Performs a real QZ decomposition of a pair of square matrices. More...

#include <RealQZ.h>

Public Types
enum	{ RowsAtCompileTime = MatrixType::RowsAtCompileTime , ColsAtCompileTime = MatrixType::ColsAtCompileTime , Options = MatrixType::Options , MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime , MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }

typedef _MatrixType	MatrixType

typedef MatrixType::Scalar	Scalar

typedef std::complex< typename NumTraits< Scalar >::Real >	ComplexScalar

typedef Eigen::Index	Index

typedef Matrix< ComplexScalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 >	EigenvalueType

typedef Matrix< Scalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 >	ColumnVectorType

Public Member Functions
	RealQZ (Index size=RowsAtCompileTime==Dynamic ? 1 :RowsAtCompileTime)
	Default constructor. More...

	RealQZ (const MatrixType &A, const MatrixType &B, bool computeQZ=true)
	Constructor; computes real QZ decomposition of given matrices. More...

const MatrixType &	matrixQ () const
	Returns matrix Q in the QZ decomposition. More...

const MatrixType &	matrixZ () const
	Returns matrix Z in the QZ decomposition. More...

const MatrixType &	matrixS () const
	Returns matrix S in the QZ decomposition. More...

const MatrixType &	matrixT () const
	Returns matrix S in the QZ decomposition. More...

RealQZ &	compute (const MatrixType &A, const MatrixType &B, bool computeQZ=true)
	Computes QZ decomposition of given matrix. More...

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

Index	iterations () const
	Returns number of performed QR-like iterations. More...

RealQZ &	setMaxIterations (Index maxIters)

Private Types
typedef Matrix< Scalar, 3, 1 >	Vector3s

typedef Matrix< Scalar, 2, 1 >	Vector2s

typedef Matrix< Scalar, 2, 2 >	Matrix2s

typedef JacobiRotation< Scalar >	JRs

Private Member Functions
void	hessenbergTriangular ()

void	computeNorms ()

Index	findSmallSubdiagEntry (Index iu)

Index	findSmallDiagEntry (Index f, Index l)

void	splitOffTwoRows (Index i)

void	pushDownZero (Index z, Index f, Index l)

void	step (Index f, Index l, Index iter)

Private Attributes
MatrixType	m_S

MatrixType	m_T

MatrixType	m_Q

MatrixType	m_Z

Matrix< Scalar, Dynamic, 1 >	m_workspace

ComputationInfo	m_info

Index	m_maxIters

bool	m_isInitialized

bool	m_computeQZ

Scalar	m_normOfT

Scalar	m_normOfS

Index	m_global_iter

Detailed Description

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

Performs a real QZ decomposition of a pair of square matrices.

\eigenvalues_module

Template Parameters

_MatrixType the type of the matrix of which we are computing the real QZ decomposition; this is expected to be an instantiation of the Matrix class template.

Given a real square matrices A and B, this class computes the real QZ decomposition: $ A = Q S Z $ , $ B = Q T Z $ where Q and Z are real orthogonal matrixes, T is upper-triangular matrix, and S is upper quasi-triangular matrix. An orthogonal matrix is a matrix whose inverse is equal to its transpose, $U^{-1} = U^T$ . A quasi-triangular matrix is a block-triangular matrix whose diagonal consists of 1-by-1 blocks and 2-by-2 blocks where further reduction is impossible due to complex eigenvalues.

The eigenvalues of the pencil $ A - z B $ can be obtained from 1x1 and 2x2 blocks on the diagonals of S and T.

Call the function compute() to compute the real QZ decomposition of a given pair of matrices. Alternatively, you can use the RealQZ(const MatrixType& B, const MatrixType& B, bool computeQZ) constructor which computes the real QZ decomposition at construction time. Once the decomposition is computed, you can use the matrixS(), matrixT(), matrixQ() and matrixZ() functions to retrieve the matrices S, T, Q and Z in the decomposition. If computeQZ==false, some time is saved by not computing matrices Q and Z.

Example:

Output:

Note: The implementation is based on the algorithm in "Matrix Computations" by Gene H. Golub and Charles F. Van Loan, and a paper "An algorithm for generalized eigenvalue problems" by C.B.Moler and G.W.Stewart.

