Performs a real Schur decomposition of a square matrix. More...

#include <RealSchur.h>

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

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

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

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

typedef Eigen::Index	Index

typedef _MatrixType	MatrixType

typedef MatrixType::Scalar	Scalar

Public Member Functions
template<typename InputType >
RealSchur &	compute (const EigenBase< InputType > &matrix, bool computeU=true)
	Computes Schur decomposition of given matrix. More...

template<typename InputType >
RealSchur< MatrixType > &	compute (const EigenBase< InputType > &matrix, bool computeU)

template<typename HessMatrixType , typename OrthMatrixType >
RealSchur &	computeFromHessenberg (const HessMatrixType &matrixH, const OrthMatrixType &matrixQ, bool computeU)
	Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T. More...

template<typename HessMatrixType , typename OrthMatrixType >
RealSchur< MatrixType > &	computeFromHessenberg (const HessMatrixType &matrixH, const OrthMatrixType &matrixQ, bool computeU)

Index	getMaxIterations ()
	Returns the maximum number of iterations. More...

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

const MatrixType &	matrixT () const
	Returns the quasi-triangular matrix in the Schur decomposition. More...

const MatrixType &	matrixU () const
	Returns the orthogonal matrix in the Schur decomposition. More...

	RealSchur (Index size=RowsAtCompileTime==Dynamic ? 1 :RowsAtCompileTime)
	Default constructor. More...

template<typename InputType >
	RealSchur (const EigenBase< InputType > &matrix, bool computeU=true)
	Constructor; computes real Schur decomposition of given matrix. More...

RealSchur &	setMaxIterations (Index maxIters)
	Sets the maximum number of iterations allowed. More...

Static Public Attributes
static const int	m_maxIterationsPerRow = 40
	Maximum number of iterations per row. More...

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

Private Member Functions
Scalar	computeNormOfT ()

void	computeShift (Index iu, Index iter, Scalar &exshift, Vector3s &shiftInfo)

Index	findSmallSubdiagEntry (Index iu, const Scalar &considerAsZero)

void	initFrancisQRStep (Index il, Index iu, const Vector3s &shiftInfo, Index &im, Vector3s &firstHouseholderVector)

void	performFrancisQRStep (Index il, Index im, Index iu, bool computeU, const Vector3s &firstHouseholderVector, Scalar *workspace)

void	splitOffTwoRows (Index iu, bool computeU, const Scalar &exshift)

Private Attributes
HessenbergDecomposition< MatrixType >	m_hess

ComputationInfo	m_info

bool	m_isInitialized

MatrixType	m_matT

MatrixType	m_matU

bool	m_matUisUptodate

Index	m_maxIters

ColumnVectorType	m_workspaceVector

Detailed Description

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

Performs a real Schur decomposition of a square matrix.

Template Parameters

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

Given a real square matrix A, this class computes the real Schur decomposition: $ A = U T U^T $ where U is a real orthogonal matrix and T is a real 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 with complex eigenvalues. The eigenvalues of the blocks on the diagonal of T are the same as the eigenvalues of the matrix A, and thus the real Schur decomposition is used in EigenSolver to compute the eigendecomposition of a matrix.

Call the function compute() to compute the real Schur decomposition of a given matrix. Alternatively, you can use the RealSchur(const MatrixType&, bool) constructor which computes the real Schur decomposition at construction time. Once the decomposition is computed, you can use the matrixU() and matrixT() functions to retrieve the matrices U and T in the decomposition.

The documentation of RealSchur(const MatrixType&, bool) contains an example of the typical use of this class.

Note: The implementation is adapted from JAMA (public domain). Their code is based on EISPACK.

See also: class ComplexSchur, class EigenSolver, class ComplexEigenSolver

Definition at line 54 of file RealSchur.h.

Member Typedef Documentation

◆ ColumnVectorType

template<typename _MatrixType>

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

Definition at line 70 of file RealSchur.h.

