1c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// This file is part of Eigen, a lightweight C++ template library 2c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// for linear algebra. 3c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// 4c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// Copyright (C) 2011 Jitse Niesen <jitse@maths.leeds.ac.uk> 5c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net> 6c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// 7c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// This Source Code Form is subject to the terms of the Mozilla 8c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// Public License v. 2.0. If a copy of the MPL was not distributed 9c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath// with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 10c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 11c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#ifndef EIGEN_MATRIX_LOGARITHM 12c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#define EIGEN_MATRIX_LOGARITHM 13c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 14c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#ifndef M_PI 15c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#define M_PI 3.141592653589793238462643383279503L 16c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#endif 17c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 18c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathnamespace Eigen { 19c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 20c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \ingroup MatrixFunctions_Module 21c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \class MatrixLogarithmAtomic 22c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \brief Helper class for computing matrix logarithm of atomic matrices. 23c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 24c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \internal 25c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * Here, an atomic matrix is a triangular matrix whose diagonal 26c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * entries are close to each other. 27c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 28c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \sa class MatrixFunctionAtomic, MatrixBase::log() 29c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 30c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 31c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathclass MatrixLogarithmAtomic 32c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 33c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathpublic: 34c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 35c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename MatrixType::Scalar Scalar; 36c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // typedef typename MatrixType::Index Index; 37c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename NumTraits<Scalar>::Real RealScalar; 38c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // typedef typename internal::stem_function<Scalar>::type StemFunction; 39c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType; 40c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 41c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \brief Constructor. */ 42c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixLogarithmAtomic() { } 43c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 44c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \brief Compute matrix logarithm of atomic matrix 45c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param[in] A argument of matrix logarithm, should be upper triangular and atomic 46c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \returns The logarithm of \p A. 47c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 48c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType compute(const MatrixType& A); 49c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 50c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathprivate: 51c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 52c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath void compute2x2(const MatrixType& A, MatrixType& result); 53c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath void computeBig(const MatrixType& A, MatrixType& result); 54c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath int getPadeDegree(float normTminusI); 55c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath int getPadeDegree(double normTminusI); 56c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath int getPadeDegree(long double normTminusI); 57c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath void computePade(MatrixType& result, const MatrixType& T, int degree); 58c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath void computePade3(MatrixType& result, const MatrixType& T); 59c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath void computePade4(MatrixType& result, const MatrixType& T); 60c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath void computePade5(MatrixType& result, const MatrixType& T); 61c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath void computePade6(MatrixType& result, const MatrixType& T); 62c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath void computePade7(MatrixType& result, const MatrixType& T); 63c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath void computePade8(MatrixType& result, const MatrixType& T); 64c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath void computePade9(MatrixType& result, const MatrixType& T); 65c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath void computePade10(MatrixType& result, const MatrixType& T); 66c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath void computePade11(MatrixType& result, const MatrixType& T); 67c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 68c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath static const int minPadeDegree = 3; 697faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez static const int maxPadeDegree = std::numeric_limits<RealScalar>::digits<= 24? 5: // single precision 707faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez std::numeric_limits<RealScalar>::digits<= 53? 7: // double precision 717faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez std::numeric_limits<RealScalar>::digits<= 64? 8: // extended precision 727faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez std::numeric_limits<RealScalar>::digits<=106? 