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 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
 // Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
 // Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
 //
 // This Source Code Form is subject to the terms of the Mozilla
 // Public License v. 2.0. If a copy of the MPL was not distributed
 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
 
 #ifndef EIGEN_MATRIXBASEEIGENVALUES_H
 #define EIGEN_MATRIXBASEEIGENVALUES_H
 
 namespace Eigen { 
 
 namespace internal {
 
 template<typename Derived, bool IsComplex>
 struct eigenvalues_selector
 {
   // this is the implementation for the case IsComplex = true
   static inline typename MatrixBase<Derived>::EigenvaluesReturnType const
   run(const MatrixBase<Derived>& m)
   {
     typedef typename Derived::PlainObject PlainObject;
     PlainObject m_eval(m);
     return ComplexEigenSolver<PlainObject>(m_eval, false).eigenvalues();
   }
 };
 
 template<typename Derived>
 struct eigenvalues_selector<Derived, false>
 {
   static inline typename MatrixBase<Derived>::EigenvaluesReturnType const
   run(const MatrixBase<Derived>& m)
   {
     typedef typename Derived::PlainObject PlainObject;
     PlainObject m_eval(m);
     return EigenSolver<PlainObject>(m_eval, false).eigenvalues();
   }
 };
 
 } // end namespace internal
 
 /** \brief Computes the eigenvalues of a matrix 
   * \returns Column vector containing the eigenvalues.
   *
   * \eigenvalues_module
   * This function computes the eigenvalues with the help of the EigenSolver
   * class (for real matrices) or the ComplexEigenSolver class (for complex
   * matrices). 
   *
   * The eigenvalues are repeated according to their algebraic multiplicity,
   * so there are as many eigenvalues as rows in the matrix.
   *
   * The SelfAdjointView class provides a better algorithm for selfadjoint
   * matrices.
   *
   * Example: \include MatrixBase_eigenvalues.cpp
   * Output: \verbinclude MatrixBase_eigenvalues.out
   *
   * \sa EigenSolver::eigenvalues(), ComplexEigenSolver::eigenvalues(),
   *     SelfAdjointView::eigenvalues()
   */
 template<typename Derived>
 inline typename MatrixBase<Derived>::EigenvaluesReturnType
 MatrixBase<Derived>::eigenvalues() const
 {
   return internal::eigenvalues_selector<Derived, NumTraits<Scalar>::IsComplex>::run(derived());
 }
 
 /** \brief Computes the eigenvalues of a matrix
   * \returns Column vector containing the eigenvalues.
   *
   * \eigenvalues_module
   * This function computes the eigenvalues with the help of the
   * SelfAdjointEigenSolver class.  The eigenvalues are repeated according to
   * their algebraic multiplicity, so there are as many eigenvalues as rows in
   * the matrix.
   *
   * Example: \include SelfAdjointView_eigenvalues.cpp
   * Output: \verbinclude SelfAdjointView_eigenvalues.out
   *
   * \sa SelfAdjointEigenSolver::eigenvalues(), MatrixBase::eigenvalues()
   */
 template<typename MatrixType, unsigned int UpLo> 
 EIGEN_DEVICE_FUNC inline typename SelfAdjointView<MatrixType, UpLo>::EigenvaluesReturnType
 SelfAdjointView<MatrixType, UpLo>::eigenvalues() const
 {
   PlainObject thisAsMatrix(*this);
   return SelfAdjointEigenSolver<PlainObject>(thisAsMatrix, false).eigenvalues();
 }
 
 
 
 /** \brief Computes the L2 operator norm
   * \returns Operator norm of the matrix.
   *
   * \eigenvalues_module
   * This function computes the L2 operator norm of a matrix, which is also
   * known as the spectral norm. The norm of a matrix \f$ A \f$ is defined to be
   * \f[ \|A\|_2 = \max_x \frac{\|Ax\|_2}{\|x\|_2} \f]
   * where the maximum is over all vectors and the norm on the right is the
   * Euclidean vector norm. The norm equals the largest singular value, which is
   * the square root of the largest eigenvalue of the positive semi-definite
   * matrix \f$ A^*A \f$.
   *
   * The current implementation uses the eigenvalues of \f$ A^*A \f$, as computed
   * by SelfAdjointView::eigenvalues(), to compute the operator norm of a
   * matrix.  The SelfAdjointView class provides a better algorithm for
   * selfadjoint matrices.
   *
   * Example: \include MatrixBase_operatorNorm.cpp
   * Output: \verbinclude MatrixBase_operatorNorm.out
   *
   * \sa SelfAdjointView::eigenvalues(), SelfAdjointView::operatorNorm()
   */
 template<typename Derived>
 inline typename MatrixBase<Derived>::RealScalar
 MatrixBase<Derived>::operatorNorm() const
 {
   using std::sqrt;
   typename Derived::PlainObject m_eval(derived());
   // FIXME if it is really guaranteed that the eigenvalues are already sorted,
   // then we don't need to compute a maxCoeff() here, comparing the 1st and last ones is enough.
   return sqrt((m_eval*m_eval.adjoint())
                  .eval()
          .template selfadjointView<Lower>()
          .eigenvalues()
          .maxCoeff()
          );
 }
 
 /** \brief Computes the L2 operator norm
   * \returns Operator norm of the matrix.
   *
   * \eigenvalues_module
   * This function computes the L2 operator norm of a self-adjoint matrix. For a
   * self-adjoint matrix, the operator norm is the largest eigenvalue.
   *
   * The current implementation uses the eigenvalues of the matrix, as computed
   * by eigenvalues(), to compute the operator norm of the matrix.
   *
   * Example: \include SelfAdjointView_operatorNorm.cpp
   * Output: \verbinclude SelfAdjointView_operatorNorm.out
   *
   * \sa eigenvalues(), MatrixBase::operatorNorm()
   */
 template<typename MatrixType, unsigned int UpLo>
 EIGEN_DEVICE_FUNC inline typename SelfAdjointView<MatrixType, UpLo>::RealScalar
 SelfAdjointView<MatrixType, UpLo>::operatorNorm() const
 {
   return eigenvalues().cwiseAbs().maxCoeff();
 }
 
 } // end namespace Eigen
 
 #endif

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