17faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// This file is part of Eigen, a lightweight C++ template library 27faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// for linear algebra. 37faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// 47faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// Copyright (C) 2012, 2013 Chen-Pang He <jdh8@ms63.hinet.net> 57faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// 67faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// This Source Code Form is subject to the terms of the Mozilla 77faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// Public License v. 2.0. If a copy of the MPL was not distributed 87faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 97faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 107faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez#ifndef EIGEN_MATRIX_POWER 117faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez#define EIGEN_MATRIX_POWER 127faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 137faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeznamespace Eigen { 147faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 157faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> class MatrixPower; 167faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 172b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang/** 182b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \ingroup MatrixFunctions_Module 192b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 202b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \brief Proxy for the matrix power of some matrix. 212b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 222b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \tparam MatrixType type of the base, a matrix. 232b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 242b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * This class holds the arguments to the matrix power until it is 252b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * assigned or evaluated for some other reason (so the argument 262b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * should not be changed in the meantime). It is the return type of 272b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * MatrixPower::operator() and related functions and most of the 282b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * time this is the only way it is used. 292b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang */ 302b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang/* TODO This class is only used by MatrixPower, so it should be nested 312b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * into MatrixPower, like MatrixPower::ReturnValue. However, my 322b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * compiler complained about unused template parameter in the 332b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * following declaration in namespace internal. 342b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 352b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * template<typename MatrixType> 362b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * struct traits<MatrixPower<MatrixType>::ReturnValue>; 372b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang */ 387faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 392b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangclass MatrixPowerParenthesesReturnValue : public ReturnByValue< MatrixPowerParenthesesReturnValue<MatrixType> > 407faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 417faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez public: 427faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename MatrixType::RealScalar RealScalar; 437faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename MatrixType::Index Index; 447faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 452b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang /** 462b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \brief Constructor. 472b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 482b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \param[in] pow %MatrixPower storing the base. 492b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \param[in] p scalar, the exponent of the matrix power. 502b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang */ 512b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang MatrixPowerParenthesesReturnValue(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p) 527faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez { } 537faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 542b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang /** 552b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \brief Compute the matrix power. 562b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 572b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \param[out] result 582b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang */ 597faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez template<typename ResultType> 607faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez inline void evalTo(ResultType& res) const 617faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez { m_pow.compute(res, m_p); } 627faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 637faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index rows() const { return m_pow.rows(); } 647faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index cols() const { return m_pow.cols(); } 657faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 667faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez private: 677faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MatrixPower<MatrixType>& m_pow; 687faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const RealScalar m_p; 697faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez}; 707faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 712b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang/** 722b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \ingroup MatrixFunctions_Module 732b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 742b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \brief Class for computing matrix powers. 752b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 762b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \tparam MatrixType type of the base, expected to be an instantiation 772b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * of the Matrix class template. 782b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 792b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * This class is capable of computing triangular real/complex matrices 802b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * raised to a power in the interval \f$ (-1, 1) \f$. 812b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 822b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \note Currently this class is only used by MatrixPower. One may 832b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * insist that this be nested into MatrixPower. This class is here to 842b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * faciliate future development of triangular matrix functions. 852b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang */ 867faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 872b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangclass MatrixPowerAtomic : internal::noncopyable 887faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 897faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez private: 907faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez enum { 917faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RowsAtCompileTime = MatrixType::RowsAtCompileTime, 927faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime 937faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez }; 947faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename MatrixType::Scalar Scalar; 957faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename MatrixType::RealScalar RealScalar; 967faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef std::complex<RealScalar> ComplexScalar; 977faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename MatrixType::Index Index; 982b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang typedef Block<MatrixType,Dynamic,Dynamic> ResultType; 997faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1007faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const MatrixType& m_A; 1017faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RealScalar m_p; 1027faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1032b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang void computePade(int degree, const MatrixType& IminusT, ResultType& res) const; 1042b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang void compute2x2(ResultType& res, RealScalar p) const; 1052b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang void computeBig(ResultType& res) const; 1067faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez static int getPadeDegree(float normIminusT); 1077faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez static int getPadeDegree(double normIminusT); 1087faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez static int getPadeDegree(long double normIminusT); 1097faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez static ComplexScalar computeSuperDiag(const ComplexScalar&, const ComplexScalar&, RealScalar p); 1107faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p); 1117faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1127faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez public: 1132b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang /** 1142b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \brief Constructor. 