0015 template<typename MatrixType> class MatrixPower;
0016
0017 /**
0018  * \ingroup MatrixFunctions_Module
0019  *
0020  * \brief Proxy for the matrix power of some matrix.
0021  *
0022  * \tparam MatrixType  type of the base, a matrix.
0023  *
0024  * This class holds the arguments to the matrix power until it is
0025  * assigned or evaluated for some other reason (so the argument
0026  * should not be changed in the meantime). It is the return type of
0027  * MatrixPower::operator() and related functions and most of the
0028  * time this is the only way it is used.
0029  */
0030 /* TODO This class is only used by MatrixPower, so it should be nested
0031  * into MatrixPower, like MatrixPower::ReturnValue. However, my
0032  * compiler complained about unused template parameter in the
0033  * following declaration in namespace internal.
0034  *
0035  * template<typename MatrixType>
0036  * struct traits<MatrixPower<MatrixType>::ReturnValue>;
0037  */
0038 template<typename MatrixType>
0039 class MatrixPowerParenthesesReturnValue : public ReturnByValue< MatrixPowerParenthesesReturnValue<MatrixType> >
0040 {
0041   public:
0042     typedef typename MatrixType::RealScalar RealScalar;
0043
0044     /**
0045      * \brief Constructor.
0046      *
0047      * \param[in] pow  %MatrixPower storing the base.
0048      * \param[in] p    scalar, the exponent of the matrix power.
0049      */
0050     MatrixPowerParenthesesReturnValue(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p)
0051     { }
0052
0053     /**
0054      * \brief Compute the matrix power.
0055      *
0056      * \param[out] result
0057      */
0058     template<typename ResultType>
0059     inline void evalTo(ResultType& result) const
0060     { m_pow.compute(result, m_p); }
0061
0062     Index rows() const { return m_pow.rows(); }
0063     Index cols() const { return m_pow.cols(); }
0064
0065   private:
0066     MatrixPower<MatrixType>& m_pow;
0067     const RealScalar m_p;
0068 };
0069
0070 /**
0071  * \ingroup MatrixFunctions_Module
0072  *
0073  * \brief Class for computing matrix powers.
0074  *
0075  * \tparam MatrixType  type of the base, expected to be an instantiation
0076  * of the Matrix class template.
0077  *
0078  * This class is capable of computing triangular real/complex matrices
0079  * raised to a power in the interval \f$ (-1, 1) \f$.
0080  *
0081  * \note Currently this class is only used by MatrixPower. One may
0082  * insist that this be nested into MatrixPower. This class is here to
0083  * facilitate future development of triangular matrix functions.
0084  */
0085 template<typename MatrixType>
0086 class MatrixPowerAtomic : internal::noncopyable
0087 {
0088   private:
0089     enum {
0090       RowsAtCompileTime = MatrixType::RowsAtCompileTime,
0091       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
0092     };
0093     typedef typename MatrixType::Scalar Scalar;
0094     typedef typename MatrixType::RealScalar RealScalar;
0095     typedef std::complex<RealScalar> ComplexScalar;
0096     typedef Block<MatrixType,Dynamic,Dynamic> ResultType;
0097
0098     const MatrixType& m_A;
0099     RealScalar m_p;
0100
0101     void computePade(int degree, const MatrixType& IminusT, ResultType& res) const;
0102     void compute2x2(ResultType& res, RealScalar p) const;
0103     void computeBig(ResultType& res) const;
0104     static int getPadeDegree(float normIminusT);
0105     static int getPadeDegree(double normIminusT);
0106     static int getPadeDegree(long double normIminusT);
0107     static ComplexScalar computeSuperDiag(const ComplexScalar&, const ComplexScalar&, RealScalar p);
0108     static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p);
0109
0110   public:
0111     /**
0112      * \brief Constructor.
0113      *
0114      * \param[in] T  the base of the matrix power.
0115      * \param[in] p  the exponent of the matrix power, should be in
0116      * \f$ (-1, 1) \f$.
0117      *
0118      * The class stores a reference to T, so it should not be changed
0119      * (or destroyed) before evaluation. Only the upper triangular
0120      * part of T is read.
0121      */
0122     MatrixPowerAtomic(const MatrixType& T, RealScalar p);
0123
0124     /**
0125      * \brief Compute the matrix power.
0126      *
0127      * \param[out] res  \f$ A^p \f$ where A and p are specified in the
0128      * constructor.
