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705 lines
23 KiB
705 lines
23 KiB
// This file is part of Eigen, a lightweight C++ template library |
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// for linear algebra. |
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// |
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// Copyright (C) 2012, 2013 Chen-Pang He <jdh8@ms63.hinet.net> |
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// |
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// This Source Code Form is subject to the terms of the Mozilla |
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// Public License v. 2.0. If a copy of the MPL was not distributed |
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// with this file, You can obtain one at the mozilla.org home page |
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#ifndef EIGEN_MATRIX_POWER |
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#define EIGEN_MATRIX_POWER |
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namespace Eigen { |
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|
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template<typename MatrixType> class MatrixPower; |
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|
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/** |
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* \ingroup MatrixFunctions_Module |
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* |
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* \brief Proxy for the matrix power of some matrix. |
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* |
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* \tparam MatrixType type of the base, a matrix. |
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* |
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* This class holds the arguments to the matrix power until it is |
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* assigned or evaluated for some other reason (so the argument |
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* should not be changed in the meantime). It is the return type of |
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* MatrixPower::operator() and related functions and most of the |
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* time this is the only way it is used. |
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*/ |
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/* TODO This class is only used by MatrixPower, so it should be nested |
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* into MatrixPower, like MatrixPower::ReturnValue. However, my |
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* compiler complained about unused template parameter in the |
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* following declaration in namespace internal. |
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* |
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* template<typename MatrixType> |
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* struct traits<MatrixPower<MatrixType>::ReturnValue>; |
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*/ |
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template<typename MatrixType> |
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class MatrixPowerParenthesesReturnValue : public ReturnByValue< MatrixPowerParenthesesReturnValue<MatrixType> > |
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{ |
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public: |
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typedef typename MatrixType::RealScalar RealScalar; |
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/** |
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* \brief Constructor. |
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* |
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* \param[in] pow %MatrixPower storing the base. |
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* \param[in] p scalar, the exponent of the matrix power. |
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*/ |
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MatrixPowerParenthesesReturnValue(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p) |
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{ } |
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/** |
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* \brief Compute the matrix power. |
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* |
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* \param[out] result |
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*/ |
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template<typename ResultType> |
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inline void evalTo(ResultType& result) const |
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{ m_pow.compute(result, m_p); } |
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Index rows() const { return m_pow.rows(); } |
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Index cols() const { return m_pow.cols(); } |
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private: |
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MatrixPower<MatrixType>& m_pow; |
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const RealScalar m_p; |
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}; |
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/** |
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* \ingroup MatrixFunctions_Module |
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* |
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* \brief Class for computing matrix powers. |
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* |
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* \tparam MatrixType type of the base, expected to be an instantiation |
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* of the Matrix class template. |
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* |
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* This class is capable of computing triangular real/complex matrices |
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* raised to a power in the interval \f$ (-1, 1) \f$. |
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* |
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* \note Currently this class is only used by MatrixPower. One may |
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* insist that this be nested into MatrixPower. This class is here to |
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* facilitate future development of triangular matrix functions. |
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*/ |
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template<typename MatrixType> |
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class MatrixPowerAtomic : internal::noncopyable |
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{ |
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private: |
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enum { |
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RowsAtCompileTime = MatrixType::RowsAtCompileTime, |
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MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime |
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}; |
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typedef typename MatrixType::Scalar Scalar; |
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typedef typename MatrixType::RealScalar RealScalar; |
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typedef std::complex<RealScalar> ComplexScalar; |