1152b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 1162b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \param[in] T the base of the matrix power. 1172b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \param[in] p the exponent of the matrix power, should be in 1182b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \f$ (-1, 1) \f$. 1192b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 1202b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * The class stores a reference to T, so it should not be changed 1212b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * (or destroyed) before evaluation. Only the upper triangular 1222b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * part of T is read. 1232b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang */ 1247faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MatrixPowerAtomic(const MatrixType& T, RealScalar p); 1252b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 1262b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang /** 1272b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \brief Compute the matrix power. 1282b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 1292b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \param[out] res \f$ A^p \f$ where A and p are specified in the 1302b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * constructor. 1312b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang */ 1322b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang void compute(ResultType& res) const; 1337faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez}; 1347faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1357faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 1367faaa9f3f0df9d23790277834d426c3d992ac3baCarlos HernandezMatrixPowerAtomic<MatrixType>::MatrixPowerAtomic(const MatrixType& T, RealScalar p) : 1377faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez m_A(T), m_p(p) 1382b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang{ 1392b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang eigen_assert(T.rows() == T.cols()); 1402b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang eigen_assert(p > -1 && p < 1); 1412b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang} 1427faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1437faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 1442b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangvoid MatrixPowerAtomic<MatrixType>::compute(ResultType& res) const 1457faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 1462b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang using std::pow; 1477faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez switch (m_A.rows()) { 1487faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez case 0: 1497faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 1507faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez case 1: 1512b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang res(0,0) = pow(m_A(0,0), m_p); 1527faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 1537faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez case 2: 1547faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez compute2x2(res, m_p); 1557faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 1567faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez default: 1577faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez computeBig(res); 1587faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 1597faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 1607faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1617faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 1622b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangvoid MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, ResultType& res) const 1637faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 1642b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang int i = 2*degree; 1652b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang res = (m_p-degree) / (2*i-2) * IminusT; 1662b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 1677faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (--i; i; --i) { 1687faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res = (MatrixType::Identity(IminusT.rows(), IminusT.cols()) + res).template triangularView<Upper>() 1692b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang .solve((i==1 ? -m_p : i&1 ? (-m_p-i/2)/(2*i) : (m_p-i/2)/(2*i-2)) * IminusT).eval(); 1707faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 1717faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res += MatrixType::Identity(IminusT.rows(), IminusT.cols()); 1727faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 1737faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1747faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// This function assumes that res has the correct size (see bug 614) 1757faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 1762b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangvoid MatrixPowerAtomic<MatrixType>::compute2x2(ResultType& res, RealScalar p) const 1777faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 1787faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez using std::abs; 1797faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez using std::pow; 1807faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res.coeffRef(0,0) = pow(m_A.coeff(0,0), p); 1817faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1827faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (Index i=1; i < m_A.cols(); ++i) { 1837faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res.coeffRef(i,i) = pow(m_A.coeff(i,i), p); 1847faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (m_A.coeff(i-1,i-1) == m_A.coeff(i,i)) 1857faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res.coeffRef(i-1,i) = p * pow(m_A.coeff(i,i), p-1); 1867faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 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))) 1877faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 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)); 1887faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez else 1897faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res.coeffRef(i-1,i) = computeSuperDiag(m_A.coeff(i,i), m_A.coeff(i-1,i-1), p); 1907faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res.coeffRef(i-1,i) *= m_A.coeff(i-1,i); 1917faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 1927faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 1937faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1947faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 1952b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangvoid MatrixPowerAtomic<MatrixType>::computeBig(ResultType& res) const 1967faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 1972b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang using std::ldexp; 1987faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const int digits = std::numeric_limits<RealScalar>::digits; 1992b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang const RealScalar maxNormForPade = digits <= 24? 4.3386528e-1L // single precision 2002b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang : digits <= 53? 