0129      */
0130     void compute(ResultType& res) const;
0131 };
0132
0133 template<typename MatrixType>
0134 MatrixPowerAtomic<MatrixType>::MatrixPowerAtomic(const MatrixType& T, RealScalar p) :
0135   m_A(T), m_p(p)
0136 {
0137   eigen_assert(T.rows() == T.cols());
0138   eigen_assert(p > -1 && p < 1);
0139 }
0140
0141 template<typename MatrixType>
0142 void MatrixPowerAtomic<MatrixType>::compute(ResultType& res) const
0143 {
0144   using std::pow;
0145   switch (m_A.rows()) {
0146     case 0:
0147       break;
0148     case 1:
0149       res(0,0) = pow(m_A(0,0), m_p);
0150       break;
0151     case 2:
0152       compute2x2(res, m_p);
0153       break;
0154     default:
0155       computeBig(res);
0156   }
0157 }
0158
0159 template<typename MatrixType>
0160 void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, ResultType& res) const
0161 {
0162   int i = 2*degree;
0163   res = (m_p-RealScalar(degree)) / RealScalar(2*i-2) * IminusT;
0164
0165   for (--i; i; --i) {
0166     res = (MatrixType::Identity(IminusT.rows(), IminusT.cols()) + res).template triangularView<Upper>()
0167     .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();
0168   }
0169   res += MatrixType::Identity(IminusT.rows(), IminusT.cols());
0170 }
0171
0172 // This function assumes that res has the correct size (see bug 614)
0173 template<typename MatrixType>
0174 void MatrixPowerAtomic<MatrixType>::compute2x2(ResultType& res, RealScalar p) const
0175 {
0176   using std::abs;
0177   using std::pow;
0178   res.coeffRef(0,0) = pow(m_A.coeff(0,0), p);
0179
0180   for (Index i=1; i < m_A.cols(); ++i) {
0181     res.coeffRef(i,i) = pow(m_A.coeff(i,i), p);
0182     if (m_A.coeff(i-1,i-1) == m_A.coeff(i,i))
0183       res.coeffRef(i-1,i) = p * pow(m_A.coeff(i,i), p-1);
0184     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)))
0185       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));
0186     else
0187       res.coeffRef(i-1,i) = computeSuperDiag(m_A.coeff(i,i), m_A.coeff(i-1,i-1), p);
0188     res.coeffRef(i-1,i) *= m_A.coeff(i-1,i);
0189   }
0190 }
0191
0192 template<typename MatrixType>
0193 void MatrixPowerAtomic<MatrixType>::computeBig(ResultType& res) const
0194 {
0195   using std::ldexp;
0196   const int digits = std::numeric_limits<RealScalar>::digits;
0197   const RealScalar maxNormForPade = RealScalar(
0198                                     digits <=  24? 4.3386528e-1L                            // single precision
0199                                   : digits <=  53? 2.789358995219730e-1L                    // double precision
0200                                   : digits <=  64? 2.4471944416607995472e-1L                // extended precision
0201                                   : digits <= 106? 1.1016843812851143391275867258512e-1L    // double-double
0202                                   :                9.134603732914548552537150753385375e-2L); // quadruple precision
0203   MatrixType IminusT, sqrtT, T = m_A.template triangularView<Upper>();
0204   RealScalar normIminusT;
0205   int degree, degree2, numberOfSquareRoots = 0;
0206   bool hasExtraSquareRoot = false;
0207
0208   for (Index i=0; i < m_A.cols(); ++i)
0209     eigen_assert(m_A(i,i) != RealScalar(0));
0210
0211   while (true) {
0212     IminusT = MatrixType::Identity(m_A.rows(), m_A.cols()) - T;
0213     normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff();
0214     if (normIminusT < maxNormForPade) {
0215       degree = getPadeDegree(normIminusT);
0216       degree2 = getPadeDegree(normIminusT/2);
0217       if (degree - degree2 <= 1 || hasExtraSquareRoot)
0218     break;
0219       hasExtraSquareRoot = true;
0220     }
0221     matrix_sqrt_triangular(T, sqrtT);
0222     T = sqrtT.template triangularView<Upper>();
0223     ++numberOfSquareRoots;
0224   }
0225   computePade(degree, IminusT, res);
0226
0227   for (; numberOfSquareRoots; --numberOfSquareRoots) {
0228     compute2x2(res, ldexp(m_p, -numberOfSquareRoots));
0229     res = res.template triangularView<Upper>() * res;
0230   }
0231   compute2x2(res, m_p);
0232 }
0233
0234 template<typename MatrixType>
0235 inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(float normIminusT)
0236 {
0237   const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f };