10: // double-double 737faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 11; // quadruple precision 74c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 75c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // Prevent copying 76c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixLogarithmAtomic(const MatrixLogarithmAtomic&); 77c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixLogarithmAtomic& operator=(const MatrixLogarithmAtomic&); 78c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath}; 79c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 80c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \brief Compute logarithm of triangular matrix with clustered eigenvalues. */ 81c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 82c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan KamathMatrixType MatrixLogarithmAtomic<MatrixType>::compute(const MatrixType& A) 83c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 84c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath using std::log; 85c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType result(A.rows(), A.rows()); 86c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (A.rows() == 1) 87c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result(0,0) = log(A(0,0)); 88c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath else if (A.rows() == 2) 89c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath compute2x2(A, result); 90c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath else 91c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath computeBig(A, result); 92c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return result; 93c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 94c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 95c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \brief Compute logarithm of 2x2 triangular matrix. */ 96c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 97c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathvoid MatrixLogarithmAtomic<MatrixType>::compute2x2(const MatrixType& A, MatrixType& result) 98c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 99c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath using std::abs; 100c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath using std::ceil; 101c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath using std::imag; 102c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath using std::log; 103c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 104c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Scalar logA00 = log(A(0,0)); 105c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Scalar logA11 = log(A(1,1)); 106c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 107c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result(0,0) = logA00; 108c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result(1,0) = Scalar(0); 109c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result(1,1) = logA11; 110c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 111c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (A(0,0) == A(1,1)) { 112c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result(0,1) = A(0,1) / A(0,0); 113c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } else if ((abs(A(0,0)) < 0.5*abs(A(1,1))) || (abs(A(0,0)) > 2*abs(A(1,1)))) { 114c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result(0,1) = A(0,1) * (logA11 - logA00) / (A(1,1) - A(0,0)); 115c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } else { 116c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath // computation in previous branch is inaccurate if A(1,1) \approx A(0,0) 117c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath int unwindingNumber = static_cast<int>(ceil((imag(logA11 - logA00) - M_PI) / (2*M_PI))); 1187faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Scalar y = A(1,1) - A(0,0), x = A(1,1) + A(0,0); 1197faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez result(0,1) = A(0,1) * (Scalar(2) * numext::atanh2(y,x) + Scalar(0,2*M_PI*unwindingNumber)) / y; 120c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 121c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 122c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 123c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \brief Compute logarithm of triangular matrices with size > 2. 124c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \details This uses a inverse scale-and-square algorithm. */ 125c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 126c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathvoid MatrixLogarithmAtomic<MatrixType>::computeBig(const MatrixType& A, MatrixType& result) 127c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 1287faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez using std::pow; 129c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath int numberOfSquareRoots = 0; 130c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath int numberOfExtraSquareRoots = 0; 131c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath int degree; 1327faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MatrixType T = A, sqrtT; 133c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar maxNormForPade = maxPadeDegree<= 5? 5.3149729967117310e-1: // single precision 134c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath maxPadeDegree<= 7? 2.6429608311114350e-1: // double precision 135c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath maxPadeDegree<= 8? 2.32777776523703892094e-1L: // extended precision 136c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath maxPadeDegree<=10? 1.05026503471351080481093652651105e-1L: // double-double 137c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 1.1880960220216759245467951592883642e-1L; // quadruple precision 138c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 139c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath while (true) { 140c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff(); 141c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (normTminusI < maxNormForPade) { 142c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath degree = getPadeDegree(normTminusI); 143c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath int degree2 = getPadeDegree(normTminusI / RealScalar(2)); 144c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1)) 1457faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 146c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath ++numberOfExtraSquareRoots; 147c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 148c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT); 1497faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez T = sqrtT.template triangularView<Upper>(); 150c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath ++numberOfSquareRoots; 151c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 152c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 153c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath computePade(result, T, degree); 154c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result *= pow(RealScalar(2), numberOfSquareRoots); 155c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 156c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 157c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = float) */ 158c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 159c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathint MatrixLogarithmAtomic<MatrixType>::getPadeDegree(float normTminusI) 160c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 161c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const float maxNormForPade[] = { 2.5111573934555054e-1 /* degree = 3 */ , 4.0535837411880493e-1, 162c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 5.3149729967117310e-1 }; 1637faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez int degree = 3; 1647faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (; degree <= maxPadeDegree; ++degree) 165c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (normTminusI <= maxNormForPade[degree - minPadeDegree]) 1667faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 1677faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez return degree; 168c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 169c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 170c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = double) */ 171c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 172c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathint MatrixLogarithmAtomic<MatrixType>::getPadeDegree(double normTminusI) 