2.789358995219730e-1L // double precision 2012b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang : digits <= 64? 2.4471944416607995472e-1L // extended precision 2022b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang : digits <= 106? 1.1016843812851143391275867258512e-1L // double-double 2032b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang : 9.134603732914548552537150753385375e-2L; // quadruple precision 2047faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MatrixType IminusT, sqrtT, T = m_A.template triangularView<Upper>(); 2057faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RealScalar normIminusT; 2067faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez int degree, degree2, numberOfSquareRoots = 0; 2077faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez bool hasExtraSquareRoot = false; 2087faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2097faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (Index i=0; i < m_A.cols(); ++i) 2107faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert(m_A(i,i) != RealScalar(0)); 2117faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2127faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez while (true) { 2137faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez IminusT = MatrixType::Identity(m_A.rows(), m_A.cols()) - T; 2147faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff(); 2157faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (normIminusT < maxNormForPade) { 2167faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez degree = getPadeDegree(normIminusT); 2177faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez degree2 = getPadeDegree(normIminusT/2); 2187faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (degree - degree2 <= 1 || hasExtraSquareRoot) 2197faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 2207faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez hasExtraSquareRoot = true; 2217faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 2222b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang matrix_sqrt_triangular(T, sqrtT); 2237faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez T = sqrtT.template triangularView<Upper>(); 2247faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez ++numberOfSquareRoots; 2257faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 2267faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez computePade(degree, IminusT, res); 2277faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2287faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (; numberOfSquareRoots; --numberOfSquareRoots) { 2292b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang compute2x2(res, ldexp(m_p, -numberOfSquareRoots)); 2307faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res = res.template triangularView<Upper>() * res; 2317faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 2327faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez compute2x2(res, m_p); 2337faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 2347faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2357faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 2367faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezinline int MatrixPowerAtomic<MatrixType>::getPadeDegree(float normIminusT) 2377faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 2387faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f }; 2397faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez int degree = 3; 2407faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (; degree <= 4; ++degree) 2417faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (normIminusT <= maxNormForPade[degree - 3]) 2427faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 2437faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez return degree; 2447faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 2457faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2467faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 2477faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezinline int MatrixPowerAtomic<MatrixType>::getPadeDegree(double normIminusT) 2487faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 2497faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1, 2507faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1.999045567181744e-1, 2.789358995219730e-1 }; 2517faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez int degree = 3; 2527faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (; degree <= 7; ++degree) 2537faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (normIminusT <= maxNormForPade[degree - 3]) 2547faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 2557faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez return degree; 2567faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 2577faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2587faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 2597faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezinline int MatrixPowerAtomic<MatrixType>::getPadeDegree(long double normIminusT) 2607faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 2617faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez#if LDBL_MANT_DIG == 53 2627faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const int maxPadeDegree = 7; 2637faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L, 2647faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1.999045567181744e-1L, 2.789358995219730e-1L }; 2657faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez#elif LDBL_MANT_DIG <= 64 2667faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const int maxPadeDegree = 8; 2672b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang const long double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L, 2687faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L }; 2697faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez#elif LDBL_MANT_DIG <= 106 2707faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const int maxPadeDegree = 10; 2717faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ , 2727faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L, 2737faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L, 2747faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1.1016843812851143391275867258512e-1L }; 2757faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez#else 2767faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const int maxPadeDegree = 10; 2777faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ , 2787faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L, 2797faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L, 2807faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L, 2817faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 9.134603732914548552537150753385375e-2L }; 2827faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez#endif 2837faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez int degree = 3; 2847faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (; degree <= maxPadeDegree; ++degree) 2857faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (normIminusT <= maxNormForPade[degree - 3]) 2867faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 2877faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez return degree; 2887faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 2897faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2907faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 2917faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezinline typename MatrixPowerAtomic<MatrixType>::ComplexScalar 2927faaa9f3f0df9d23790277834d426c3d992ac3baCarlos HernandezMatrixPowerAtomic<MatrixType>::computeSuperDiag(const ComplexScalar& curr, const ComplexScalar& prev, RealScalar p) 2937faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 2942b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang using std::ceil; 2952b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang using std::exp; 2962b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang using std::log; 2972b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang using std::sinh; 2982b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 2992b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang ComplexScalar logCurr = log(curr); 3002b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang ComplexScalar logPrev = log(prev); 3012b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang int unwindingNumber = ceil((numext::imag(logCurr - logPrev) - RealScalar(EIGEN_PI)) / RealScalar(2*EIGEN_PI)); 3022b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang ComplexScalar w = numext::log1p((curr-prev)/prev)/RealScalar(2) + ComplexScalar(0, EIGEN_PI*unwindingNumber); 3032b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang return RealScalar(2) * exp(RealScalar(0.5) * p * (logCurr + logPrev)) * sinh(p * w) / (curr - prev); 3047faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 3057faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3067faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 3077faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezinline typename MatrixPowerAtomic<MatrixType>::RealScalar 3087faaa9f3f0df9d23790277834d426c3d992ac3baCarlos HernandezMatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p) 3097faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 3102b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang using std::exp; 3112b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang using std::log; 3122b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang using std::sinh; 3132b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 3142b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang RealScalar w = numext::log1p((curr-prev)/prev)/RealScalar(2); 3152b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang return 2 * exp(p * (log(curr) + log(prev)) / 2) * sinh(p * w) / (curr - prev); 3167faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 3177faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3187faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez/** 3197faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \ingroup MatrixFunctions_Module 3207faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 3217faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \brief Class for computing matrix powers. 3227faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 3237faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \tparam MatrixType type of the base, expected to be an instantiation 3247faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * of the Matrix class template. 3257faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 3267faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * This class is capable of computing real/complex matrices raised to 3277faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * an arbitrary real power. Meanwhile, it saves the result of Schur 3287faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * decomposition if an non-integral power has even been calculated. 3297faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * Therefore, if you want to compute multiple (>= 2) matrix powers 3307faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * for the same matrix, using the class directly is more efficient than 3317faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * calling MatrixBase::pow(). 3327faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 3337faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * Example: 3347faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \include MatrixPower_optimal.cpp 3357faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * Output: \verbinclude MatrixPower_optimal.out 3367faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 3377faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 3382b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangclass MatrixPower : internal::noncopyable 3397faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 3407faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez private: 3417faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename MatrixType::Scalar Scalar; 3427faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename MatrixType::RealScalar RealScalar; 3437faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename MatrixType::Index Index; 3447faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3457faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez public: 3467faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez /** 3477faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \brief Constructor. 3487faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 3497faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \param[in] A the base of the matrix power. 3507faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 3517faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * The class stores a reference to A, so it should not be changed 3527faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * (or destroyed) before evaluation. 3537faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 3542b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang explicit MatrixPower(const MatrixType& A) : 3552b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_A(A), 3562b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_conditionNumber(0), 3572b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_rank(A.cols()), 3582b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_nulls(0) 3597faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez { eigen_assert(A.rows() == A.cols()); } 3607faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3617faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez /** 3627faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \brief Returns the matrix power. 3637faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 3647faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \param[in] p exponent, a real scalar. 3657faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \return The expression \f$ A^p \f$, where A is specified in the 3667faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * constructor. 3677faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 3682b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang const MatrixPowerParenthesesReturnValue<MatrixType> operator()(RealScalar p) 3692b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang { return MatrixPowerParenthesesReturnValue<MatrixType>(*this, p); } 3707faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3717faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez /** 3727faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \brief Compute the matrix power. 3737faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 3747faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \param[in] p exponent, a real scalar. 3757faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \param[out] res \f$ A^p \f$ where A is specified in the 3767faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * constructor. 3777faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 3787faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez template<typename ResultType> 3797faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez void compute(ResultType& res, RealScalar p); 3807faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3817faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index rows() const { return m_A.rows(); } 3827faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index cols() const { return m_A.cols(); } 3837faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3847faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez private: 3857faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef std::complex<RealScalar> ComplexScalar; 3862b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang typedef Matrix<ComplexScalar, Dynamic, Dynamic, 0, 3872b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime> ComplexMatrix; 3887faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3892b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang /** \brief Reference to the base of matrix power. */ 3907faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typename MatrixType::Nested m_A; 3912b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 3922b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang /** \brief Temporary storage. */ 3937faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MatrixType m_tmp; 3942b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 3952b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang /** \brief Store the result of Schur decomposition. */ 3962b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang ComplexMatrix m_T, m_U; 3972b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 3982b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang /** \brief Store fractional power of m_T. */ 3992b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang ComplexMatrix m_fT; 4002b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 4012b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang /** 4022b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \brief Condition number of m_A. 4032b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 4042b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * It is initialized as 0 to avoid performing unnecessary Schur 4052b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * decomposition, which is the bottleneck. 