0238   int degree = 3;
0239   for (; degree <= 4; ++degree)
0240     if (normIminusT <= maxNormForPade[degree - 3])
0241       break;
0242   return degree;
0243 }
0244
0245 template<typename MatrixType>
0246 inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(double normIminusT)
0247 {
0248   const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1,
0249       1.999045567181744e-1, 2.789358995219730e-1 };
0250   int degree = 3;
0251   for (; degree <= 7; ++degree)
0252     if (normIminusT <= maxNormForPade[degree - 3])
0253       break;
0254   return degree;
0255 }
0256
0257 template<typename MatrixType>
0258 inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(long double normIminusT)
0259 {
0260 #if   LDBL_MANT_DIG == 53
0261   const int maxPadeDegree = 7;
0262   const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L,
0263       1.999045567181744e-1L, 2.789358995219730e-1L };
0264 #elif LDBL_MANT_DIG <= 64
0265   const int maxPadeDegree = 8;
0266   const long double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L,
0267       6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L };
0268 #elif LDBL_MANT_DIG <= 106
0269   const int maxPadeDegree = 10;
0270   const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ ,
0271       1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L,
0272       2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L,
0273       1.1016843812851143391275867258512e-1L };
0274 #else
0275   const int maxPadeDegree = 10;
0276   const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ ,
0277       6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L,
0278       9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L,
0279       3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L,
0280       9.134603732914548552537150753385375e-2L };
0281 #endif
0282   int degree = 3;
0283   for (; degree <= maxPadeDegree; ++degree)
0284     if (normIminusT <= maxNormForPade[degree - 3])
0285       break;
0286   return degree;
0287 }
0288
0289 template<typename MatrixType>
0290 inline typename MatrixPowerAtomic<MatrixType>::ComplexScalar
0291 MatrixPowerAtomic<MatrixType>::computeSuperDiag(const ComplexScalar& curr, const ComplexScalar& prev, RealScalar p)
0292 {
0293   using std::ceil;
0294   using std::exp;
0295   using std::log;
0296   using std::sinh;
0297
0298   ComplexScalar logCurr = log(curr);
0299   ComplexScalar logPrev = log(prev);
0300   RealScalar unwindingNumber = ceil((numext::imag(logCurr - logPrev) - RealScalar(EIGEN_PI)) / RealScalar(2*EIGEN_PI));
0301   ComplexScalar w = numext::log1p((curr-prev)/prev)/RealScalar(2) + ComplexScalar(0, RealScalar(EIGEN_PI)*unwindingNumber);
0302   return RealScalar(2) * exp(RealScalar(0.5) * p * (logCurr + logPrev)) * sinh(p * w) / (curr - prev);
0303 }
0304
0305 template<typename MatrixType>
0306 inline typename MatrixPowerAtomic<MatrixType>::RealScalar
0307 MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p)
0308 {
0309   using std::exp;
0310   using std::log;
0311   using std::sinh;
0312
0313   RealScalar w = numext::log1p((curr-prev)/prev)/RealScalar(2);
0314   return 2 * exp(p * (log(curr) + log(prev)) / 2) * sinh(p * w) / (curr - prev);
0315 }
0316
0317 /**
0318  * \ingroup MatrixFunctions_Module
0319  *
0320  * \brief Class for computing matrix powers.
0321  *
0322  * \tparam MatrixType  type of the base, expected to be an instantiation
0323  * of the Matrix class template.
0324  *
0325  * This class is capable of computing real/complex matrices raised to
0326  * an arbitrary real power. Meanwhile, it saves the result of Schur
0327  * decomposition if an non-integral power has even been calculated.
0328  * Therefore, if you want to compute multiple (>= 2) matrix powers
0329  * for the same matrix, using the class directly is more efficient than
0330  * calling MatrixBase::pow().
0331  *
0332  * Example:
0333  * \include MatrixPower_optimal.cpp
0334  * Output: \verbinclude MatrixPower_optimal.out
0335  */
0336 template<typename MatrixType>
0337 class MatrixPower : internal::noncopyable
0338 {
0339   private:
0340     typedef typename MatrixType::Scalar Scalar;
0341     typedef typename MatrixType::RealScalar RealScalar;
0342
0343   public:
0344     /**
0345      * \brief Constructor.
