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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
*
* Accoring 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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