MatrixBaseEigenvalues.h (5575B)
1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr> 5 // Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk> 6 // 7 // This Source Code Form is subject to the terms of the Mozilla 8 // Public License v. 2.0. If a copy of the MPL was not distributed 9 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 10 11 #ifndef EIGEN_MATRIXBASEEIGENVALUES_H 12 #define EIGEN_MATRIXBASEEIGENVALUES_H 13 14 namespace Eigen { 15 16 namespace internal { 17 18 template<typename Derived, bool IsComplex> 19 struct eigenvalues_selector 20 { 21 // this is the implementation for the case IsComplex = true 22 static inline typename MatrixBase<Derived>::EigenvaluesReturnType const 23 run(const MatrixBase<Derived>& m) 24 { 25 typedef typename Derived::PlainObject PlainObject; 26 PlainObject m_eval(m); 27 return ComplexEigenSolver<PlainObject>(m_eval, false).eigenvalues(); 28 } 29 }; 30 31 template<typename Derived> 32 struct eigenvalues_selector<Derived, false> 33 { 34 static inline typename MatrixBase<Derived>::EigenvaluesReturnType const 35 run(const MatrixBase<Derived>& m) 36 { 37 typedef typename Derived::PlainObject PlainObject; 38 PlainObject m_eval(m); 39 return EigenSolver<PlainObject>(m_eval, false).eigenvalues(); 40 } 41 }; 42 43 } // end namespace internal 44 45 /** \brief Computes the eigenvalues of a matrix 46 * \returns Column vector containing the eigenvalues. 47 * 48 * \eigenvalues_module 49 * This function computes the eigenvalues with the help of the EigenSolver 50 * class (for real matrices) or the ComplexEigenSolver class (for complex 51 * matrices). 52 * 53 * The eigenvalues are repeated according to their algebraic multiplicity, 54 * so there are as many eigenvalues as rows in the matrix. 55 * 56 * The SelfAdjointView class provides a better algorithm for selfadjoint 57 * matrices. 58 * 59 * Example: \include MatrixBase_eigenvalues.cpp 60 * Output: \verbinclude MatrixBase_eigenvalues.out 61 * 62 * \sa EigenSolver::eigenvalues(), ComplexEigenSolver::eigenvalues(), 63 * SelfAdjointView::eigenvalues() 64 */ 65 template<typename Derived> 66 inline typename MatrixBase<Derived>::EigenvaluesReturnType 67 MatrixBase<Derived>::eigenvalues() const 68 { 69 return internal::eigenvalues_selector<Derived, NumTraits<Scalar>::IsComplex>::run(derived()); 70 } 71 72 /** \brief Computes the eigenvalues of a matrix 73 * \returns Column vector containing the eigenvalues. 74 * 75 * \eigenvalues_module 76 * This function computes the eigenvalues with the help of the 77 * SelfAdjointEigenSolver class. The eigenvalues are repeated according to 78 * their algebraic multiplicity, so there are as many eigenvalues as rows in 79 * the matrix. 80 * 81 * Example: \include SelfAdjointView_eigenvalues.cpp 82 * Output: \verbinclude SelfAdjointView_eigenvalues.out 83 * 84 * \sa SelfAdjointEigenSolver::eigenvalues(), MatrixBase::eigenvalues() 85 */ 86 template<typename MatrixType, unsigned int UpLo> 87 EIGEN_DEVICE_FUNC inline typename SelfAdjointView<MatrixType, UpLo>::EigenvaluesReturnType 88 SelfAdjointView<MatrixType, UpLo>::eigenvalues() const 89 { 90 PlainObject thisAsMatrix(*this); 91 return SelfAdjointEigenSolver<PlainObject>(thisAsMatrix, false).eigenvalues(); 92 } 93 94 95 96 /** \brief Computes the L2 operator norm 97 * \returns Operator norm of the matrix. 98 * 99 * \eigenvalues_module 100 * This function computes the L2 operator norm of a matrix, which is also 101 * known as the spectral norm. The norm of a matrix \f$ A \f$ is defined to be 102 * \f[ \|A\|_2 = \max_x \frac{\|Ax\|_2}{\|x\|_2} \f] 103 * where the maximum is over all vectors and the norm on the right is the 104 * Euclidean vector norm. The norm equals the largest singular value, which is 105 * the square root of the largest eigenvalue of the positive semi-definite 106 * matrix \f$ A^*A \f$. 107 * 108 * The current implementation uses the eigenvalues of \f$ A^*A \f$, as computed 109 * by SelfAdjointView::eigenvalues(), to compute the operator norm of a 110 * matrix. The SelfAdjointView class provides a better algorithm for 111 * selfadjoint matrices. 112 * 113 * Example: \include MatrixBase_operatorNorm.cpp 114 * Output: \verbinclude MatrixBase_operatorNorm.out 115 * 116 * \sa SelfAdjointView::eigenvalues(), SelfAdjointView::operatorNorm() 117 */ 118 template<typename Derived> 119 inline typename MatrixBase<Derived>::RealScalar 120 MatrixBase<Derived>::operatorNorm() const 121 { 122 using std::sqrt; 123 typename Derived::PlainObject m_eval(derived()); 124 // FIXME if it is really guaranteed that the eigenvalues are already sorted, 125 // then we don't need to compute a maxCoeff() here, comparing the 1st and last ones is enough. 126 return sqrt((m_eval*m_eval.adjoint()) 127 .eval() 128 .template selfadjointView<Lower>() 129 .eigenvalues() 130 .maxCoeff() 131 ); 132 } 133 134 /** \brief Computes the L2 operator norm 135 * \returns Operator norm of the matrix. 136 * 137 * \eigenvalues_module 138 * This function computes the L2 operator norm of a self-adjoint matrix. For a 139 * self-adjoint matrix, the operator norm is the largest eigenvalue. 140 * 141 * The current implementation uses the eigenvalues of the matrix, as computed 142 * by eigenvalues(), to compute the operator norm of the matrix. 143 * 144 * Example: \include SelfAdjointView_operatorNorm.cpp 145 * Output: \verbinclude SelfAdjointView_operatorNorm.out 146 * 147 * \sa eigenvalues(), MatrixBase::operatorNorm() 148 */ 149 template<typename MatrixType, unsigned int UpLo> 150 EIGEN_DEVICE_FUNC inline typename SelfAdjointView<MatrixType, UpLo>::RealScalar 151 SelfAdjointView<MatrixType, UpLo>::operatorNorm() const 152 { 153 return eigenvalues().cwiseAbs().maxCoeff(); 154 } 155 156 } // end namespace Eigen 157 158 #endif