\eigenvalues_module 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.

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

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

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

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

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

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

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

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

RealQZ &	setMaxIterations (Index maxIters)
	Sets the maximal number of iterations allowed to converge to one eigenvalue or decouple the problem.

Detailed Description

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

\eigenvalues_module

Performs a real QZ decomposition of a pair of square matrices

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

◆ Index

template<typename _MatrixType >

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

Deprecated:: since Eigen 3.3

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

◆ info()

template<typename _MatrixType >

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

inline

Reports whether previous computation was successful.

Returns: Success if computation was succesful, NoConvergence otherwise.

◆ 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.

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

External/Eigen/src/Eigenvalues/RealQZ.h

Public Types

Public Member Functions

Detailed Description

Member Typedef Documentation

◆ Index

Constructor & Destructor Documentation

◆ RealQZ() [1/2]

◆ RealQZ() [2/2]

Member Function Documentation

◆ compute()

◆ info()

◆ matrixQ()

◆ matrixS()

◆ matrixT()

◆ matrixZ()