\eigenvalues_module More...

#include </home/runner/work/allwpilib/allwpilib/wpimath/src/main/native/thirdparty/eigen/include/Eigen/src/Eigenvalues/RealSchur.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
	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...

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

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

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

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...

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

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

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

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

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

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

Detailed Description

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

\eigenvalues_module

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

Member Typedef Documentation

◆ ColumnVectorType

template<typename _MatrixType >

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

◆ ComplexScalar

template<typename _MatrixType >

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

◆ EigenvalueType

template<typename _MatrixType >

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

◆ Index

template<typename _MatrixType >

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

Deprecated:: since Eigen 3.3

◆ MatrixType

template<typename _MatrixType >

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

◆ Scalar

template<typename _MatrixType >

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

Member Enumeration Documentation

◆ anonymous enum

template<typename _MatrixType >

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

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.

◆ 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:

Output:

Member Function Documentation

◆ compute() [1/2]

template<typename _MatrixType >

template<typename InputType >

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

◆ compute() [2/2]

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:

Output:

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

◆ 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
	)

◆ getMaxIterations()

template<typename _MatrixType >

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

inline

Returns the maximum number of iterations.

◆ 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.

◆ 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

◆ 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

◆ 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.

Member Data Documentation

◆ 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.

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

/home/runner/work/allwpilib/allwpilib/wpimath/src/main/native/thirdparty/eigen/include/Eigen/src/Eigenvalues/RealSchur.h

Public Types

Public Member Functions

Static Public Attributes

Detailed Description

Member Typedef Documentation

◆ ColumnVectorType

◆ ComplexScalar

◆ EigenvalueType

◆ Index

◆ MatrixType

◆ Scalar

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]

◆ getMaxIterations()

◆ info()

◆ matrixT()

◆ matrixU()

◆ setMaxIterations()

Member Data Documentation

◆ m_maxIterationsPerRow