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typedef Block<MatrixType,Dynamic,Dynamic> ResultType; |
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const MatrixType& m_A; |
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RealScalar m_p; |
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void computePade(int degree, const MatrixType& IminusT, ResultType& res) const; |
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void compute2x2(ResultType& res, RealScalar p) const; |
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void computeBig(ResultType& res) const; |
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static int getPadeDegree(float normIminusT); |
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static int getPadeDegree(double normIminusT); |
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static int getPadeDegree(long double normIminusT); |
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static ComplexScalar computeSuperDiag(const ComplexScalar&, const ComplexScalar&, RealScalar p); |
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static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p); |
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public: |
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/** |
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* \brief Constructor. |
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* |
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* \param[in] T the base of the matrix power. |
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* \param[in] p the exponent of the matrix power, should be in |
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* \f$ (-1, 1) \f$. |
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* |
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* The class stores a reference to T, so it should not be changed |
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* (or destroyed) before evaluation. Only the upper triangular |
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* part of T is read. |
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*/ |
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MatrixPowerAtomic(const MatrixType& T, RealScalar p); |
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/** |
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* \brief Compute the matrix power. |
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* |
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* \param[out] res \f$ A^p \f$ where A and p are specified in the |
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* constructor. |
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*/ |
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void compute(ResultType& res) const; |
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}; |
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template<typename MatrixType> |
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MatrixPowerAtomic<MatrixType>::MatrixPowerAtomic(const MatrixType& T, RealScalar p) : |
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m_A(T), m_p(p) |
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{ |
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eigen_assert(T.rows() == T.cols()); |
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eigen_assert(p > -1 && p < 1); |
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} |
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template<typename MatrixType> |
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void MatrixPowerAtomic<MatrixType>::compute(ResultType& res) const |
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{ |
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using std::pow; |
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switch (m_A.rows()) { |
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case 0: |
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break; |
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case 1: |
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res(0,0) = pow(m_A(0,0), m_p); |
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break; |
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case 2: |
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compute2x2(res, m_p); |
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break; |
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default: |
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computeBig(res); |
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} |
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} |
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template<typename MatrixType> |
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void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, ResultType& res) const |
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{ |
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int i = 2*degree; |
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res = (m_p-RealScalar(degree)) / RealScalar(2*i-2) * IminusT; |
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for (--i; i; --i) { |
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res = (MatrixType::Identity(IminusT.rows(), IminusT.cols()) + res).template triangularView<Upper>() |
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.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(); |
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} |
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res += MatrixType::Identity(IminusT.rows(), IminusT.cols()); |
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} |
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// This function assumes that res has the correct size (see bug 614) |
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template<typename MatrixType> |
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void MatrixPowerAtomic<MatrixType>::compute2x2(ResultType& res, RealScalar p) const |
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{ |
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using std::abs; |
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using std::pow; |
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res.coeffRef(0,0) = pow(m_A.coeff(0,0), p); |
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for (Index i=1; i < m_A.cols(); ++i) { |
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res.coeffRef(i,i) = pow(m_A.coeff(i,i), p); |
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if (m_A.coeff(i-1,i-1) == m_A.coeff(i,i)) |
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res.coeffRef(i-1,i) = p * pow(m_A.coeff(i,i), p-1); |
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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))) |
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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)); |
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else |
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res.coeffRef(i-1,i) = computeSuperDiag(m_A.coeff(i,i), m_A.coeff(i-1,i-1), p); |
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res.coeffRef(i-1,i) *= m_A.coeff(i-1,i); |
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} |
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} |
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template<typename MatrixType> |
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void MatrixPowerAtomic<MatrixType>::computeBig(ResultType& res) const |
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{ |
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using std::ldexp; |
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const int digits = std::numeric_limits<RealScalar>::digits; |
