Computes eigenvalues and eigenvectors of the generalized selfadjoint eigen problem. More...

#include <GeneralizedSelfAdjointEigenSolver.h>

Inheritance diagram for Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >:

Public Types
typedef _MatrixType	MatrixType

Public Types inherited from Eigen::SelfAdjointEigenSolver< _MatrixType >
enum	{ Size = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }

typedef Matrix< Scalar, Size, Size, ColMajor, MaxColsAtCompileTime, MaxColsAtCompileTime >	EigenvectorsType

typedef Eigen::Index	Index

typedef _MatrixType	MatrixType

typedef NumTraits< Scalar >::Real	RealScalar
	Real scalar type for `_MatrixType`. More...

typedef internal::plain_col_type< MatrixType, RealScalar >::type	RealVectorType
	Type for vector of eigenvalues as returned by eigenvalues(). More...

typedef MatrixType::Scalar	Scalar
	Scalar type for matrices of type `_MatrixType`. More...

typedef TridiagonalizationType::SubDiagonalType	SubDiagonalType

typedef Tridiagonalization< MatrixType >	TridiagonalizationType

Public Member Functions
GeneralizedSelfAdjointEigenSolver &	compute (const MatrixType &matA, const MatrixType &matB, int options=ComputeEigenvectors\|Ax_lBx)
	Computes generalized eigendecomposition of given matrix pencil. More...

	GeneralizedSelfAdjointEigenSolver ()
	Default constructor for fixed-size matrices. More...

	GeneralizedSelfAdjointEigenSolver (const MatrixType &matA, const MatrixType &matB, int options=ComputeEigenvectors\|Ax_lBx)
	Constructor; computes generalized eigendecomposition of given matrix pencil. More...

	GeneralizedSelfAdjointEigenSolver (Index size)
	Constructor, pre-allocates memory for dynamic-size matrices. More...

Public Member Functions inherited from Eigen::SelfAdjointEigenSolver< _MatrixType >
template<typename InputType >
EIGEN_DEVICE_FUNC SelfAdjointEigenSolver< MatrixType > &	compute (const EigenBase< InputType > &a_matrix, int options)

template<typename InputType >
EIGEN_DEVICE_FUNC SelfAdjointEigenSolver &	compute (const EigenBase< InputType > &matrix, int options=ComputeEigenvectors)
	Computes eigendecomposition of given matrix. More...

EIGEN_DEVICE_FUNC SelfAdjointEigenSolver &	computeDirect (const MatrixType &matrix, int options=ComputeEigenvectors)
	Computes eigendecomposition of given matrix using a closed-form algorithm. More...

SelfAdjointEigenSolver &	computeFromTridiagonal (const RealVectorType &diag, const SubDiagonalType &subdiag, int options=ComputeEigenvectors)
	Computes the eigen decomposition from a tridiagonal symmetric matrix. More...

const EIGEN_DEVICE_FUNC RealVectorType &	eigenvalues () const
	Returns the eigenvalues of given matrix. More...

const EIGEN_DEVICE_FUNC EigenvectorsType &	eigenvectors () const
	Returns the eigenvectors of given matrix. More...

EIGEN_DEVICE_FUNC ComputationInfo	info () const
	Reports whether previous computation was successful. More...

EIGEN_DEVICE_FUNC MatrixType	operatorInverseSqrt () const
	Computes the inverse square root of the matrix. More...

EIGEN_DEVICE_FUNC MatrixType	operatorSqrt () const
	Computes the positive-definite square root of the matrix. More...

EIGEN_DEVICE_FUNC	SelfAdjointEigenSolver ()
	Default constructor for fixed-size matrices. More...

template<typename InputType >
EIGEN_DEVICE_FUNC	SelfAdjointEigenSolver (const EigenBase< InputType > &matrix, int options=ComputeEigenvectors)
	Constructor; computes eigendecomposition of given matrix. More...

EIGEN_DEVICE_FUNC	SelfAdjointEigenSolver (Index size)
	Constructor, pre-allocates memory for dynamic-size matrices. More...

Private Types
typedef SelfAdjointEigenSolver< _MatrixType >	Base

Additional Inherited Members
Static Public Attributes inherited from Eigen::SelfAdjointEigenSolver< _MatrixType >
static const int	m_maxIterations = 30
	Maximum number of iterations. More...

Static Protected Member Functions inherited from Eigen::SelfAdjointEigenSolver< _MatrixType >
static void	check_template_parameters ()

Protected Attributes inherited from Eigen::SelfAdjointEigenSolver< _MatrixType >
bool	m_eigenvectorsOk

RealVectorType	m_eivalues

EigenvectorsType	m_eivec

ComputationInfo	m_info

bool	m_isInitialized

TridiagonalizationType::SubDiagonalType	m_subdiag

Detailed Description

template<typename _MatrixType>
class Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >

Computes eigenvalues and eigenvectors of the generalized selfadjoint eigen problem.