173c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 174c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const double maxNormForPade[] = { 1.6206284795015624e-2 /* degree = 3 */ , 5.3873532631381171e-2, 175c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 }; 1767faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez int degree = 3; 1777faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (; degree <= maxPadeDegree; ++degree) 178c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (normTminusI <= maxNormForPade[degree - minPadeDegree]) 1797faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 1807faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez return degree; 181c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 182c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 183c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = long double) */ 184c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 185c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathint MatrixLogarithmAtomic<MatrixType>::getPadeDegree(long double normTminusI) 186c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 187c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#if LDBL_MANT_DIG == 53 // double precision 1887faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const long double maxNormForPade[] = { 1.6206284795015624e-2L /* degree = 3 */ , 5.3873532631381171e-2L, 1897faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1.1352802267628681e-1L, 1.8662860613541288e-1L, 2.642960831111435e-1L }; 190c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#elif LDBL_MANT_DIG <= 64 // extended precision 1917faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const long double maxNormForPade[] = { 5.48256690357782863103e-3L /* degree = 3 */, 2.34559162387971167321e-2L, 1927faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 5.84603923897347449857e-2L, 1.08486423756725170223e-1L, 1.68385767881294446649e-1L, 1937faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2.32777776523703892094e-1L }; 194c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#elif LDBL_MANT_DIG <= 106 // double-double 1957faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const long double maxNormForPade[] = { 8.58970550342939562202529664318890e-5L /* degree = 3 */, 1967faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 9.34074328446359654039446552677759e-4L, 4.26117194647672175773064114582860e-3L, 1977faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1.21546224740281848743149666560464e-2L, 2.61100544998339436713088248557444e-2L, 1987faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4.66170074627052749243018566390567e-2L, 7.32585144444135027565872014932387e-2L, 1997faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1.05026503471351080481093652651105e-1L }; 200c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#else // quadruple precision 2017faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const long double maxNormForPade[] = { 4.7419931187193005048501568167858103e-5L /* degree = 3 */, 2027faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 5.8853168473544560470387769480192666e-4L, 2.9216120366601315391789493628113520e-3L, 2037faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 8.8415758124319434347116734705174308e-3L, 1.9850836029449446668518049562565291e-2L, 2047faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3.6688019729653446926585242192447447e-2L, 5.9290962294020186998954055264528393e-2L, 2057faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 8.6998436081634343903250580992127677e-2L, 1.1880960220216759245467951592883642e-1L }; 206c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#endif 2077faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez int degree = 3; 2087faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (; degree <= maxPadeDegree; ++degree) 209c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath if (normTminusI <= maxNormForPade[degree - minPadeDegree]) 2107faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 2117faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez return degree; 212c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 213c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 214c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/* \brief Compute Pade approximation to matrix logarithm */ 215c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 216c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathvoid MatrixLogarithmAtomic<MatrixType>::computePade(MatrixType& result, const MatrixType& T, int degree) 217c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 218c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath switch (degree) { 219c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath case 3: computePade3(result, T); break; 220c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath case 4: computePade4(result, T); break; 221c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath case 5: computePade5(result, T); break; 222c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath case 6: computePade6(result, T); break; 223c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath case 7: computePade7(result, T); break; 224c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath case 8: computePade8(result, T); break; 225c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath case 9: computePade9(result, T); break; 226c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath case 10: computePade10(result, T); break; 227c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath case 11: computePade11(result, T); break; 228c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath default: assert(false); // should never happen 229c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 230c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 231c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 232c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 233c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathvoid MatrixLogarithmAtomic<MatrixType>::computePade3(MatrixType& result, const MatrixType& T) 234c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 235c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const int degree = 3; 236c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar nodes[] = { 0.1127016653792583114820734600217600L, 0.5000000000000000000000000000000000L, 237c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.8872983346207416885179265399782400L }; 238c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar weights[] = { 0.2777777777777777777777777777777778L, 0.4444444444444444444444444444444444L, 239c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.2777777777777777777777777777777778L }; 2407faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert(degree <= maxPadeDegree); 241c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows()); 242c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result.setZero(T.rows(), T.rows()); 243c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for (int k = 0; k < degree; ++k) 244c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI) 245c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .template triangularView<Upper>().solve(TminusI); 246c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 247c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 248c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 249c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathvoid MatrixLogarithmAtomic<MatrixType>::computePade4(MatrixType& result, const MatrixType& T) 250c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 