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const RealScalar maxNormForPade = RealScalar( |
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digits <= 24? 4.3386528e-1L // single precision |
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: digits <= 53? 2.789358995219730e-1L // double precision |
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: digits <= 64? 2.4471944416607995472e-1L // extended precision |
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: digits <= 106? 1.1016843812851143391275867258512e-1L // double-double |
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: 9.134603732914548552537150753385375e-2L); // quadruple precision |
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MatrixType IminusT, sqrtT, T = m_A.template triangularView<Upper>(); |
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RealScalar normIminusT; |
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int degree, degree2, numberOfSquareRoots = 0; |
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bool hasExtraSquareRoot = false; |
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for (Index i=0; i < m_A.cols(); ++i) |
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eigen_assert(m_A(i,i) != RealScalar(0)); |
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while (true) { |
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IminusT = MatrixType::Identity(m_A.rows(), m_A.cols()) - T; |
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normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff(); |
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if (normIminusT < maxNormForPade) { |
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degree = getPadeDegree(normIminusT); |
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degree2 = getPadeDegree(normIminusT/2); |
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if (degree - degree2 <= 1 || hasExtraSquareRoot) |
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break; |
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hasExtraSquareRoot = true; |
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} |
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matrix_sqrt_triangular(T, sqrtT); |
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T = sqrtT.template triangularView<Upper>(); |
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++numberOfSquareRoots; |
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} |
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computePade(degree, IminusT, res); |
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for (; numberOfSquareRoots; --numberOfSquareRoots) { |
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compute2x2(res, ldexp(m_p, -numberOfSquareRoots)); |
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res = res.template triangularView<Upper>() * res; |
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} |
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compute2x2(res, m_p); |
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} |
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template<typename MatrixType> |
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inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(float normIminusT) |
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{ |
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const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f }; |
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int degree = 3; |
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for (; degree <= 4; ++degree) |
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if (normIminusT <= maxNormForPade[degree - 3]) |
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break; |
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return degree; |
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} |
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template<typename MatrixType> |
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inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(double normIminusT) |
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{ |
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const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1, |
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1.999045567181744e-1, 2.789358995219730e-1 }; |
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int degree = 3; |
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for (; degree <= 7; ++degree) |
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if (normIminusT <= maxNormForPade[degree - 3]) |
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break; |
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return degree; |
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} |
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template<typename MatrixType> |
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inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(long double normIminusT) |
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{ |
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#if LDBL_MANT_DIG == 53 |
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const int maxPadeDegree = 7; |
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const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L, |
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1.999045567181744e-1L, 2.789358995219730e-1L }; |
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#elif LDBL_MANT_DIG <= 64 |
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const int maxPadeDegree = 8; |
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const long double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L, |
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6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L }; |
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#elif LDBL_MANT_DIG <= 106 |
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const int maxPadeDegree = 10; |
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const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ , |
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1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L, |
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2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L, |
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1.1016843812851143391275867258512e-1L }; |
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#else |
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const int maxPadeDegree = 10; |
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const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ , |
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6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L, |
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9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L, |
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3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L, |
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9.134603732914548552537150753385375e-2L }; |
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#endif |
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int degree = 3; |
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for (; degree <= maxPadeDegree; ++degree) |
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if (normIminusT <= maxNormForPade[degree - 3]) |
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break; |
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return degree; |
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} |
