Computes the generalized eigenvalues and eigenvectors of a pair of general matrices. More...

#include <GeneralizedEigenSolver.h>

Public Types
enum	{ RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }
typedef std::complex< RealScalar >	ComplexScalar
	Complex scalar type for MatrixType.
typedef Matrix< ComplexScalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 >	ComplexVectorType
	Type for vector of complex scalar values eigenvalues as returned by betas().
typedef CwiseBinaryOp < internal::scalar_quotient_op < ComplexScalar, Scalar > , ComplexVectorType, VectorType >	EigenvalueType
	Expression type for the eigenvalues as returned by eigenvalues().
typedef Matrix< ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime >	EigenvectorsType
	Type for matrix of eigenvectors as returned by eigenvectors().
typedef MatrixType::Index	Index
typedef _MatrixType	MatrixType
	Synonym for the template parameter `_MatrixType`.
typedef NumTraits< Scalar >::Real	RealScalar
typedef MatrixType::Scalar	Scalar
	Scalar type for matrices of type MatrixType.
typedef Matrix< Scalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 >	VectorType
	Type for vector of real scalar values eigenvalues as returned by betas().
Public Member Functions
ComplexVectorType	alphas () const
VectorType	betas () const
GeneralizedEigenSolver &	compute (const MatrixType &A, const MatrixType &B, bool computeEigenvectors=true)
	Computes generalized eigendecomposition of given matrix.
EigenvalueType	eigenvalues () const
	Returns an expression of the computed generalized eigenvalues.
	GeneralizedEigenSolver ()
	Default constructor.
	GeneralizedEigenSolver (Index size)
	Default constructor with memory preallocation.
	GeneralizedEigenSolver (const MatrixType &A, const MatrixType &B, bool computeEigenvectors=true)
	Constructor; computes the generalized eigendecomposition of given matrix pair.
ComputationInfo	info () const
GeneralizedEigenSolver &	setMaxIterations (Index maxIters)
Protected Types
typedef Matrix< Scalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 >	ColumnVectorType
Protected Attributes
ComplexVectorType	m_alphas
VectorType	m_betas
bool	m_eigenvectorsOk
MatrixType	m_eivec
bool	m_isInitialized
MatrixType	m_matS
RealQZ< MatrixType >	m_realQZ
ColumnVectorType	m_tmp

Detailed Description

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

Computes the generalized eigenvalues and eigenvectors of a pair of general matrices.

Template Parameters:

_MatrixType the type of the matrices of which we are computing the eigen-decomposition; this is expected to be an instantiation of the Matrix class template. Currently, only real matrices are supported.

The generalized eigenvalues and eigenvectors of a matrix pair $ A $ and $ B $ are scalars $\lambda$ and vectors $ v $ such that $Av = \lambda Bv$ . If $ D $ is a diagonal matrix with the eigenvalues on the diagonal, and $ V $ is a matrix with the eigenvectors as its columns, then $ A V = B V D $ . The matrix $ V $ is almost always invertible, in which case we have $A = B V D V^{-1}$ . This is called the generalized eigen-decomposition.

The generalized eigenvalues and eigenvectors of a matrix pair may be complex, even when the matrices are real. Moreover, the generalized eigenvalue might be infinite if the matrix B is singular. To workaround this difficulty, the eigenvalues are provided as a pair of complex $\alpha$ and real $\beta$ such that: $\lambda_i = \alpha_i / \beta_i$ . If $\beta_i$ is (nearly) zero, then one can consider the well defined left eigenvalue $\mu = \beta_i / \alpha_i$ such that: $\mu_i A v_i = B v_i$ , or even $\mu_i u_i^T A = u_i^T B$ where $ u_i $ is called the left eigenvector.

Call the function compute() to compute the generalized eigenvalues and eigenvectors of a given matrix pair. Alternatively, you can use the GeneralizedEigenSolver(const MatrixType&, const MatrixType&, bool) 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.

Here is an usage example of this class: Example:

Output:

See also:: MatrixBase::eigenvalues(), class ComplexEigenSolver, class SelfAdjointEigenSolver

Definition at line 57 of file GeneralizedEigenSolver.h.

Member Typedef Documentation

template<typename _MatrixType >

typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::GeneralizedEigenSolver< _MatrixType >::ColumnVectorType [protected]

Definition at line 274 of file GeneralizedEigenSolver.h.

template<typename _MatrixType >

typedef std::complex<RealScalar> Eigen::GeneralizedEigenSolver< _MatrixType >::ComplexScalar

Complex scalar type for MatrixType.

This is std::complex<Scalar> if Scalar is real (e.g., float or double) and just Scalar if Scalar is complex.

Definition at line 83 of file GeneralizedEigenSolver.h.

template<typename _MatrixType >

typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::GeneralizedEigenSolver< _MatrixType >::ComplexVectorType

Type for vector of complex scalar values eigenvalues as returned by betas().

This is a column vector with entries of type ComplexScalar. The length of the vector is the size of MatrixType.

Definition at line 97 of file GeneralizedEigenSolver.h.

template<typename _MatrixType >

typedef CwiseBinaryOp<internal::scalar_quotient_op<ComplexScalar,Scalar>,ComplexVectorType,VectorType> Eigen::GeneralizedEigenSolver< _MatrixType >::EigenvalueType

Expression type for the eigenvalues as returned by eigenvalues().