251c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const int degree = 4; 252c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar nodes[] = { 0.0694318442029737123880267555535953L, 0.3300094782075718675986671204483777L, 253c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.6699905217924281324013328795516223L, 0.9305681557970262876119732444464048L }; 254c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar weights[] = { 0.1739274225687269286865319746109997L, 0.3260725774312730713134680253890003L, 255c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.3260725774312730713134680253890003L, 0.1739274225687269286865319746109997L }; 2567faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert(degree <= maxPadeDegree); 257c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows()); 258c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result.setZero(T.rows(), T.rows()); 259c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for (int k = 0; k < degree; ++k) 260c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI) 261c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .template triangularView<Upper>().solve(TminusI); 262c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 263c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 264c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 265c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathvoid MatrixLogarithmAtomic<MatrixType>::computePade5(MatrixType& result, const MatrixType& T) 266c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 267c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const int degree = 5; 268c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar nodes[] = { 0.0469100770306680036011865608503035L, 0.2307653449471584544818427896498956L, 269c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.5000000000000000000000000000000000L, 0.7692346550528415455181572103501044L, 270c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.9530899229693319963988134391496965L }; 271c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar weights[] = { 0.1184634425280945437571320203599587L, 0.2393143352496832340206457574178191L, 272c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.2844444444444444444444444444444444L, 0.2393143352496832340206457574178191L, 273c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1184634425280945437571320203599587L }; 2747faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert(degree <= maxPadeDegree); 275c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows()); 276c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result.setZero(T.rows(), T.rows()); 277c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for (int k = 0; k < degree; ++k) 278c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI) 279c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .template triangularView<Upper>().solve(TminusI); 280c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 281c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 282c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 283c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathvoid MatrixLogarithmAtomic<MatrixType>::computePade6(MatrixType& result, const MatrixType& T) 284c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 285c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const int degree = 6; 286c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar nodes[] = { 0.0337652428984239860938492227530027L, 0.1693953067668677431693002024900473L, 287c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.3806904069584015456847491391596440L, 0.6193095930415984543152508608403560L, 2887faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 0.8306046932331322568306997975099527L, 0.9662347571015760139061507772469973L }; 289c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar weights[] = { 0.0856622461895851725201480710863665L, 0.1803807865240693037849167569188581L, 290c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.2339569672863455236949351719947755L, 0.2339569672863455236949351719947755L, 2917faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 0.1803807865240693037849167569188581L, 0.0856622461895851725201480710863665L }; 2927faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert(degree <= maxPadeDegree); 293c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows()); 294c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result.setZero(T.rows(), T.rows()); 295c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for (int k = 0; k < degree; ++k) 296c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI) 297c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .template triangularView<Upper>().solve(TminusI); 298c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 299c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 300c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 301c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathvoid MatrixLogarithmAtomic<MatrixType>::computePade7(MatrixType& result, const MatrixType& T) 302c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 303c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const int degree = 7; 304c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar nodes[] = { 0.0254460438286207377369051579760744L, 0.1292344072003027800680676133596058L, 305c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.2970774243113014165466967939615193L, 0.5000000000000000000000000000000000L, 306c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.7029225756886985834533032060384807L, 0.8707655927996972199319323866403942L, 307c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.9745539561713792622630948420239256L }; 308c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar weights[] = { 0.0647424830844348466353057163395410L, 0.1398526957446383339507338857118898L, 309c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1909150252525594724751848877444876L, 0.2089795918367346938775510204081633L, 310c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1909150252525594724751848877444876L, 0.1398526957446383339507338857118898L, 311c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.0647424830844348466353057163395410L }; 3127faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert(degree <= maxPadeDegree); 313c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows()); 314c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result.setZero(T.rows(), T.rows()); 315c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for (int k = 0; k < degree; ++k) 316c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI) 317c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .template triangularView<Upper>().solve(TminusI); 318c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 319c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 320c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 321c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathvoid MatrixLogarithmAtomic<MatrixType>::computePade8(MatrixType& result, const MatrixType& T) 322c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 323c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const int degree = 8; 324c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar nodes[] = { 0.0198550717512318841582195657152635L, 0.1016667612931866302042230317620848L, 325c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.2372337950418355070911304754053768L, 0.4082826787521750975302619288199080L, 326c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.5917173212478249024697380711800920L, 0.7627662049581644929088695245946232L, 