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650 lines
21 KiB
C++
650 lines
21 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 MatrixPowerType>
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class MatrixPowerRetval : public ReturnByValue< MatrixPowerRetval<MatrixPowerType> >
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{
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public:
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typedef typename MatrixPowerType::PlainObject::RealScalar RealScalar;
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typedef typename MatrixPowerType::PlainObject::Index Index;
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MatrixPowerRetval(MatrixPowerType& pow, RealScalar p) : m_pow(pow), m_p(p)
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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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MatrixPowerType& m_pow;
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const RealScalar m_p;
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MatrixPowerRetval& operator=(const MatrixPowerRetval&);
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};
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template<typename MatrixType>
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class MatrixPowerAtomic
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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 Array< Scalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime > ArrayType;
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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, MatrixType& res) const;
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void compute2x2(MatrixType& res, RealScalar p) const;
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void computeBig(MatrixType& 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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MatrixPowerAtomic(const MatrixType& T, RealScalar p);
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void compute(MatrixType& 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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{ eigen_assert(T.rows() == T.cols()); }
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template<typename MatrixType>
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void MatrixPowerAtomic<MatrixType>::compute(MatrixType& res) const
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{
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res.resizeLike(m_A);
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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) = std::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, MatrixType& res) const
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{
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int i = degree<<1;
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res = (m_p-degree) / ((i-1)<<1) * 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>>1))/(i<<1) : (m_p-(i>>1))/((i-1)<<1)) * 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(MatrixType& 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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ArrayType logTdiag = m_A.diagonal().array().log();
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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(MatrixType& res) const
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{
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const int digits = std::numeric_limits<RealScalar>::digits;
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const RealScalar maxNormForPade = digits <= 24? 4.3386528e-1f: // sigle precision
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digits <= 53? 2.789358995219730e-1: // 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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/* FIXME
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* For singular T, norm(I - T) >= 1 but maxNormForPade < 1, leads to infinite
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* loop. We should move 0 eigenvalues to bottom right corner. We need not
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* worry about tiny values (e.g. 1e-300) because they will reach 1 if
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* repetitively sqrt'ed.
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*
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* If the 0 eigenvalues are semisimple, they can form a 0 matrix at the
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* bottom right corner.
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*
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* [ T A ]^p [ T^p (T^-1 T^p A) ]
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* [ ] = [ ]
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* [ 0 0 ] [ 0 0 ]
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*/
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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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MatrixSquareRootTriangular<MatrixType>(T).compute(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, std::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 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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ComplexScalar logCurr = std::log(curr);
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ComplexScalar logPrev = std::log(prev);
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int unwindingNumber = std::ceil((numext::imag(logCurr - logPrev) - M_PI) / (2*M_PI));
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ComplexScalar w = numext::atanh2(curr - prev, curr + prev) + ComplexScalar(0, M_PI*unwindingNumber);
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return RealScalar(2) * std::exp(RealScalar(0.5) * p * (logCurr + logPrev)) * std::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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RealScalar w = numext::atanh2(curr - prev, curr + prev);
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return 2 * std::exp(p * (std::log(curr) + std::log(prev)) / 2) * std::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 upper triangular matrices raised
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* to an arbitrary real power.
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*/
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template<typename MatrixType>
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class MatrixPowerTriangular
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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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ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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Options = MatrixType::Options,
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MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
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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 typename MatrixType::Index Index;
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public:
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typedef MatrixType PlainObject;
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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 MatrixPowerTriangular(const MatrixType& A) : m_A(A), m_conditionNumber(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$, where A is specified in the
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* constructor.
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*/
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const MatrixPowerRetval<MatrixPowerTriangular> operator()(RealScalar p)
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{ return MatrixPowerRetval<MatrixPowerTriangular>(*this, p); }
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/**
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* \brief Compute the matrix power.
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*
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* \param[in] p exponent, a real scalar.
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* \param[out] res \f$ A^p \f$ where A is specified in the
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* constructor.