0346      *
0347      * \param[in] A  the base of the matrix power.
0348      *
0349      * The class stores a reference to A, so it should not be changed
0350      * (or destroyed) before evaluation.
0351      */
0352     explicit MatrixPower(const MatrixType& A) :
0353       m_A(A),
0354       m_conditionNumber(0),
0355       m_rank(A.cols()),
0356       m_nulls(0)
0357     { eigen_assert(A.rows() == A.cols()); }
0358
0359     /**
0360      * \brief Returns the matrix power.
0361      *
0362      * \param[in] p  exponent, a real scalar.
0363      * \return The expression \f$ A^p \f$, where A is specified in the
0364      * constructor.
0365      */
0366     const MatrixPowerParenthesesReturnValue<MatrixType> operator()(RealScalar p)
0367     { return MatrixPowerParenthesesReturnValue<MatrixType>(*this, p); }
0368
0369     /**
0370      * \brief Compute the matrix power.
0371      *
0372      * \param[in]  p    exponent, a real scalar.
0373      * \param[out] res  \f$ A^p \f$ where A is specified in the
0374      * constructor.
0375      */
0376     template<typename ResultType>
0377     void compute(ResultType& res, RealScalar p);
0378
0379     Index rows() const { return m_A.rows(); }
0380     Index cols() const { return m_A.cols(); }
0381
0382   private:
0383     typedef std::complex<RealScalar> ComplexScalar;
0384     typedef Matrix<ComplexScalar, Dynamic, Dynamic, 0,
0385               MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime> ComplexMatrix;
0386
0387     /** \brief Reference to the base of matrix power. */
0388     typename MatrixType::Nested m_A;
0389
0390     /** \brief Temporary storage. */
0391     MatrixType m_tmp;
0392
0393     /** \brief Store the result of Schur decomposition. */
0394     ComplexMatrix m_T, m_U;
0395
0396     /** \brief Store fractional power of m_T. */
0397     ComplexMatrix m_fT;
0398
0399     /**
0400      * \brief Condition number of m_A.
0401      *
0402      * It is initialized as 0 to avoid performing unnecessary Schur
0403      * decomposition, which is the bottleneck.
0404      */
0405     RealScalar m_conditionNumber;
0406
0407     /** \brief Rank of m_A. */
0408     Index m_rank;
0409
0410     /** \brief Rank deficiency of m_A. */
0411     Index m_nulls;
0412
0413     /**
0414      * \brief Split p into integral part and fractional part.
0415      *
0416      * \param[in]  p        The exponent.
0417      * \param[out] p        The fractional part ranging in \f$ (-1, 1) \f$.
0418      * \param[out] intpart  The integral part.
0419      *
0420      * Only if the fractional part is nonzero, it calls initialize().
0421      */
0422     void split(RealScalar& p, RealScalar& intpart);
0423
0424     /** \brief Perform Schur decomposition for fractional power. */
0425     void initialize();
0426
0427     template<typename ResultType>
0428     void computeIntPower(ResultType& res, RealScalar p);
0429
0430     template<typename ResultType>
0431     void computeFracPower(ResultType& res, RealScalar p);
0432
0433     template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
0434     static void revertSchur(
0435         Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
0436         const ComplexMatrix& T,
0437         const ComplexMatrix& U);
0438
0439     template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
0440     static void revertSchur(
0441         Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
0442         const ComplexMatrix& T,
0443         const ComplexMatrix& U);
0444 };
0445
0446 template<typename MatrixType>
0447 template<typename ResultType>
0448 void MatrixPower<MatrixType>::compute(ResultType& res, RealScalar p)
0449 {
0450   using std::pow;
0451   switch (cols()) {
0452     case 0:
0453       break;
0454     case 1:
0455       res(0,0) = pow(m_A.coeff(0,0), p);
0456       break;
0457     default:
0458       RealScalar intpart;
0459       split(p, intpart);
0460
0461       res = MatrixType::Identity(rows(), cols());
0462       computeIntPower(res, intpart);
0463       if (p) computeFracPower(res, p);
0464   }
0465 }
0466
0467 template<typename MatrixType>
0468 void MatrixPower<MatrixType>::split(RealScalar& p, RealScalar& intpart)
0469 {
0470   using std::floor;
0471   using std::pow;
0472
0473   intpart = floor(p);
0474   p -= intpart;
0475
0476   // Perform Schur decomposition if it is not yet performed and the power is
0477   // not an integer.