◆ ComplexScalar

template<typename _MatrixType>

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

Definition at line 66 of file RealSchur.h.

◆ EigenvalueType

template<typename _MatrixType>

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

Definition at line 69 of file RealSchur.h.

◆ Index

template<typename _MatrixType>

typedef Eigen::Index Eigen::RealSchur< _MatrixType >::Index

Deprecated:: since Eigen 3.3

Definition at line 67 of file RealSchur.h.

◆ MatrixType

template<typename _MatrixType>

typedef _MatrixType Eigen::RealSchur< _MatrixType >::MatrixType

Definition at line 57 of file RealSchur.h.

◆ Scalar

template<typename _MatrixType>

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

Definition at line 65 of file RealSchur.h.

◆ Vector3s

template<typename _MatrixType>

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

private

Definition at line 236 of file RealSchur.h.

Member Enumeration Documentation

◆ anonymous enum

template<typename _MatrixType>

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 58 of file RealSchur.h.

Constructor & Destructor Documentation

◆ RealSchur() [1/2]

template<typename _MatrixType>

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

inlineexplicit

Default constructor.

Parameters

[in] size Positive integer, size of the matrix whose Schur 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.

Definition at line 83 of file RealSchur.h.

◆ RealSchur() [2/2]

template<typename _MatrixType>

template<typename InputType >

Eigen::RealSchur< _MatrixType >::RealSchur	(	const EigenBase< InputType > &	matrix,
		bool	computeU = `true`
	)

inlineexplicit

Constructor; computes real Schur decomposition of given matrix.

Parameters

[in]	matrix	Square matrix whose Schur decomposition is to be computed.
[in]	computeU	If true, both T and U are computed; if false, only T is computed.

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

Example:

MatrixXd A = MatrixXd::Random(6,6);
cout << "Here is a random 6x6 matrix, A:" << endl << A << endl << endl;
RealSchur<MatrixXd> schur(A);
cout << "The orthogonal matrix U is:" << endl << schur.matrixU() << endl;
cout << "The quasi-triangular matrix T is:" << endl << schur.matrixT() << endl << endl;
MatrixXd U = schur.matrixU();
MatrixXd T = schur.matrixT();
cout << "U * T * U^T = " << endl << U * T * U.transpose() << endl;

Output:

Definition at line 104 of file RealSchur.h.

Member Function Documentation

◆ compute() [1/2]

template<typename _MatrixType>

template<typename InputType >

RealSchur& Eigen::RealSchur< _MatrixType >::compute	(	const EigenBase< InputType > &	matrix,
		bool	computeU = `true`
	)

Computes Schur decomposition of given matrix.

Parameters

[in]	matrix	Square matrix whose Schur decomposition is to be computed.
[in]	computeU	If true, both T and U are computed; if false, only T is computed.

Returns: Reference to *this

The Schur decomposition is computed by first reducing the matrix to Hessenberg form using the class HessenbergDecomposition. The Hessenberg matrix is then reduced to triangular form by performing Francis QR iterations with implicit double shift. The cost of computing the Schur decomposition depends on the number of iterations; as a rough guide, it may be taken to be $25n^3$ flops if computeU is true and $10n^3$ flops if computeU is false.

Example:

MatrixXf A = MatrixXf::Random(4,4);
RealSchur<MatrixXf> schur(4);
schur.compute(A, /* computeU = */ false);
cout << "The matrix T in the decomposition of A is:" << endl << schur.matrixT() << endl;
schur.compute(A.inverse(), /* computeU = */ false);
cout << "The matrix T in the decomposition of A^(-1) is:" << endl << schur.matrixT() << endl;

Output:

See also: compute(const MatrixType&, bool, Index)

◆ compute() [2/2]

template<typename _MatrixType>

template<typename InputType >

RealSchur<MatrixType>& Eigen::RealSchur< _MatrixType >::compute	(	const EigenBase< InputType > &	matrix,
		bool	computeU
	)

Definition at line 249 of file RealSchur.h.