327c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.8983332387068133697957769682379152L, 0.9801449282487681158417804342847365L }; 328c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar weights[] = { 0.0506142681451881295762656771549811L, 0.1111905172266872352721779972131204L, 329c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1568533229389436436689811009933007L, 0.1813418916891809914825752246385978L, 330c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1813418916891809914825752246385978L, 0.1568533229389436436689811009933007L, 331c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1111905172266872352721779972131204L, 0.0506142681451881295762656771549811L }; 3327faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert(degree <= maxPadeDegree); 333c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows()); 334c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result.setZero(T.rows(), T.rows()); 335c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for (int k = 0; k < degree; ++k) 336c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI) 337c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .template triangularView<Upper>().solve(TminusI); 338c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 339c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 340c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 341c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathvoid MatrixLogarithmAtomic<MatrixType>::computePade9(MatrixType& result, const MatrixType& T) 342c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 343c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const int degree = 9; 344c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar nodes[] = { 0.0159198802461869550822118985481636L, 0.0819844463366821028502851059651326L, 345c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1933142836497048013456489803292629L, 0.3378732882980955354807309926783317L, 346c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.5000000000000000000000000000000000L, 0.6621267117019044645192690073216683L, 347c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.8066857163502951986543510196707371L, 0.9180155536633178971497148940348674L, 348c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.9840801197538130449177881014518364L }; 349c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar weights[] = { 0.0406371941807872059859460790552618L, 0.0903240803474287020292360156214564L, 350c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1303053482014677311593714347093164L, 0.1561735385200014200343152032922218L, 351c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1651196775006298815822625346434870L, 0.1561735385200014200343152032922218L, 352c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1303053482014677311593714347093164L, 0.0903240803474287020292360156214564L, 353c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.0406371941807872059859460790552618L }; 3547faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert(degree <= maxPadeDegree); 355c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows()); 356c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result.setZero(T.rows(), T.rows()); 357c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for (int k = 0; k < degree; ++k) 358c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI) 359c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .template triangularView<Upper>().solve(TminusI); 360c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 361c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 362c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 363c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathvoid MatrixLogarithmAtomic<MatrixType>::computePade10(MatrixType& result, const MatrixType& T) 364c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 365c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const int degree = 10; 366c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar nodes[] = { 0.0130467357414141399610179939577740L, 0.0674683166555077446339516557882535L, 367c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1602952158504877968828363174425632L, 0.2833023029353764046003670284171079L, 368c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.4255628305091843945575869994351400L, 0.5744371694908156054424130005648600L, 369c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.7166976970646235953996329715828921L, 0.8397047841495122031171636825574368L, 370c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.9325316833444922553660483442117465L, 0.9869532642585858600389820060422260L }; 371c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar weights[] = { 0.0333356721543440687967844049466659L, 0.0747256745752902965728881698288487L, 372c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1095431812579910219977674671140816L, 0.1346333596549981775456134607847347L, 373c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1477621123573764350869464973256692L, 0.1477621123573764350869464973256692L, 374c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1346333596549981775456134607847347L, 0.1095431812579910219977674671140816L, 375c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.0747256745752902965728881698288487L, 0.0333356721543440687967844049466659L }; 3767faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert(degree <= maxPadeDegree); 377c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows()); 378c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result.setZero(T.rows(), T.rows()); 379c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for (int k = 0; k < degree; ++k) 380c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI) 381c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .template triangularView<Upper>().solve(TminusI); 382c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 383c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 384c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename MatrixType> 385c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathvoid MatrixLogarithmAtomic<MatrixType>::computePade11(MatrixType& result, const MatrixType& T) 386c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 387c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const int degree = 11; 388c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar nodes[] = { 0.0108856709269715035980309994385713L, 0.0564687001159523504624211153480364L, 389c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1349239972129753379532918739844233L, 0.2404519353965940920371371652706952L, 390c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.3652284220238275138342340072995692L, 0.5000000000000000000000000000000000L, 391c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.6347715779761724861657659927004308L, 0.7595480646034059079628628347293048L, 392c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.8650760027870246620467081260155767L, 0.9435312998840476495375788846519636L, 393c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.9891143290730284964019690005614287L }; 394c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const RealScalar weights[] = { 0.0278342835580868332413768602212743L, 0.0627901847324523123173471496119701L, 395c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.0931451054638671257130488207158280L, 0.1165968822959952399592618524215876L, 396c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1314022722551233310903444349452546L, 0.1364625433889503153572417641681711L, 397c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.1314022722551233310903444349452546L, 0.1165968822959952399592618524215876L, 398c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.0931451054638671257130488207158280L, 0.0627901847324523123173471496119701L, 399c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 0.0278342835580868332413768602212743L }; 4007faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert(degree <= maxPadeDegree); 401c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows()); 402c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result.setZero(T.rows(), T.rows()); 403c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath for (int k = 0; k < degree; ++k) 404c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI) 405c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath .template triangularView<Upper>().solve(TminusI); 406c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 407c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 408c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/** \ingroup MatrixFunctions_Module 409c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 410c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \brief Proxy for the matrix logarithm of some matrix (expression). 