4062b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang */ 4077faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RealScalar m_conditionNumber; 4087faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4092b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang /** \brief Rank of m_A. */ 4102b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang Index m_rank; 4112b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 4122b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang /** \brief Rank deficiency of m_A. */ 4132b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang Index m_nulls; 4142b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 4152b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang /** 4162b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \brief Split p into integral part and fractional part. 4172b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 4182b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \param[in] p The exponent. 4192b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \param[out] p The fractional part ranging in \f$ (-1, 1) \f$. 4202b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \param[out] intpart The integral part. 4212b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 4222b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * Only if the fractional part is nonzero, it calls initialize(). 4232b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang */ 4242b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang void split(RealScalar& p, RealScalar& intpart); 4252b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 4262b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang /** \brief Perform Schur decomposition for fractional power. */ 4272b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang void initialize(); 4287faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4297faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez template<typename ResultType> 4302b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang void computeIntPower(ResultType& res, RealScalar p); 4317faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4327faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez template<typename ResultType> 4332b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang void computeFracPower(ResultType& res, RealScalar p); 4347faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4357faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 4367faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez static void revertSchur( 4377faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 4387faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexMatrix& T, 4397faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexMatrix& U); 4407faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4417faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 4427faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez static void revertSchur( 4437faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 4447faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexMatrix& T, 4457faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexMatrix& U); 4467faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez}; 4477faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4487faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 4497faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename ResultType> 4507faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezvoid MatrixPower<MatrixType>::compute(ResultType& res, RealScalar p) 4517faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 4522b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang using std::pow; 4537faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez switch (cols()) { 4547faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez case 0: 4557faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 4567faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez case 1: 4572b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang res(0,0) = pow(m_A.coeff(0,0), p); 4587faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 4597faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez default: 4602b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang RealScalar intpart; 4612b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang split(p, intpart); 4622b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 4632b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang res = MatrixType::Identity(rows(), cols()); 4647faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez computeIntPower(res, intpart); 4652b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang if (p) computeFracPower(res, p); 4667faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 4677faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 4687faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4697faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 4702b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangvoid MatrixPower<MatrixType>::split(RealScalar& p, RealScalar& intpart) 4717faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 4722b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang using std::floor; 4732b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang using std::pow; 4747faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4752b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang intpart = floor(p); 4762b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang p -= intpart; 4777faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4782b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang // Perform Schur decomposition if it is not yet performed and the power is 4792b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang // not an integer. 4802b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang if (!m_conditionNumber && p) 4812b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang initialize(); 4822b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 4832b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang // Choose the more stable of intpart = floor(p) and intpart = ceil(p). 4842b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang if (p > RealScalar(0.5) && p > (1-p) * pow(m_conditionNumber, p)) { 4852b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang --p; 4862b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang ++intpart; 4872b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang } 4882b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang} 4892b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 4902b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangtemplate<typename MatrixType> 4912b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangvoid MatrixPower<MatrixType>::initialize() 4922b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang{ 4932b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang const ComplexSchur<MatrixType> schurOfA(m_A); 4942b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang JacobiRotation<ComplexScalar> rot; 4952b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang ComplexScalar eigenvalue; 4962b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 4972b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_fT.resizeLike(m_A); 4982b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_T = schurOfA.matrixT(); 4992b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_U = schurOfA.matrixU(); 5002b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_conditionNumber = m_T.diagonal().array().abs().maxCoeff() / m_T.diagonal().array().abs().minCoeff(); 5012b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 5022b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang // Move zero eigenvalues to the bottom right corner. 