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template<typename MatrixType> |
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inline typename MatrixPowerAtomic<MatrixType>::ComplexScalar |
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MatrixPowerAtomic<MatrixType>::computeSuperDiag(const ComplexScalar& curr, const ComplexScalar& prev, RealScalar p) |
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{ |
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using std::ceil; |
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using std::exp; |
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using std::log; |
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using std::sinh; |
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ComplexScalar logCurr = log(curr); |
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ComplexScalar logPrev = log(prev); |
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RealScalar unwindingNumber = ceil((numext::imag(logCurr - logPrev) - RealScalar(EIGEN_PI)) / RealScalar(2*EIGEN_PI)); |
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ComplexScalar w = numext::log1p((curr-prev)/prev)/RealScalar(2) + ComplexScalar(0, RealScalar(EIGEN_PI)*unwindingNumber); |
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return RealScalar(2) * exp(RealScalar(0.5) * p * (logCurr + logPrev)) * sinh(p * w) / (curr - prev); |
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} |
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template<typename MatrixType> |
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inline typename MatrixPowerAtomic<MatrixType>::RealScalar |
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MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p) |
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{ |
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using std::exp; |
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using std::log; |
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using std::sinh; |
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RealScalar w = numext::log1p((curr-prev)/prev)/RealScalar(2); |
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return 2 * exp(p * (log(curr) + log(prev)) / 2) * sinh(p * w) / (curr - prev); |
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} |
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/** |
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* \ingroup MatrixFunctions_Module |
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* |
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* \brief Class for computing matrix powers. |
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* |
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* \tparam MatrixType type of the base, expected to be an instantiation |
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* of the Matrix class template. |
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* |
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* This class is capable of computing real/complex matrices raised to |
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* an arbitrary real power. Meanwhile, it saves the result of Schur |
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* decomposition if an non-integral power has even been calculated. |
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* Therefore, if you want to compute multiple (>= 2) matrix powers |
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* for the same matrix, using the class directly is more efficient than |
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* calling MatrixBase::pow(). |
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* |
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* Example: |
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* \include MatrixPower_optimal.cpp |
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* Output: \verbinclude MatrixPower_optimal.out |
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*/ |
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template<typename MatrixType> |
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class MatrixPower : internal::noncopyable |
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{ |
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private: |
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typedef typename MatrixType::Scalar Scalar; |
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typedef typename MatrixType::RealScalar RealScalar; |
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public: |
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/** |
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* \brief Constructor. |
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* |
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* \param[in] A the base of the matrix power. |
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* |
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* The class stores a reference to A, so it should not be changed |
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* (or destroyed) before evaluation. |
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*/ |
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explicit MatrixPower(const MatrixType& A) : |
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m_A(A), |
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m_conditionNumber(0), |
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m_rank(A.cols()), |
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m_nulls(0) |
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{ eigen_assert(A.rows() == A.cols()); } |
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/** |
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* \brief Returns the matrix power. |
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* |
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* \param[in] p exponent, a real scalar. |
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* \return The expression \f$ A^p \f$, where A is specified in the |
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* constructor. |
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*/ |
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const MatrixPowerParenthesesReturnValue<MatrixType> operator()(RealScalar p) |
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{ return MatrixPowerParenthesesReturnValue<MatrixType>(*this, p); } |
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/** |
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* \brief Compute the matrix power. |
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* |
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* \param[in] p exponent, a real scalar. |
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* \param[out] res \f$ A^p \f$ where A is specified in the |
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* constructor. |
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*/ |
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template<typename ResultType> |
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void compute(ResultType& res, RealScalar p); |
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Index rows() const { return m_A.rows(); } |
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Index cols() const { return m_A.cols(); } |
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private: |
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typedef std::complex<RealScalar> ComplexScalar; |
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typedef Matrix<ComplexScalar, Dynamic, Dynamic, 0, |