See also: class RealSchur, class ComplexSchur, class EigenSolver, class ComplexEigenSolver

Member Typedef Documentation

◆ ColumnVectorType

template<typename _MatrixType >

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

◆ ComplexScalar

template<typename _MatrixType >

typedef std::complex<typename NumTraits<Scalar>::Real> Eigen::RealQZ< _MatrixType >::ComplexScalar

◆ EigenvalueType

template<typename _MatrixType >

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

◆ Index

template<typename _MatrixType >

typedef Eigen::Index Eigen::RealQZ< _MatrixType >::Index

Deprecated:: since Eigen 3.3

◆ JRs

template<typename _MatrixType >

typedef JacobiRotation<Scalar> Eigen::RealQZ< _MatrixType >::JRs

private

◆ Matrix2s

template<typename _MatrixType >

typedef Matrix<Scalar,2,2> Eigen::RealQZ< _MatrixType >::Matrix2s

private

◆ MatrixType

template<typename _MatrixType >

typedef _MatrixType Eigen::RealQZ< _MatrixType >::MatrixType

◆ Scalar

template<typename _MatrixType >

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

◆ Vector2s

template<typename _MatrixType >

typedef Matrix<Scalar,2,1> Eigen::RealQZ< _MatrixType >::Vector2s

private

◆ Vector3s

template<typename _MatrixType >

typedef Matrix<Scalar,3,1> Eigen::RealQZ< _MatrixType >::Vector3s

private

Member Enumeration Documentation

◆ anonymous enum

template<typename _MatrixType >

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

Constructor & Destructor Documentation

◆ RealQZ() [1/2]

template<typename _MatrixType >

Eigen::RealQZ< _MatrixType >::RealQZ ( Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime )

inlineexplicit

Default constructor.

Parameters

[in] size Positive integer, size of the matrix whose QZ decomposition will be computed.

The default constructor is useful in cases in which 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.

◆ RealQZ() [2/2]

template<typename _MatrixType >

Eigen::RealQZ< _MatrixType >::RealQZ	(	const MatrixType &	A,
		const MatrixType &	B,
		bool	computeQZ = `true`
	)

inline

Constructor; computes real QZ decomposition of given matrices.

Parameters

[in]	A	Matrix A.
[in]	B	Matrix B.
[in]	computeQZ	If false, A and Z are not computed.

This constructor calls compute() to compute the QZ decomposition.

Member Function Documentation

◆ compute()

template<typename MatrixType >

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

Computes QZ decomposition of given matrix.

Parameters

[in]	A	Matrix A.
[in]	B	Matrix B.
[in]	computeQZ	If false, A and Z are not computed.

Returns: Reference to *this

◆ computeNorms()

template<typename MatrixType >

void Eigen::RealQZ< MatrixType >::computeNorms

inlineprivate

◆ findSmallDiagEntry()

template<typename MatrixType >

Index Eigen::RealQZ< MatrixType >::findSmallDiagEntry	(	Index	f,
		Index	l
	)

inlineprivate

◆ findSmallSubdiagEntry()

template<typename MatrixType >

Index Eigen::RealQZ< MatrixType >::findSmallSubdiagEntry ( Index iu )

inlineprivate

◆ hessenbergTriangular()

template<typename MatrixType >

void Eigen::RealQZ< MatrixType >::hessenbergTriangular

private

◆ info()

template<typename _MatrixType >

ComputationInfo Eigen::RealQZ< _MatrixType >::info ( ) const

inline

Reports whether previous computation was successful.

Returns: Success if computation was successful, NoConvergence otherwise.

◆ iterations()

template<typename _MatrixType >

Index Eigen::RealQZ< _MatrixType >::iterations ( ) const

inline

Returns number of performed QR-like iterations.

◆ matrixQ()

template<typename _MatrixType >

const MatrixType& Eigen::RealQZ< _MatrixType >::matrixQ ( ) const

inline

Returns matrix Q in the QZ decomposition.

Returns: A const reference to the matrix Q.

◆ matrixS()

template<typename _MatrixType >

const MatrixType& Eigen::RealQZ< _MatrixType >::matrixS ( ) const

inline

Returns matrix S in the QZ decomposition.

Returns: A const reference to the matrix S.