0478   if (!m_conditionNumber && p)
0479     initialize();
0480
0481   // Choose the more stable of intpart = floor(p) and intpart = ceil(p).
0482   if (p > RealScalar(0.5) && p > (1-p) * pow(m_conditionNumber, p)) {
0483     --p;
0484     ++intpart;
0485   }
0486 }
0487
0488 template<typename MatrixType>
0489 void MatrixPower<MatrixType>::initialize()
0490 {
0491   const ComplexSchur<MatrixType> schurOfA(m_A);
0492   JacobiRotation<ComplexScalar> rot;
0493   ComplexScalar eigenvalue;
0494
0495   m_fT.resizeLike(m_A);
0496   m_T = schurOfA.matrixT();
0497   m_U = schurOfA.matrixU();
0498   m_conditionNumber = m_T.diagonal().array().abs().maxCoeff() / m_T.diagonal().array().abs().minCoeff();
0499
0500   // Move zero eigenvalues to the bottom right corner.
0501   for (Index i = cols()-1; i>=0; --i) {
0502     if (m_rank <= 2)
0503       return;
0504     if (m_T.coeff(i,i) == RealScalar(0)) {
0505       for (Index j=i+1; j < m_rank; ++j) {
0506         eigenvalue = m_T.coeff(j,j);
0507         rot.makeGivens(m_T.coeff(j-1,j), eigenvalue);
0508         m_T.applyOnTheRight(j-1, j, rot);
0509         m_T.applyOnTheLeft(j-1, j, rot.adjoint());
0510         m_T.coeffRef(j-1,j-1) = eigenvalue;
0511         m_T.coeffRef(j,j) = RealScalar(0);
0512         m_U.applyOnTheRight(j-1, j, rot);
0513       }
0514       --m_rank;
0515     }
0516   }
0517
0518   m_nulls = rows() - m_rank;
0519   if (m_nulls) {
0520     eigen_assert(m_T.bottomRightCorner(m_nulls, m_nulls).isZero()
0521         && "Base of matrix power should be invertible or with a semisimple zero eigenvalue.");
0522     m_fT.bottomRows(m_nulls).fill(RealScalar(0));
0523   }
0524 }
0525
0526 template<typename MatrixType>
0527 template<typename ResultType>
0528 void MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p)
0529 {
0530   using std::abs;
0531   using std::fmod;
0532   RealScalar pp = abs(p);
0533
0534   if (p<0)
0535     m_tmp = m_A.inverse();
0536   else
0537     m_tmp = m_A;
0538
0539   while (true) {
0540     if (fmod(pp, 2) >= 1)
0541       res = m_tmp * res;
0542     pp /= 2;
0543     if (pp < 1)
0544       break;
0545     m_tmp *= m_tmp;
0546   }
0547 }
0548
0549 template<typename MatrixType>
0550 template<typename ResultType>
0551 void MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p)
0552 {
0553   Block<ComplexMatrix,Dynamic,Dynamic> blockTp(m_fT, 0, 0, m_rank, m_rank);
0554   eigen_assert(m_conditionNumber);
0555   eigen_assert(m_rank + m_nulls == rows());
0556
0557   MatrixPowerAtomic<ComplexMatrix>(m_T.topLeftCorner(m_rank, m_rank), p).compute(blockTp);
0558   if (m_nulls) {
0559     m_fT.topRightCorner(m_rank, m_nulls) = m_T.topLeftCorner(m_rank, m_rank).template triangularView<Upper>()
0560         .solve(blockTp * m_T.topRightCorner(m_rank, m_nulls));
0561   }
0562   revertSchur(m_tmp, m_fT, m_U);
0563   res = m_tmp * res;
0564 }
0565
0566 template<typename MatrixType>
0567 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
0568 inline void MatrixPower<MatrixType>::revertSchur(
0569     Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
0570     const ComplexMatrix& T,
0571     const ComplexMatrix& U)
0572 { res.noalias() = U * (T.template triangularView<Upper>() * U.adjoint()); }
0573
0574 template<typename MatrixType>
0575 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
0576 inline void MatrixPower<MatrixType>::revertSchur(
0577     Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
0578     const ComplexMatrix& T,
0579     const ComplexMatrix& U)
0580 { res.noalias() = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); }
0581
0582 /**
0583  * \ingroup MatrixFunctions_Module
0584  *
0585  * \brief Proxy for the matrix power of some matrix (expression).
0586  *
0587  * \tparam Derived  type of the base, a matrix (expression).