\eigenvalues_module

Template Parameters

_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 $Av = \lambda Bv$ . In this case, the matrix $ A $ should be selfadjoint and the matrix $ B $ should be positive definite.

Only the lower triangular 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.

See also: class SelfAdjointEigenSolver, class EigenSolver, class ComplexEigenSolver

Definition at line 48 of file GeneralizedSelfAdjointEigenSolver.h.

Member Typedef Documentation

◆ Base

template<typename _MatrixType >

typedef SelfAdjointEigenSolver<_MatrixType> Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >::Base

private

Definition at line 50 of file GeneralizedSelfAdjointEigenSolver.h.

◆ MatrixType

template<typename _MatrixType >

typedef _MatrixType Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >::MatrixType

Definition at line 53 of file GeneralizedSelfAdjointEigenSolver.h.

Constructor & Destructor Documentation

◆ GeneralizedSelfAdjointEigenSolver() [1/3]

template<typename _MatrixType >

Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >::GeneralizedSelfAdjointEigenSolver ( )

inline

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 _MatrixType is a fixed-size matrix; use GeneralizedSelfAdjointEigenSolver(Index) for dynamic-size matrices.

Definition at line 62 of file GeneralizedSelfAdjointEigenSolver.h.

◆ GeneralizedSelfAdjointEigenSolver() [2/3]

template<typename _MatrixType >

Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >::GeneralizedSelfAdjointEigenSolver ( Index size )

inlineexplicit

Constructor, pre-allocates memory for dynamic-size matrices.

Parameters

[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 size parameter is only used as a hint. It is not an error to give a wrong size, but it may impair performance.

See also: compute() for an example

Definition at line 76 of file GeneralizedSelfAdjointEigenSolver.h.

◆ GeneralizedSelfAdjointEigenSolver() [3/3]

template<typename _MatrixType >

Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >::GeneralizedSelfAdjointEigenSolver	(	const MatrixType &	matA,
		const MatrixType &	matB,
		int	options = `ComputeEigenvectors\|Ax_lBx`
	)

inline

Constructor; computes generalized eigendecomposition of given matrix pencil.

Parameters

[in]	matA	Selfadjoint matrix in matrix pencil. Only the lower triangular part of the matrix is referenced.
[in]	matB	Positive-definite matrix in matrix pencil. Only the lower triangular part of the matrix is referenced.
[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 $Ax = \lambda B x$ with matA the selfadjoint matrix $ A $ and matB the positive definite matrix $ B $ . Each eigenvector $ x $ satisfies the property $ x^* B x = 1 $ . The eigenvectors are computed if options contains ComputeEigenvectors.

In addition, the two following variants can be solved via options:

ABx_lx: $ABx = \lambda x$
BAx_lx: $BAx = \lambda x$

Example:

Output:

See also: compute(const MatrixType&, const MatrixType&, int)

Definition at line 106 of file GeneralizedSelfAdjointEigenSolver.h.

Member Function Documentation

◆ compute()

template<typename MatrixType >

GeneralizedSelfAdjointEigenSolver< MatrixType > & Eigen::GeneralizedSelfAdjointEigenSolver< MatrixType >::compute	(	const MatrixType &	matA,
		const MatrixType &	matB,
		int	options = `ComputeEigenvectors\|Ax_lBx`
	)

Computes generalized eigendecomposition of given matrix pencil.

Parameters

[in]	matA	Selfadjoint matrix in matrix pencil. Only the lower triangular part of the matrix is referenced.
[in]	matB	Positive-definite matrix in matrix pencil. Only the lower triangular part of the matrix is referenced.
[in]	options	A or-ed set of flags {ComputeEigenvectors,EigenvaluesOnly} \| {Ax_lBx,ABx_lx,BAx_lx}. Default is ComputeEigenvectors\|Ax_lBx.

Returns: Reference to *this

Accoring to options, this function computes eigenvalues and (if requested) the eigenvectors of one of the following three generalized eigenproblems:

Ax_lBx: $Ax = \lambda B x$
ABx_lx: $ABx = \lambda x$
BAx_lx: $BAx = \lambda x$ with matA the selfadjoint matrix and matB the positive definite matrix . In addition, each eigenvector satisfies the property .

The eigenvalues() function can be used to retrieve the eigenvalues. If 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 $ B = LL^* $ and computes the classical eigendecomposition of the selfadjoint matrix $L^{-1} A (L^*)^{-1}$ if options contains Ax_lBx and of $L^{*} A L$ otherwise. This solves the generalized eigenproblem, because any solution of the generalized eigenproblem $Ax = \lambda B x$ corresponds to a solution $L^{-1} A (L^*)^{-1} (L^* x) = \lambda (L^* x)$ of the eigenproblem for $L^{-1} A (L^*)^{-1}$ . Similar statements can be made for the two other variants.