◆ matrixT()

template<typename _MatrixType >

const MatrixType& Eigen::RealQZ< _MatrixType >::matrixT ( ) const

inline

Returns matrix S in the QZ decomposition.

Returns: A const reference to the matrix S.

◆ matrixZ()

template<typename _MatrixType >

const MatrixType& Eigen::RealQZ< _MatrixType >::matrixZ ( ) const

inline

Returns matrix Z in the QZ decomposition.

Returns: A const reference to the matrix Z.

◆ pushDownZero()

template<typename MatrixType >

void Eigen::RealQZ< MatrixType >::pushDownZero	(	Index	z,
		Index	f,
		Index	l
	)

inlineprivate

◆ setMaxIterations()

template<typename _MatrixType >

RealQZ& Eigen::RealQZ< _MatrixType >::setMaxIterations ( Index maxIters )

inline

Sets the maximal number of iterations allowed to converge to one eigenvalue or decouple the problem.

◆ splitOffTwoRows()

template<typename MatrixType >

void Eigen::RealQZ< MatrixType >::splitOffTwoRows ( Index i )

inlineprivate

◆ step()

template<typename MatrixType >

void Eigen::RealQZ< MatrixType >::step	(	Index	f,
		Index	l,
		Index	iter
	)

inlineprivate

Member Data Documentation

◆ m_computeQZ

template<typename _MatrixType >

bool Eigen::RealQZ< _MatrixType >::m_computeQZ

private

◆ m_global_iter

template<typename _MatrixType >

Index Eigen::RealQZ< _MatrixType >::m_global_iter

private

◆ m_info

template<typename _MatrixType >

ComputationInfo Eigen::RealQZ< _MatrixType >::m_info

private

◆ m_isInitialized

template<typename _MatrixType >

bool Eigen::RealQZ< _MatrixType >::m_isInitialized

private

◆ m_maxIters

template<typename _MatrixType >

Index Eigen::RealQZ< _MatrixType >::m_maxIters

private

◆ m_normOfS

template<typename _MatrixType >

Scalar Eigen::RealQZ< _MatrixType >::m_normOfS

private

◆ m_normOfT

template<typename _MatrixType >

Scalar Eigen::RealQZ< _MatrixType >::m_normOfT

private

◆ m_Q

template<typename _MatrixType >

MatrixType Eigen::RealQZ< _MatrixType >::m_Q

private

◆ m_S

template<typename _MatrixType >

MatrixType Eigen::RealQZ< _MatrixType >::m_S

private

◆ m_T

template<typename _MatrixType >

MatrixType Eigen::RealQZ< _MatrixType >::m_T

private

◆ m_workspace

template<typename _MatrixType >

Matrix<Scalar,Dynamic,1> Eigen::RealQZ< _MatrixType >::m_workspace

private

◆ m_Z

template<typename _MatrixType >

MatrixType Eigen::RealQZ< _MatrixType >::m_Z

private

The documentation for this class was generated from the following file:

/home/runner/work/NDDEM/NDDEM/src/CoarseGraining/Eigen/src/Eigenvalues/RealQZ.h

Public Types

Public Member Functions

Private Types

Private Member Functions

Private Attributes

Detailed Description

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

Member Typedef Documentation

◆ ColumnVectorType

◆ ComplexScalar

◆ EigenvalueType

◆ Index

◆ JRs

◆ Matrix2s

◆ MatrixType

◆ Scalar

◆ Vector2s

◆ Vector3s

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ RealQZ() [1/2]

◆ RealQZ() [2/2]

Member Function Documentation

◆ compute()

◆ computeNorms()

◆ findSmallDiagEntry()

◆ findSmallSubdiagEntry()

◆ hessenbergTriangular()

◆ info()

◆ iterations()

◆ matrixQ()

◆ matrixS()

◆ matrixT()

◆ matrixZ()

◆ pushDownZero()

◆ setMaxIterations()

◆ splitOffTwoRows()

◆ step()

Member Data Documentation

◆ m_computeQZ

◆ m_global_iter

◆ m_info

◆ m_isInitialized

◆ m_maxIters

◆ m_normOfS

◆ m_normOfT

◆ m_Q

◆ m_S

◆ m_T

◆ m_workspace

◆ m_Z

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