5032b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang for (Index i = cols()-1; i>=0; --i) { 5042b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang if (m_rank <= 2) 5052b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang return; 5062b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang if (m_T.coeff(i,i) == RealScalar(0)) { 5072b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang for (Index j=i+1; j < m_rank; ++j) { 5082b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang eigenvalue = m_T.coeff(j,j); 5092b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang rot.makeGivens(m_T.coeff(j-1,j), eigenvalue); 5102b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_T.applyOnTheRight(j-1, j, rot); 5112b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_T.applyOnTheLeft(j-1, j, rot.adjoint()); 5122b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_T.coeffRef(j-1,j-1) = eigenvalue; 5132b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_T.coeffRef(j,j) = RealScalar(0); 5142b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_U.applyOnTheRight(j-1, j, rot); 5152b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang } 5162b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang --m_rank; 5172b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang } 5187faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 5197faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 5202b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_nulls = rows() - m_rank; 5212b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang if (m_nulls) { 5222b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang eigen_assert(m_T.bottomRightCorner(m_nulls, m_nulls).isZero() 5232b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang && "Base of matrix power should be invertible or with a semisimple zero eigenvalue."); 5242b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_fT.bottomRows(m_nulls).fill(RealScalar(0)); 5257faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 5267faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 5277faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 5287faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 5297faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename ResultType> 5307faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezvoid MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p) 5317faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 5322b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang using std::abs; 5332b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang using std::fmod; 5342b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang RealScalar pp = abs(p); 5357faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 5362b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang if (p<0) 5372b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_tmp = m_A.inverse(); 5382b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang else 5392b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_tmp = m_A; 5407faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 5412b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang while (true) { 5422b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang if (fmod(pp, 2) >= 1) 5437faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res = m_tmp * res; 5447faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez pp /= 2; 5452b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang if (pp < 1) 5462b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang break; 5472b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_tmp *= m_tmp; 5487faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 5497faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 5507faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 5517faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 5527faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename ResultType> 5537faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezvoid MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p) 5547faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 5552b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang Block<ComplexMatrix,Dynamic,Dynamic> blockTp(m_fT, 0, 0, m_rank, m_rank); 5562b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang eigen_assert(m_conditionNumber); 5572b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang eigen_assert(m_rank + m_nulls == rows()); 5582b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 5592b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang MatrixPowerAtomic<ComplexMatrix>(m_T.topLeftCorner(m_rank, m_rank), p).compute(blockTp); 5602b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang if (m_nulls) { 5612b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang m_fT.topRightCorner(m_rank, m_nulls) = m_T.topLeftCorner(m_rank, m_rank).template triangularView<Upper>() 5622b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang .solve(blockTp * m_T.topRightCorner(m_rank, m_nulls)); 5637faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 5642b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang revertSchur(m_tmp, m_fT, m_U); 5652b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang res = m_tmp * res; 5667faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 5677faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 5687faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 5697faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 5707faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezinline void MatrixPower<MatrixType>::revertSchur( 5717faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 5727faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexMatrix& T, 5737faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexMatrix& U) 5747faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ res.noalias() = U * (T.template triangularView<Upper>() * U.adjoint()); } 5757faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 5767faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 5777faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 5787faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezinline void MatrixPower<MatrixType>::revertSchur( 5797faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 5807faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexMatrix& T, 5817faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexMatrix& U) 5827faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ res.noalias() = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); } 5837faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 5847faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez/** 5857faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \ingroup MatrixFunctions_Module 5867faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 5877faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \brief Proxy for the matrix power of some matrix (expression). 5887faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 5897faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \tparam Derived type of the base, a matrix (expression). 5907faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 5917faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * This class holds the arguments to the matrix power until it is 5927faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * assigned or evaluated for some other reason (so the argument 5937faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * should not be changed in the meantime). It is the return type of 5947faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * MatrixBase::pow() and related functions and most of the 5957faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * time this is the only way it is used. 5967faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 5977faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename Derived> 5987faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezclass MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Derived> > 5997faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 6007faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez public: 6017faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename Derived::PlainObject PlainObject; 6027faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename Derived::RealScalar RealScalar; 6037faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename Derived::Index Index; 6047faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 6057faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez /** 6067faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \brief Constructor. 6077faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 6087faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \param[in] A %Matrix (expression), the base of the matrix power. 6092b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \param[in] p real scalar, the exponent of the matrix power. 