◆ computeFromHessenberg() [1/2]

template<typename _MatrixType>

template<typename HessMatrixType , typename OrthMatrixType >

RealSchur& Eigen::RealSchur< _MatrixType >::computeFromHessenberg	(	const HessMatrixType &	matrixH,
		const OrthMatrixType &	matrixQ,
		bool	computeU
	)

Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T.

Parameters

[in]	matrixH	Matrix in Hessenberg form H
[in]	matrixQ	orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T
	computeU	Computes the matriX U of the Schur vectors

Returns: Reference to *this

This routine assumes that the matrix is already reduced in Hessenberg form matrixH using either the class HessenbergDecomposition or another mean. It computes the upper quasi-triangular matrix T of the Schur decomposition of H When computeU is true, this routine computes the matrix U such that A = U T U^T = (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix

NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix is not available, the user should give an identity matrix (Q.setIdentity())

See also: compute(const MatrixType&, bool)

◆ computeFromHessenberg() [2/2]

template<typename _MatrixType>

template<typename HessMatrixType , typename OrthMatrixType >

RealSchur<MatrixType>& Eigen::RealSchur< _MatrixType >::computeFromHessenberg	(	const HessMatrixType &	matrixH,
		const OrthMatrixType &	matrixQ,
		bool	computeU
	)

Definition at line 287 of file RealSchur.h.

◆ computeNormOfT()

template<typename MatrixType >

MatrixType::Scalar Eigen::RealSchur< MatrixType >::computeNormOfT ( )

inlineprivate

Definition at line 362 of file RealSchur.h.

◆ computeShift()

template<typename MatrixType >

void Eigen::RealSchur< MatrixType >::computeShift	(	Index	iu,
		Index	iter,
		Scalar &	exshift,
		Vector3s &	shiftInfo
	)

inlineprivate

Definition at line 430 of file RealSchur.h.

◆ findSmallSubdiagEntry()

template<typename MatrixType >

Index Eigen::RealSchur< MatrixType >::findSmallSubdiagEntry	(	Index	iu,
		const Scalar &	considerAsZero
	)

inlineprivate

Definition at line 376 of file RealSchur.h.

◆ getMaxIterations()

template<typename _MatrixType>

Index Eigen::RealSchur< _MatrixType >::getMaxIterations ( )

inline

Returns the maximum number of iterations.

Definition at line 213 of file RealSchur.h.

◆ info()

template<typename _MatrixType>

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

inline

Reports whether previous computation was successful.

Returns: Success if computation was successful, NoConvergence otherwise.

Definition at line 195 of file RealSchur.h.

◆ initFrancisQRStep()

template<typename MatrixType >

void Eigen::RealSchur< MatrixType >::initFrancisQRStep	(	Index	il,
		Index	iu,
		const Vector3s &	shiftInfo,
		Index &	im,
		Vector3s &	firstHouseholderVector
	)

inlineprivate

Definition at line 472 of file RealSchur.h.

◆ matrixT()

template<typename _MatrixType>

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

inline

Returns the quasi-triangular matrix in the Schur decomposition.

Returns: A const reference to the matrix T.

Precondition: Either the constructor RealSchur(const MatrixType&, bool) or the member function compute(const MatrixType&, bool) has been called before to compute the Schur decomposition of a matrix.

See also: RealSchur(const MatrixType&, bool) for an example

Definition at line 144 of file RealSchur.h.

◆ matrixU()

template<typename _MatrixType>

const MatrixType& Eigen::RealSchur< _MatrixType >::matrixU ( ) const

inline

Returns the orthogonal matrix in the Schur decomposition.

Returns: A const reference to the matrix U.

Precondition: Either the constructor RealSchur(const MatrixType&, bool) or the member function compute(const MatrixType&, bool) has been called before to compute the Schur decomposition of a matrix, and computeU was set to true (the default value).

See also: RealSchur(const MatrixType&, bool) for an example

Definition at line 127 of file RealSchur.h.

