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 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
 // Copyright (C) 2008-2010 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_GENERALIZEDSELFADJOINTEIGENSOLVER_H
 #define EIGEN_GENERALIZEDSELFADJOINTEIGENSOLVER_H
 
 #include "./Tridiagonalization.h"
 
 namespace Eigen { 
 
 /** \eigenvalues_module \ingroup Eigenvalues_Module
   *
   *
   * \class GeneralizedSelfAdjointEigenSolver
   *
   * \brief Computes eigenvalues and eigenvectors of the generalized selfadjoint eigen problem
   *
   * \tparam _MatrixType the type of the matrix of which we are computing the
   * eigendecomposition; this is expected to be an instantiation of the Matrix
   * class template.
   *
   * This class solves the generalized eigenvalue problem
   * \f$ Av = \lambda Bv \f$. In this case, the matrix \f$ A \f$ should be
   * selfadjoint and the matrix \f$ B \f$ should be positive definite.
   *
   * Only the \b lower \b triangular \b part of the input matrix is referenced.
   *
   * Call the function compute() to compute the eigenvalues and eigenvectors of
   * a given matrix. Alternatively, you can use the
   * GeneralizedSelfAdjointEigenSolver(const MatrixType&, const MatrixType&, int)
   * constructor which computes the eigenvalues and eigenvectors at construction time.
   * Once the eigenvalue and eigenvectors are computed, they can be retrieved with the eigenvalues()
   * and eigenvectors() functions.
   *
   * The documentation for GeneralizedSelfAdjointEigenSolver(const MatrixType&, const MatrixType&, int)
   * contains an example of the typical use of this class.
   *
   * \sa class SelfAdjointEigenSolver, class EigenSolver, class ComplexEigenSolver
   */
 template<typename _MatrixType>
 class GeneralizedSelfAdjointEigenSolver : public SelfAdjointEigenSolver<_MatrixType>
 {
     typedef SelfAdjointEigenSolver<_MatrixType> Base;
   public:
 
     typedef _MatrixType MatrixType;
 
     /** \brief Default constructor for fixed-size matrices.
       *
       * The default constructor is useful in cases in which the user intends to
       * perform decompositions via compute(). This constructor
       * can only be used if \p _MatrixType is a fixed-size matrix; use
       * GeneralizedSelfAdjointEigenSolver(Index) for dynamic-size matrices.
       */
     GeneralizedSelfAdjointEigenSolver() : Base() {}
 
     /** \brief Constructor, pre-allocates memory for dynamic-size matrices.
       *
       * \param [in]  size  Positive integer, size of the matrix whose
       * eigenvalues and eigenvectors will be computed.
       *
       * This constructor is useful for dynamic-size matrices, when the user
       * intends to perform decompositions via compute(). The \p size
       * parameter is only used as a hint. It is not an error to give a wrong
       * \p size, but it may impair performance.
       *
       * \sa compute() for an example
       */
     explicit GeneralizedSelfAdjointEigenSolver(Index size)
         : Base(size)
     {}
 
     /** \brief Constructor; computes generalized eigendecomposition of given matrix pencil.
       *
       * \param[in]  matA  Selfadjoint matrix in matrix pencil.
       *                   Only the lower triangular part of the matrix is referenced.
       * \param[in]  matB  Positive-definite matrix in matrix pencil.
       *                   Only the lower triangular part of the matrix is referenced.
       * \param[in]  options A or-ed set of flags {#ComputeEigenvectors,#EigenvaluesOnly} | {#Ax_lBx,#ABx_lx,#BAx_lx}.
       *                     Default is #ComputeEigenvectors|#Ax_lBx.
       *
       * This constructor calls compute(const MatrixType&, const MatrixType&, int)
       * to compute the eigenvalues and (if requested) the eigenvectors of the
       * generalized eigenproblem \f$ Ax = \lambda B x \f$ with \a matA the
       * selfadjoint matrix \f$ A \f$ and \a matB the positive definite matrix
       * \f$ B \f$. Each eigenvector \f$ x \f$ satisfies the property
       * \f$ x^* B x = 1 \f$. The eigenvectors are computed if
       * \a options contains ComputeEigenvectors.
       *
       * In addition, the two following variants can be solved via \p options:
       * - \c ABx_lx: \f$ ABx = \lambda x \f$
       * - \c BAx_lx: \f$ BAx = \lambda x \f$
       *
       * Example: \include SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType2.cpp
       * Output: \verbinclude SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType2.out
       *
       * \sa compute(const MatrixType&, const MatrixType&, int)
       */
     GeneralizedSelfAdjointEigenSolver(const MatrixType& matA, const MatrixType& matB,
                                       int options = ComputeEigenvectors|Ax_lBx)
       : Base(matA.cols())
     {
       compute(matA, matB, options);
     }
 