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*/
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void compute(MatrixType& 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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typename MatrixType::Nested m_A;
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MatrixType m_tmp;
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RealScalar m_conditionNumber;
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RealScalar modfAndInit(RealScalar, RealScalar*);
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template<typename ResultType>
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void computeIntPower(ResultType&, RealScalar);
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template<typename ResultType>
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void computeFracPower(ResultType&, RealScalar);
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};
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template<typename MatrixType>
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void MatrixPowerTriangular<MatrixType>::compute(MatrixType& res, RealScalar p)
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{
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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) = std::pow(m_A.coeff(0,0), p);
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break;
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default:
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RealScalar intpart, x = modfAndInit(p, &intpart);
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computeIntPower(res, intpart);
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computeFracPower(res, x);
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}
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}
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template<typename MatrixType>
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typename MatrixPowerTriangular<MatrixType>::RealScalar
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MatrixPowerTriangular<MatrixType>::modfAndInit(RealScalar x, RealScalar* intpart)
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{
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typedef Array< RealScalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime > RealArray;
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*intpart = std::floor(x);
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RealScalar res = x - *intpart;
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if (!m_conditionNumber && res) {
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const RealArray absTdiag = m_A.diagonal().array().abs();
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m_conditionNumber = absTdiag.maxCoeff() / absTdiag.minCoeff();
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}
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if (res>RealScalar(0.5) && res>(1-res)*std::pow(m_conditionNumber, res)) {
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--res;
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++*intpart;
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}
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return res;
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}
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template<typename MatrixType>
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template<typename ResultType>
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void MatrixPowerTriangular<MatrixType>::computeIntPower(ResultType& res, RealScalar p)
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{
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RealScalar pp = std::abs(p);
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if (p<0) m_tmp = m_A.template triangularView<Upper>().solve(MatrixType::Identity(rows(), cols()));
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else m_tmp = m_A.template triangularView<Upper>();
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res = MatrixType::Identity(rows(), cols());
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while (pp >= 1) {
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if (std::fmod(pp, 2) >= 1)
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res.template triangularView<Upper>() = m_tmp.template triangularView<Upper>() * res;
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m_tmp.template triangularView<Upper>() = m_tmp.template triangularView<Upper>() * m_tmp;
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pp /= 2;
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}
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}
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template<typename MatrixType>
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template<typename ResultType>
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void MatrixPowerTriangular<MatrixType>::computeFracPower(ResultType& res, RealScalar p)
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{
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if (p) {
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eigen_assert(m_conditionNumber);
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MatrixPowerAtomic<MatrixType>(m_A, p).compute(m_tmp);
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res = m_tmp * res;
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}