◆ performFrancisQRStep()

template<typename MatrixType >

void Eigen::RealSchur< MatrixType >::performFrancisQRStep	(	Index	il,
		Index	im,
		Index	iu,
		bool	computeU,
		const Vector3s &	firstHouseholderVector,
		Scalar *	workspace
	)

inlineprivate

Definition at line 497 of file RealSchur.h.

◆ setMaxIterations()

template<typename _MatrixType>

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

inline

Sets the maximum number of iterations allowed.

If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size of the matrix.

Definition at line 206 of file RealSchur.h.

◆ splitOffTwoRows()

template<typename MatrixType >

void Eigen::RealSchur< MatrixType >::splitOffTwoRows	(	Index	iu,
		bool	computeU,
		const Scalar &	exshift
	)

inlineprivate

Definition at line 395 of file RealSchur.h.

Member Data Documentation

◆ m_hess

template<typename _MatrixType>

HessenbergDecomposition<MatrixType> Eigen::RealSchur< _MatrixType >::m_hess

private

Definition at line 230 of file RealSchur.h.

◆ m_info

template<typename _MatrixType>

ComputationInfo Eigen::RealSchur< _MatrixType >::m_info

private

Definition at line 231 of file RealSchur.h.

◆ m_isInitialized

template<typename _MatrixType>

bool Eigen::RealSchur< _MatrixType >::m_isInitialized

private

Definition at line 232 of file RealSchur.h.

◆ m_matT

template<typename _MatrixType>

MatrixType Eigen::RealSchur< _MatrixType >::m_matT

private

Definition at line 227 of file RealSchur.h.

◆ m_matU

template<typename _MatrixType>

MatrixType Eigen::RealSchur< _MatrixType >::m_matU

private

Definition at line 228 of file RealSchur.h.

◆ m_matUisUptodate

template<typename _MatrixType>

bool Eigen::RealSchur< _MatrixType >::m_matUisUptodate

private

Definition at line 233 of file RealSchur.h.

◆ m_maxIterationsPerRow

template<typename _MatrixType>

const int Eigen::RealSchur< _MatrixType >::m_maxIterationsPerRow = 40

static

Maximum number of iterations per row.

If not otherwise specified, the maximum number of iterations is this number times the size of the matrix. It is currently set to 40.

Definition at line 223 of file RealSchur.h.

◆ m_maxIters

template<typename _MatrixType>

Index Eigen::RealSchur< _MatrixType >::m_maxIters

private

Definition at line 234 of file RealSchur.h.

◆ m_workspaceVector

template<typename _MatrixType>

ColumnVectorType Eigen::RealSchur< _MatrixType >::m_workspaceVector

private

Definition at line 229 of file RealSchur.h.

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

RealSchur.h

Public Types

Public Member Functions

Static Public Attributes

Private Types

Private Member Functions

Private Attributes

Detailed Description

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

Member Typedef Documentation

◆ ColumnVectorType

◆ ComplexScalar

◆ EigenvalueType

◆ Index

◆ MatrixType

◆ Scalar

◆ Vector3s

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ RealSchur() [1/2]

◆ RealSchur() [2/2]

Member Function Documentation

◆ compute() [1/2]

◆ compute() [2/2]

◆ computeFromHessenberg() [1/2]

◆ computeFromHessenberg() [2/2]

◆ computeNormOfT()

◆ computeShift()

◆ findSmallSubdiagEntry()

◆ getMaxIterations()

◆ info()

◆ initFrancisQRStep()

◆ matrixT()

◆ matrixU()

◆ performFrancisQRStep()

◆ setMaxIterations()

◆ splitOffTwoRows()

Member Data Documentation

◆ m_hess

◆ m_info

◆ m_isInitialized

◆ m_matT

◆ m_matU

◆ m_matUisUptodate

◆ m_maxIterationsPerRow

◆ m_maxIters

◆ m_workspaceVector

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