     /** \brief Computes generalized eigendecomposition of given matrix pencil.
       *
       * \param[in]  matA  Selfadjoint matrix in matrix pencil.
       *                   Only the lower triangular part of the matrix is referenced.
       * \param[in]  matB  Positive-definite matrix in matrix pencil.
       *                   Only the lower triangular part of the matrix is referenced.
       * \param[in]  options A or-ed set of flags {#ComputeEigenvectors,#EigenvaluesOnly} | {#Ax_lBx,#ABx_lx,#BAx_lx}.
       *                     Default is #ComputeEigenvectors|#Ax_lBx.
       *
       * \returns    Reference to \c *this
       *
       * According to \p options, this function computes eigenvalues and (if requested)
       * the eigenvectors of one of the following three generalized eigenproblems:
       * - \c Ax_lBx: \f$ Ax = \lambda B x \f$
       * - \c ABx_lx: \f$ ABx = \lambda x \f$
       * - \c BAx_lx: \f$ BAx = \lambda x \f$
       * with \a matA the selfadjoint matrix \f$ A \f$ and \a matB the positive definite
       * matrix \f$ B \f$.
       * In addition, each eigenvector \f$ x \f$ satisfies the property \f$ x^* B x = 1 \f$.
       *
       * The eigenvalues() function can be used to retrieve
       * the eigenvalues. If \p options contains ComputeEigenvectors, then the
       * eigenvectors are also computed and can be retrieved by calling
       * eigenvectors().
       *
       * The implementation uses LLT to compute the Cholesky decomposition
       * \f$ B = LL^* \f$ and computes the classical eigendecomposition
       * of the selfadjoint matrix \f$ L^{-1} A (L^*)^{-1} \f$ if \p options contains Ax_lBx
       * and of \f$ L^{*} A L \f$ otherwise. This solves the
       * generalized eigenproblem, because any solution of the generalized
       * eigenproblem \f$ Ax = \lambda B x \f$ corresponds to a solution
       * \f$ L^{-1} A (L^*)^{-1} (L^* x) = \lambda (L^* x) \f$ of the
       * eigenproblem for \f$ L^{-1} A (L^*)^{-1} \f$. Similar statements
       * can be made for the two other variants.
       *
       * Example: \include SelfAdjointEigenSolver_compute_MatrixType2.cpp
       * Output: \verbinclude SelfAdjointEigenSolver_compute_MatrixType2.out
       *
       * \sa GeneralizedSelfAdjointEigenSolver(const MatrixType&, const MatrixType&, int)
       */
     GeneralizedSelfAdjointEigenSolver& compute(const MatrixType& matA, const MatrixType& matB,
                                                int options = ComputeEigenvectors|Ax_lBx);
 
   protected:
 
 };
 
 
 template<typename MatrixType>
 GeneralizedSelfAdjointEigenSolver<MatrixType>& GeneralizedSelfAdjointEigenSolver<MatrixType>::
 compute(const MatrixType& matA, const MatrixType& matB, int options)
 {
   eigen_assert(matA.cols()==matA.rows() && matB.rows()==matA.rows() && matB.cols()==matB.rows());
   eigen_assert((options&~(EigVecMask|GenEigMask))==0
           && (options&EigVecMask)!=EigVecMask
           && ((options&GenEigMask)==0 || (options&GenEigMask)==Ax_lBx
            || (options&GenEigMask)==ABx_lx || (options&GenEigMask)==BAx_lx)
           && "invalid option parameter");
 
   bool computeEigVecs = ((options&EigVecMask)==0) || ((options&EigVecMask)==ComputeEigenvectors);
 
   // Compute the cholesky decomposition of matB = L L' = U'U
   LLT<MatrixType> cholB(matB);
 
   int type = (options&GenEigMask);
   if(type==0)
     type = Ax_lBx;
 
   if(type==Ax_lBx)
   {
     // compute C = inv(L) A inv(L')
     MatrixType matC = matA.template selfadjointView<Lower>();
     cholB.matrixL().template solveInPlace<OnTheLeft>(matC);
     cholB.matrixU().template solveInPlace<OnTheRight>(matC);
 
     Base::compute(matC, computeEigVecs ? ComputeEigenvectors : EigenvaluesOnly );
 
     // transform back the eigen vectors: evecs = inv(U) * evecs
     if(computeEigVecs)
       cholB.matrixU().solveInPlace(Base::m_eivec);
   }
   else if(type==ABx_lx)
   {
     // compute C = L' A L
     MatrixType matC = matA.template selfadjointView<Lower>();
     matC = matC * cholB.matrixL();
     matC = cholB.matrixU() * matC;
 
     Base::compute(matC, computeEigVecs ? ComputeEigenvectors : EigenvaluesOnly);
 
     // transform back the eigen vectors: evecs = inv(U) * evecs
     if(computeEigVecs)
       cholB.matrixU().solveInPlace(Base::m_eivec);
   }
   else if(type==BAx_lx)
   {
     // compute C = L' A L
     MatrixType matC = matA.template selfadjointView<Lower>();
     matC = matC * cholB.matrixL();
     matC = cholB.matrixU() * matC;
 
     Base::compute(matC, computeEigVecs ? ComputeEigenvectors : EigenvaluesOnly);
 
     // transform back the eigen vectors: evecs = L * evecs
     if(computeEigVecs)
       Base::m_eivec = cholB.matrixL() * Base::m_eivec;
   }
 
   return *this;
 }
 
 } // end namespace Eigen
 
 #endif // EIGEN_GENERALIZEDSELFADJOINTEIGENSOLVER_H

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