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710 lines
23 KiB
C++
710 lines
23 KiB
C++
// 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 http://mozilla.org/MPL/2.0/.
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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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template<typename MatrixType> class MatrixPower;
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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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typedef typename MatrixType::Index Index;
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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& res) const
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{ m_pow.compute(res, 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 typename MatrixType::Index Index;
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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-degree) / (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-i/2)/(2*i) : (m_p-i/2)/(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 = 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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int 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, 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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typedef typename MatrixType::Index Index;
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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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/** \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;
|
|
split(p, intpart);
|
|
|
|
res = MatrixType::Identity(rows(), cols());
|
|
computeIntPower(res, intpart);
|
|
if (p) computeFracPower(res, p);
|
|
}
|
|
}
|
|
|
|
template<typename MatrixType>
|
|
void MatrixPower<MatrixType>::split(RealScalar& p, RealScalar& intpart)
|
|
{
|
|
using std::floor;
|
|
using std::pow;
|
|
|
|
intpart = floor(p);
|
|
p -= intpart;
|
|
|
|
// Perform Schur decomposition if it is not yet performed and the power is
|
|
// not an integer.
|
|
if (!m_conditionNumber && p)
|
|
initialize();
|
|
|
|
// Choose the more stable of intpart = floor(p) and intpart = ceil(p).
|
|
if (p > RealScalar(0.5) && p > (1-p) * pow(m_conditionNumber, p)) {
|
|
--p;
|
|
++intpart;
|
|
}
|
|
}
|
|
|
|
template<typename MatrixType>
|
|
void MatrixPower<MatrixType>::initialize()
|
|
{
|
|
const ComplexSchur<MatrixType> schurOfA(m_A);
|
|
JacobiRotation<ComplexScalar> rot;
|
|
ComplexScalar eigenvalue;
|
|
|
|
m_fT.resizeLike(m_A);
|
|
m_T = schurOfA.matrixT();
|
|
m_U = schurOfA.matrixU();
|
|
m_conditionNumber = m_T.diagonal().array().abs().maxCoeff() / m_T.diagonal().array().abs().minCoeff();
|
|
|
|
// Move zero eigenvalues to the bottom right corner.
|
|
for (Index i = cols()-1; i>=0; --i) {
|
|
if (m_rank <= 2)
|
|
return;
|
|
if (m_T.coeff(i,i) == RealScalar(0)) {
|
|
for (Index j=i+1; j < m_rank; ++j) {
|
|
eigenvalue = m_T.coeff(j,j);
|
|
rot.makeGivens(m_T.coeff(j-1,j), eigenvalue);
|
|
m_T.applyOnTheRight(j-1, j, rot);
|
|
m_T.applyOnTheLeft(j-1, j, rot.adjoint());
|
|
m_T.coeffRef(j-1,j-1) = eigenvalue;
|
|
m_T.coeffRef(j,j) = RealScalar(0);
|
|
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;
|
|
typedef typename Derived::Index Index;
|
|
|
|
/**
|
|
* \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& res) const
|
|
{ MatrixPower<PlainObject>(m_A.eval()).compute(res, 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;
|
|
typedef typename Derived::Index Index;
|
|
|
|
/**
|
|
* \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& res) const
|
|
{ res = (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
|