Definition at line 101 of file GeneralizedEigenSolver.h.

template<typename _MatrixType >

typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> Eigen::GeneralizedEigenSolver< _MatrixType >::EigenvectorsType

Type for matrix of eigenvectors as returned by eigenvectors().

This is a square matrix with entries of type ComplexScalar. The size is the same as the size of MatrixType.

Definition at line 108 of file GeneralizedEigenSolver.h.

template<typename _MatrixType >

typedef MatrixType::Index Eigen::GeneralizedEigenSolver< _MatrixType >::Index

Definition at line 75 of file GeneralizedEigenSolver.h.

template<typename _MatrixType >

typedef _MatrixType Eigen::GeneralizedEigenSolver< _MatrixType >::MatrixType

Synonym for the template parameter _MatrixType.

Definition at line 62 of file GeneralizedEigenSolver.h.

template<typename _MatrixType >

typedef NumTraits<Scalar>::Real Eigen::GeneralizedEigenSolver< _MatrixType >::RealScalar

Definition at line 74 of file GeneralizedEigenSolver.h.

template<typename _MatrixType >

typedef MatrixType::Scalar Eigen::GeneralizedEigenSolver< _MatrixType >::Scalar

Scalar type for matrices of type MatrixType.

Definition at line 73 of file GeneralizedEigenSolver.h.

template<typename _MatrixType >

typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::GeneralizedEigenSolver< _MatrixType >::VectorType

Type for vector of real scalar values eigenvalues as returned by betas().

This is a column vector with entries of type Scalar. The length of the vector is the size of MatrixType.

Definition at line 90 of file GeneralizedEigenSolver.h.

Member Enumeration Documentation

template<typename _MatrixType >

anonymous enum

Enumerator:

RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 64 of file GeneralizedEigenSolver.h.

Constructor & Destructor Documentation

template<typename _MatrixType >

Eigen::GeneralizedEigenSolver< _MatrixType >::GeneralizedEigenSolver ( ) [inline]

Default constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via EigenSolver::compute(const MatrixType&, bool).

See also:: compute() for an example.

Definition at line 117 of file GeneralizedEigenSolver.h.

template<typename _MatrixType >

Eigen::GeneralizedEigenSolver< _MatrixType >::GeneralizedEigenSolver ( Index size ) [inline]

Default constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also:: GeneralizedEigenSolver()

Definition at line 125 of file GeneralizedEigenSolver.h.

template<typename _MatrixType >

Eigen::GeneralizedEigenSolver< _MatrixType >::GeneralizedEigenSolver	(	const MatrixType &	A,
		const MatrixType &	B,
		bool	computeEigenvectors = `true`
	)		`[inline]`

Constructor; computes the generalized eigendecomposition of given matrix pair.

Parameters:

[in]	A	Square matrix whose eigendecomposition is to be computed.
[in]	B	Square matrix whose eigendecomposition is to be computed.
[in]	computeEigenvectors	If true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.

This constructor calls compute() to compute the generalized eigenvalues and eigenvectors.

See also:: compute()

Definition at line 148 of file GeneralizedEigenSolver.h.

Member Function Documentation

template<typename _MatrixType >

ComplexVectorType Eigen::GeneralizedEigenSolver< _MatrixType >::alphas ( ) const [inline]

Returns:: A const reference to the vectors containing the alpha values

This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j).

See also:: betas(), eigenvalues()

Definition at line 209 of file GeneralizedEigenSolver.h.

template<typename _MatrixType >

VectorType Eigen::GeneralizedEigenSolver< _MatrixType >::betas ( ) const [inline]

Returns:: A const reference to the vectors containing the beta values

This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j).

See also:: alphas(), eigenvalues()

Definition at line 220 of file GeneralizedEigenSolver.h.

template<typename MatrixType >

GeneralizedEigenSolver< MatrixType > & Eigen::GeneralizedEigenSolver< MatrixType >::compute	(	const MatrixType &	A,
		const MatrixType &	B,
		bool	computeEigenvectors = `true`
	)

Computes generalized eigendecomposition of given matrix.

Parameters:

[in]	A	Square matrix whose eigendecomposition is to be computed.
[in]	B	Square matrix whose eigendecomposition is to be computed.
[in]	computeEigenvectors	If true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.

Returns:: Reference to *this

This function computes the eigenvalues of the real matrix matrix. The eigenvalues() function can be used to retrieve them. If computeEigenvectors is true, then the eigenvectors are also computed and can be retrieved by calling eigenvectors().

The matrix is first reduced to real generalized Schur form using the RealQZ class. The generalized Schur decomposition is then used to compute the eigenvalues and eigenvectors.

The cost of the computation is dominated by the cost of the generalized Schur decomposition.

This method reuses of the allocated data in the GeneralizedEigenSolver object.

Definition at line 291 of file GeneralizedEigenSolver.h.

template<typename _MatrixType >

EigenvalueType Eigen::GeneralizedEigenSolver< _MatrixType >::eigenvalues ( ) const [inline]

Returns an expression of the computed generalized eigenvalues.

Returns:: An expression of the column vector containing the eigenvalues.

It is a shortcut for

 this->alphas().cwiseQuotient(this->betas());

Not that betas might contain zeros. It is therefore not recommended to use this function, but rather directly deal with the alphas and betas vectors.

Precondition:: Either the constructor GeneralizedEigenSolver(const MatrixType&,const MatrixType&,bool) or the member function compute(const MatrixType&,const MatrixType&,bool) has been called before.

The eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix. The eigenvalues are not sorted in any particular order.