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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
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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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ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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Options = MatrixType::Options,
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MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
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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 typename MatrixType::Index Index;
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public:
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typedef MatrixType PlainObject;
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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) : m_A(A), m_conditionNumber(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$, where A is specified in the
|
|
* constructor.
|
|
*/
|
|
const MatrixPowerRetval<MatrixPower> operator()(RealScalar p)
|
|
{ return MatrixPowerRetval<MatrixPower>(*this, p); }
|
|
|
|
/**
|
|
* \brief Compute the matrix power.
|
|
*
|
|
* \param[in] p exponent, a real scalar.
|
|
* \param[out] res \f$ A^p \f$ where A is specified in the
|
|
* constructor.
|
|
*/
|
|
void compute(MatrixType& res, RealScalar p);
|
|
|
|
Index rows() const { return m_A.rows(); }
|
|
Index cols() const { return m_A.cols(); }
|
|
|
|
private:
|
|
typedef std::complex<RealScalar> ComplexScalar;
|
|
typedef Matrix< ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime,
|
|
MaxColsAtCompileTime > ComplexMatrix;
|
|
|
|
typename MatrixType::Nested m_A;
|
|
MatrixType m_tmp;
|
|
ComplexMatrix m_T, m_U, m_fT;
|
|
RealScalar m_conditionNumber;
|
|
|
|
RealScalar modfAndInit(RealScalar, RealScalar*);
|
|
|
|
template<typename ResultType>
|
|
void computeIntPower(ResultType&, RealScalar);
|
|
|
|
template<typename ResultType>
|
|
void computeFracPower(ResultType&, RealScalar);
|
|
|
|
template<int Rows, int Cols, int Opt, int MaxRows, int MaxCols>
|
|
static void revertSchur(
|
|
Matrix< ComplexScalar, Rows, Cols, Opt, MaxRows, MaxCols >& res,
|
|
const ComplexMatrix& T,
|
|
const ComplexMatrix& U);
|
|
|
|
template<int Rows, int Cols, int Opt, int MaxRows, int MaxCols>
|
|
static void revertSchur(
|
|
Matrix< RealScalar, Rows, Cols, Opt, MaxRows, MaxCols >& res,
|
|
const ComplexMatrix& T,
|
|
const ComplexMatrix& U);
|
|
};
|
|
|
|
template<typename MatrixType>
|
|
void MatrixPower<MatrixType>::compute(MatrixType& res, RealScalar p)
|
|
{
|
|
switch (cols()) {
|
|
case 0:
|
|
break;
|
|
case 1:
|
|
res(0,0) = std::pow(m_A.coeff(0,0), p);
|
|
break;
|
|
default:
|
|
RealScalar intpart, x = modfAndInit(p, &intpart);
|
|
computeIntPower(res, intpart);
|
|
computeFracPower(res, x);
|
|
}
|
|
}
|
|
|
|
template<typename MatrixType>
|
|
typename MatrixPower<MatrixType>::RealScalar
|
|
MatrixPower<MatrixType>::modfAndInit(RealScalar x, RealScalar* intpart)
|
|
{
|
|
typedef Array< RealScalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime > RealArray;
|
|
|
|
*intpart = std::floor(x);
|
|
RealScalar res = x - *intpart;
|
|
|
|
if (!m_conditionNumber && res) {
|
|
const ComplexSchur<MatrixType> schurOfA(m_A);
|
|
m_T = schurOfA.matrixT();
|
|
m_U = schurOfA.matrixU();
|
|
|
|
const RealArray absTdiag = m_T.diagonal().array().abs();
|
|
m_conditionNumber = absTdiag.maxCoeff() / absTdiag.minCoeff();
|
|
}
|
|
|
|
if (res>RealScalar(0.5) && res>(1-res)*std::pow(m_conditionNumber, res)) {
|
|
--res;
|
|
++*intpart;
|
|
}
|
|
return res;
|
|
}
|
|
|
|
template<typename MatrixType>
|
|
template<typename ResultType>
|
|
void MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p)
|
|
{
|
|
RealScalar pp = std::abs(p);
|
|
|
|
if (p<0) m_tmp = m_A.inverse();
|
|
else m_tmp = m_A;
|
|
|
|
res = MatrixType::Identity(rows(), cols());
|
|
while (pp >= 1) {
|
|
if (std::fmod(pp, 2) >= 1)
|
|
res = m_tmp * res;
|
|
m_tmp *= m_tmp;
|
|
pp /= 2;
|
|
}
|
|
}
|
|
|
|
template<typename MatrixType>
|
|
template<typename ResultType>
|
|
void MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p)
|
|
{
|
|
if (p) {
|
|
eigen_assert(m_conditionNumber);
|
|
MatrixPowerAtomic<ComplexMatrix>(m_T, p).compute(m_fT);
|
|
revertSchur(m_tmp, m_fT, m_U);
|
|
res = m_tmp * res;
|
|
}
|
|
}
|
|
|
|
template<typename MatrixType>
|
|
template<int Rows, int Cols, int Opt, int MaxRows, int MaxCols>
|
|
inline void MatrixPower<MatrixType>::revertSchur(
|
|
Matrix< ComplexScalar, Rows, Cols, Opt, 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 Opt, int MaxRows, int MaxCols>
|
|
inline void MatrixPower<MatrixType>::revertSchur(
|
|
Matrix< RealScalar, Rows, Cols, Opt, 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 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;
|
|
MatrixPowerReturnValue& operator=(const MatrixPowerReturnValue&);
|
|
};
|
|
|
|
namespace internal {
|
|
|
|
template<typename MatrixPowerType>
|
|
struct traits< MatrixPowerRetval<MatrixPowerType> >
|
|
{ typedef typename MatrixPowerType::PlainObject ReturnType; };
|
|
|
|
template<typename Derived>
|
|
struct traits< MatrixPowerReturnValue<Derived> >
|
|
{ typedef typename Derived::PlainObject ReturnType; };
|
|
|
|
}
|
|
|
|
template<typename Derived>
|
|
const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const
|
|
{ return MatrixPowerReturnValue<Derived>(derived(), p); }
|
|
|
|
} // namespace Eigen
|
|
|
|
#endif // EIGEN_MATRIX_POWER
|