0588  *
0589  * This class holds the arguments to the matrix power until it is
0590  * assigned or evaluated for some other reason (so the argument
0591  * should not be changed in the meantime). It is the return type of
0592  * MatrixBase::pow() and related functions and most of the
0593  * time this is the only way it is used.
0594  */
0595 template<typename Derived>
0596 class MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Derived> >
0597 {
0598   public:
0599     typedef typename Derived::PlainObject PlainObject;
0600     typedef typename Derived::RealScalar RealScalar;
0601
0602     /**
0603      * \brief Constructor.
0604      *
0605      * \param[in] A  %Matrix (expression), the base of the matrix power.
0606      * \param[in] p  real scalar, the exponent of the matrix power.
0607      */
0608     MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p)
0609     { }
0610
0611     /**
0612      * \brief Compute the matrix power.
0613      *
0614      * \param[out] result  \f$ A^p \f$ where \p A and \p p are as in the
0615      * constructor.
0616      */
0617     template<typename ResultType>
0618     inline void evalTo(ResultType& result) const
0619     { MatrixPower<PlainObject>(m_A.eval()).compute(result, m_p); }
0620
0621     Index rows() const { return m_A.rows(); }
0622     Index cols() const { return m_A.cols(); }
0623
0624   private:
0625     const Derived& m_A;
0626     const RealScalar m_p;
0627 };
0628
0629 /**
0630  * \ingroup MatrixFunctions_Module
0631  *
0632  * \brief Proxy for the matrix power of some matrix (expression).
0633  *
0634  * \tparam Derived  type of the base, a matrix (expression).
0635  *
0636  * This class holds the arguments to the matrix power until it is
0637  * assigned or evaluated for some other reason (so the argument
0638  * should not be changed in the meantime). It is the return type of
0639  * MatrixBase::pow() and related functions and most of the
0640  * time this is the only way it is used.
0641  */
0642 template<typename Derived>
0643 class MatrixComplexPowerReturnValue : public ReturnByValue< MatrixComplexPowerReturnValue<Derived> >
0644 {
0645   public:
0646     typedef typename Derived::PlainObject PlainObject;
0647     typedef typename std::complex<typename Derived::RealScalar> ComplexScalar;
0648
0649     /**
0650      * \brief Constructor.
0651      *
0652      * \param[in] A  %Matrix (expression), the base of the matrix power.
0653      * \param[in] p  complex scalar, the exponent of the matrix power.
0654      */
0655     MatrixComplexPowerReturnValue(const Derived& A, const ComplexScalar& p) : m_A(A), m_p(p)
0656     { }
0657
0658     /**
0659      * \brief Compute the matrix power.
0660      *
0661      * Because \p p is complex, \f$ A^p \f$ is simply evaluated as \f$
0662      * \exp(p \log(A)) \f$.
0663      *
0664      * \param[out] result  \f$ A^p \f$ where \p A and \p p are as in the
0665      * constructor.
0666      */
0667     template<typename ResultType>
0668     inline void evalTo(ResultType& result) const
0669     { result = (m_p * m_A.log()).exp(); }
0670
0671     Index rows() const { return m_A.rows(); }
0672     Index cols() const { return m_A.cols(); }
0673
0674   private:
0675     const Derived& m_A;
0676     const ComplexScalar m_p;
0677 };
0678
0679 namespace internal {
0680
0681 template<typename MatrixPowerType>
0682 struct traits< MatrixPowerParenthesesReturnValue<MatrixPowerType> >
0683 { typedef typename MatrixPowerType::PlainObject ReturnType; };
0684
0685 template<typename Derived>
0686 struct traits< MatrixPowerReturnValue<Derived> >
0687 { typedef typename Derived::PlainObject ReturnType; };
0688
0689 template<typename Derived>
0690 struct traits< MatrixComplexPowerReturnValue<Derived> >
0691 { typedef typename Derived::PlainObject ReturnType; };
0692
0693 }
0694
0695 template<typename Derived>
0696 const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const
0697 { return MatrixPowerReturnValue<Derived>(derived(), p); }
0698
0699 template<typename Derived>
0700 const MatrixComplexPowerReturnValue<Derived> MatrixBase<Derived>::pow(const std::complex<RealScalar>& p) const
0701 { return MatrixComplexPowerReturnValue<Derived>(derived(), p); }
0702
0703 } // namespace Eigen
0704
0705 #endif // EIGEN_MATRIX_POWER