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MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime> ComplexMatrix; |
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/** \brief Reference to the base of matrix power. */ |
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typename MatrixType::Nested m_A; |
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/** \brief Temporary storage. */ |
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MatrixType m_tmp; |
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/** \brief Store the result of Schur decomposition. */ |
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ComplexMatrix m_T, m_U; |
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/** \brief Store fractional power of m_T. */ |
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ComplexMatrix m_fT; |
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/** |
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* \brief Condition number of m_A. |
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* |
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* It is initialized as 0 to avoid performing unnecessary Schur |
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* decomposition, which is the bottleneck. |
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*/ |
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RealScalar m_conditionNumber; |
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/** \brief Rank of m_A. */ |
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Index m_rank; |
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/** \brief Rank deficiency of m_A. */ |
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Index m_nulls; |
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/** |
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* \brief Split p into integral part and fractional part. |
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* |
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* \param[in] p The exponent. |
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* \param[out] p The fractional part ranging in \f$ (-1, 1) \f$. |
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* \param[out] intpart The integral part. |
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* |
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* Only if the fractional part is nonzero, it calls initialize(). |
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*/ |
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void split(RealScalar& p, RealScalar& intpart); |
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|
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/** \brief Perform Schur decomposition for fractional power. */ |
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void initialize(); |
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template<typename ResultType> |
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void computeIntPower(ResultType& res, RealScalar p); |
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template<typename ResultType> |
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void computeFracPower(ResultType& res, RealScalar p); |
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template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> |
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static void revertSchur( |
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Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, |
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const ComplexMatrix& T, |
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const ComplexMatrix& U); |
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template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> |
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static void revertSchur( |
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Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, |
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const ComplexMatrix& T, |
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const ComplexMatrix& U); |
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}; |
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template<typename MatrixType> |
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template<typename ResultType> |
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void MatrixPower<MatrixType>::compute(ResultType& res, RealScalar p) |
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{ |
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using std::pow; |
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switch (cols()) { |
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case 0: |
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break; |
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case 1: |
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res(0,0) = pow(m_A.coeff(0,0), p); |
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break; |
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default: |
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RealScalar intpart; |
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split(p, intpart); |
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res = MatrixType::Identity(rows(), cols()); |
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computeIntPower(res, intpart); |
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if (p) computeFracPower(res, p); |
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} |
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} |
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template<typename MatrixType> |
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void MatrixPower<MatrixType>::split(RealScalar& p, RealScalar& intpart) |
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{ |
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using std::floor; |
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using std::pow; |
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intpart = floor(p); |
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p -= intpart; |
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// Perform Schur decomposition if it is not yet performed and the power is |
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// not an integer. |
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if (!m_conditionNumber && p) |
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initialize(); |
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|
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// Choose the more stable of intpart = floor(p) and intpart = ceil(p). |
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if (p > RealScalar(0.5) && p > (1-p) * pow(m_conditionNumber, p)) { |
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--p; |
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++intpart; |
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} |
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} |
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|
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template<typename MatrixType> |
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void MatrixPower<MatrixType>::initialize() |
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{ |
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const ComplexSchur<MatrixType> schurOfA(m_A); |
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JacobiRotation<ComplexScalar> rot; |
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ComplexScalar eigenvalue; |
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|
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m_fT.resizeLike(m_A); |