411c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 412c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \tparam Derived Type of the argument to the matrix function. 413c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 414c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * This class holds the argument to the matrix function until it is 415c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * assigned or evaluated for some other reason (so the argument 416c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * should not be changed in the meantime). It is the return type of 4177faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * MatrixBase::log() and most of the time this is the only way it 4187faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * is used. 419c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 420c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate<typename Derived> class MatrixLogarithmReturnValue 421c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath: public ReturnByValue<MatrixLogarithmReturnValue<Derived> > 422c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 423c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathpublic: 424c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 425c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename Derived::Scalar Scalar; 426c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename Derived::Index Index; 427c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 428c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \brief Constructor. 429c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 430c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param[in] A %Matrix (expression) forming the argument of the matrix logarithm. 431c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 432c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { } 433c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 434c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath /** \brief Compute the matrix logarithm. 435c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * 436c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath * \param[out] result Logarithm of \p A, where \A is as specified in the constructor. 437c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath */ 438c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath template <typename ResultType> 439c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath inline void evalTo(ResultType& result) const 440c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 441c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename Derived::PlainObject PlainObject; 442c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef internal::traits<PlainObject> Traits; 443c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath static const int RowsAtCompileTime = Traits::RowsAtCompileTime; 444c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath static const int ColsAtCompileTime = Traits::ColsAtCompileTime; 445c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath static const int Options = PlainObject::Options; 446c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar; 447c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef Matrix<ComplexScalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType; 448c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef MatrixLogarithmAtomic<DynMatrixType> AtomicType; 449c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath AtomicType atomic; 450c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 451c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath const PlainObject Aevaluated = m_A.eval(); 452c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixFunction<PlainObject, AtomicType> mf(Aevaluated, atomic); 453c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath mf.compute(result); 454c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath } 455c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 456c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Index rows() const { return m_A.rows(); } 457c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath Index cols() const { return m_A.cols(); } 458c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 459c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathprivate: 460c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typename internal::nested<Derived>::type m_A; 461c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 462c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath MatrixLogarithmReturnValue& operator=(const MatrixLogarithmReturnValue&); 463c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath}; 464c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 465c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathnamespace internal { 466c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath template<typename Derived> 467c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath struct traits<MatrixLogarithmReturnValue<Derived> > 468c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath { 469c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath typedef typename Derived::PlainObject ReturnType; 470c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath }; 471c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 472c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 473c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 474c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath/********** MatrixBase method **********/ 475c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 476c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 477c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathtemplate <typename Derived> 478c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamathconst MatrixLogarithmReturnValue<Derived> MatrixBase<Derived>::log() const 479c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath{ 480c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath eigen_assert(rows() == cols()); 481c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath return MatrixLogarithmReturnValue<Derived>(derived()); 482c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} 483c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 484c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath} // end namespace Eigen 485c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath 486c981c48f5bc9aefeffc0bcb0cc3934c2fae179ddNarayan Kamath#endif // EIGEN_MATRIX_LOGARITHM 487