6107faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 6117faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p) 6127faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez { } 6137faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 6147faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez /** 6157faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \brief Compute the matrix power. 6167faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 6177faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \param[out] result \f$ A^p \f$ where \p A and \p p are as in the 6187faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * constructor. 6197faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 6207faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez template<typename ResultType> 6217faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez inline void evalTo(ResultType& res) const 6227faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez { MatrixPower<PlainObject>(m_A.eval()).compute(res, m_p); } 6237faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 6247faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index rows() const { return m_A.rows(); } 6257faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index cols() const { return m_A.cols(); } 6267faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 6277faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez private: 6287faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const Derived& m_A; 6297faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const RealScalar m_p; 6302b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang}; 6312b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 6322b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang/** 6332b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \ingroup MatrixFunctions_Module 6342b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 6352b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \brief Proxy for the matrix power of some matrix (expression). 6362b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 6372b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \tparam Derived type of the base, a matrix (expression). 6382b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 6392b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * This class holds the arguments to the matrix power until it is 6402b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * assigned or evaluated for some other reason (so the argument 6412b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * should not be changed in the meantime). It is the return type of 6422b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * MatrixBase::pow() and related functions and most of the 6432b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * time this is the only way it is used. 6442b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang */ 6452b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangtemplate<typename Derived> 6462b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangclass MatrixComplexPowerReturnValue : public ReturnByValue< MatrixComplexPowerReturnValue<Derived> > 6472b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang{ 6482b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang public: 6492b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang typedef typename Derived::PlainObject PlainObject; 6502b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang typedef typename std::complex<typename Derived::RealScalar> ComplexScalar; 6512b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang typedef typename Derived::Index Index; 6522b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 6532b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang /** 6542b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \brief Constructor. 6552b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 6562b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \param[in] A %Matrix (expression), the base of the matrix power. 6572b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \param[in] p complex scalar, the exponent of the matrix power. 6582b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang */ 6592b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang MatrixComplexPowerReturnValue(const Derived& A, const ComplexScalar& p) : m_A(A), m_p(p) 6602b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang { } 6612b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 6622b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang /** 6632b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \brief Compute the matrix power. 6642b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 6652b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * Because \p p is complex, \f$ A^p \f$ is simply evaluated as \f$ 6662b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \exp(p \log(A)) \f$. 6672b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * 6682b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * \param[out] result \f$ A^p \f$ where \p A and \p p are as in the 6692b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang * constructor. 6702b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang */ 6712b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang template<typename ResultType> 6722b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang inline void evalTo(ResultType& res) const 6732b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang { res = (m_p * m_A.log()).exp(); } 6742b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 6752b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang Index rows() const { return m_A.rows(); } 6762b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang Index cols() const { return m_A.cols(); } 6772b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 6782b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang private: 6792b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang const Derived& m_A; 6802b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang const ComplexScalar m_p; 6817faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez}; 6827faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 6837faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeznamespace internal { 6847faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 6857faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixPowerType> 6862b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangstruct traits< MatrixPowerParenthesesReturnValue<MatrixPowerType> > 6877faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ typedef typename MatrixPowerType::PlainObject ReturnType; }; 6887faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 6897faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename Derived> 6907faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezstruct traits< MatrixPowerReturnValue<Derived> > 6917faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ typedef typename Derived::PlainObject ReturnType; }; 6927faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 6932b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangtemplate<typename Derived> 6942b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangstruct traits< MatrixComplexPowerReturnValue<Derived> > 6952b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang{ typedef typename Derived::PlainObject ReturnType; }; 6962b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 6977faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 6987faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 6997faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename Derived> 7007faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezconst MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const 7017faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ return MatrixPowerReturnValue<Derived>(derived(), p); } 7027faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 7032b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangtemplate<typename Derived> 7042b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wangconst MatrixComplexPowerReturnValue<Derived> MatrixBase<Derived>::pow(const std::complex<RealScalar>& p) const 7052b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang{ return MatrixComplexPowerReturnValue<Derived>(derived(), p); } 7062b8756b6f1de65d3f8bffab45be6c44ceb7411fcMiao Wang 7077faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} // namespace Eigen 7087faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 7097faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez#endif // EIGEN_MATRIX_POWER 710