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m_T = schurOfA.matrixT(); |
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m_U = schurOfA.matrixU(); |
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m_conditionNumber = m_T.diagonal().array().abs().maxCoeff() / m_T.diagonal().array().abs().minCoeff(); |
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|
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// Move zero eigenvalues to the bottom right corner. |
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for (Index i = cols()-1; i>=0; --i) { |
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if (m_rank <= 2) |
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return; |
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if (m_T.coeff(i,i) == RealScalar(0)) { |
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for (Index j=i+1; j < m_rank; ++j) { |
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eigenvalue = m_T.coeff(j,j); |
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rot.makeGivens(m_T.coeff(j-1,j), eigenvalue); |
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m_T.applyOnTheRight(j-1, j, rot); |
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m_T.applyOnTheLeft(j-1, j, rot.adjoint()); |
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m_T.coeffRef(j-1,j-1) = eigenvalue; |
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m_T.coeffRef(j,j) = RealScalar(0); |
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m_U.applyOnTheRight(j-1, j, rot); |
|
} |
|
--m_rank; |
|
} |
|
} |
|
|
|
m_nulls = rows() - m_rank; |
|
if (m_nulls) { |
|
eigen_assert(m_T.bottomRightCorner(m_nulls, m_nulls).isZero() |
|
&& "Base of matrix power should be invertible or with a semisimple zero eigenvalue."); |
|
m_fT.bottomRows(m_nulls).fill(RealScalar(0)); |
|
} |
|
} |
|
|
|
template<typename MatrixType> |
|
template<typename ResultType> |
|
void MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p) |
|
{ |
|
using std::abs; |
|
using std::fmod; |
|
RealScalar pp = abs(p); |
|
|
|
if (p<0) |
|
m_tmp = m_A.inverse(); |
|
else |
|
m_tmp = m_A; |
|
|
|
while (true) { |
|
if (fmod(pp, 2) >= 1) |
|
res = m_tmp * res; |
|
pp /= 2; |
|
if (pp < 1) |
|
break; |
|
m_tmp *= m_tmp; |
|
} |
|
} |
|
|
|
template<typename MatrixType> |
|
template<typename ResultType> |
|
void MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p) |
|
{ |
|
Block<ComplexMatrix,Dynamic,Dynamic> blockTp(m_fT, 0, 0, m_rank, m_rank); |
|
eigen_assert(m_conditionNumber); |
|
eigen_assert(m_rank + m_nulls == rows()); |
|
|
|
MatrixPowerAtomic<ComplexMatrix>(m_T.topLeftCorner(m_rank, m_rank), p).compute(blockTp); |
|
if (m_nulls) { |
|
m_fT.topRightCorner(m_rank, m_nulls) = m_T.topLeftCorner(m_rank, m_rank).template triangularView<Upper>() |
|
.solve(blockTp * m_T.topRightCorner(m_rank, m_nulls)); |
|
} |
|
revertSchur(m_tmp, m_fT, m_U); |
|
res = m_tmp * res; |
|
} |
|
|
|
template<typename MatrixType> |
|
template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> |
|
inline void MatrixPower<MatrixType>::revertSchur( |
|
Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, |
|
const ComplexMatrix& T, |
|
const ComplexMatrix& U) |
|
{ res.noalias() = U * (T.template triangularView<Upper>() * U.adjoint()); } |
|
|
|
template<typename MatrixType> |
|
template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> |
|
inline void MatrixPower<MatrixType>::revertSchur( |
|
Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, |
|
const ComplexMatrix& T, |
|
const ComplexMatrix& U) |
|
{ res.noalias() = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); } |
|
|
|
/** |
|
* \ingroup MatrixFunctions_Module |
|
* |
|
* \brief Proxy for the matrix power of some matrix (expression). |
|
* |
|
* \tparam Derived type of the base, a matrix (expression). |
|
* |
|
* This class holds the arguments to the matrix power until it is |
|
* assigned or evaluated for some other reason (so the argument |
|
* should not be changed in the meantime). It is the return type of |
|
* MatrixBase::pow() and related functions and most of the |
|
* time this is the only way it is used. |
|
*/ |
|
template<typename Derived> |
|
class MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Derived> > |
|
{ |
|
public: |
|
typedef typename Derived::PlainObject PlainObject; |
|
typedef typename Derived::RealScalar RealScalar; |
|
|
|
/** |
|
* \brief Constructor. |
|
* |
|
* \param[in] A %Matrix (expression), the base of the matrix power. |
|
* \param[in] p real scalar, the exponent of the matrix power. |
|
*/ |
|
MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p) |
|
{ } |
|
|
|
/** |
|
* \brief Compute the matrix power. |
|
* |
|
* \param[out] result \f$ A^p \f$ where \p A and \p p are as in the |
|
* constructor. |
|
*/ |
|
template<typename ResultType> |
|
inline void evalTo(ResultType& result) const |
|
{ MatrixPower<PlainObject>(m_A.eval()).compute(result, m_p); } |
|
|
|
Index rows() const { return m_A.rows(); } |
|
Index cols() const { return m_A.cols(); } |
|
|
|
private: |
|
const Derived& m_A; |
|
const RealScalar m_p; |
|
}; |
|
|
|
/** |
|
* \ingroup MatrixFunctions_Module |
|
* |
|
* \brief Proxy for the matrix power of some matrix (expression). |
|
* |
|
* \tparam Derived type of the base, a matrix (expression). |
|
* |
|
* This class holds the arguments to the matrix power until it is |
|
* assigned or evaluated for some other reason (so the argument |
|
* should not be changed in the meantime). It is the return type of |
|
* MatrixBase::pow() and related functions and most of the |
|
* time this is the only way it is used. |
|
*/ |
|
template<typename Derived> |
|
class MatrixComplexPowerReturnValue : public ReturnByValue< MatrixComplexPowerReturnValue<Derived> > |
|
{ |
|
public: |
|
typedef typename Derived::PlainObject PlainObject; |
|
typedef typename std::complex<typename Derived::RealScalar> ComplexScalar; |
|
|
|
/** |
|
* \brief Constructor. |
|
* |
|
* \param[in] A %Matrix (expression), the base of the matrix power. |
|
* \param[in] p complex scalar, the exponent of the matrix power. |
|
*/ |
|
MatrixComplexPowerReturnValue(const Derived& A, const ComplexScalar& p) : m_A(A), m_p(p) |
|
{ } |
|
|
|
/** |
|
* \brief Compute the matrix power. |
|
* |
|
* Because \p p is complex, \f$ A^p \f$ is simply evaluated as \f$ |
|
* \exp(p \log(A)) \f$. |
|
* |
|
* \param[out] result \f$ A^p \f$ where \p A and \p p are as in the |
|
* constructor. |
|
*/ |
|
template<typename ResultType> |
|
inline void evalTo(ResultType& result) const |
|
{ result = (m_p * m_A.log()).exp(); } |
|
|
|
Index rows() const { return m_A.rows(); } |
|
Index cols() const { return m_A.cols(); } |
|
|
|
private: |
|
const Derived& m_A; |
|
const ComplexScalar m_p; |
|
}; |
|
|
|
namespace internal { |
|
|
|
template<typename MatrixPowerType> |
|
struct traits< MatrixPowerParenthesesReturnValue<MatrixPowerType> > |
|
{ typedef typename MatrixPowerType::PlainObject ReturnType; }; |
|
|
|
template<typename Derived> |
|
struct traits< MatrixPowerReturnValue<Derived> > |
|
{ typedef typename Derived::PlainObject ReturnType; }; |
|
|
|
template<typename Derived> |
|
struct traits< MatrixComplexPowerReturnValue<Derived> > |
|
{ typedef typename Derived::PlainObject ReturnType; }; |
|
|
|
} |
|
|
|
template<typename Derived> |
|
const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const |
|
{ return MatrixPowerReturnValue<Derived>(derived(), p); } |
|
|
|
template<typename Derived> |
|
const MatrixComplexPowerReturnValue<Derived> MatrixBase<Derived>::pow(const std::complex<RealScalar>& p) const |
|
{ return MatrixComplexPowerReturnValue<Derived>(derived(), p); } |
|
|
|
} // namespace Eigen |
|
|
|
#endif // EIGEN_MATRIX_POWER
|
|
|