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

#include <ComplexSchur.h>

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

typedef Matrix< ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime >	ComplexMatrixType
	Type for the matrices in the Schur decomposition. More...

typedef std::complex< RealScalar >	ComplexScalar
	Complex scalar type for `_MatrixType`. More...

typedef Eigen::Index	Index

typedef _MatrixType	MatrixType

typedef NumTraits< Scalar >::Real	RealScalar

typedef MatrixType::Scalar	Scalar
	Scalar type for matrices of type `_MatrixType`. More...

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

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

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

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

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

template<typename HessMatrixType , typename OrthMatrixType >
ComplexSchur &	computeFromHessenberg (const HessMatrixType &matrixH, const OrthMatrixType &matrixQ, bool computeU=true)
	Compute Schur decomposition from a given Hessenberg matrix. More...

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

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

const ComplexMatrixType &	matrixT () const
	Returns the triangular matrix in the Schur decomposition. More...

const ComplexMatrixType &	matrixU () const
	Returns the unitary matrix in the Schur decomposition. More...

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

Static Public Attributes
static const int	m_maxIterationsPerRow = 30
	Maximum number of iterations per row. More...

Protected Attributes
HessenbergDecomposition< MatrixType >	m_hess

ComputationInfo	m_info

bool	m_isInitialized

ComplexMatrixType	m_matT

ComplexMatrixType	m_matU

bool	m_matUisUptodate

Index	m_maxIters

Private Member Functions
ComplexScalar	computeShift (Index iu, Index iter)

void	reduceToTriangularForm (bool computeU)

bool	subdiagonalEntryIsNeglegible (Index i)

Friends
struct	internal::complex_schur_reduce_to_hessenberg< MatrixType, NumTraits< Scalar >::IsComplex >

Detailed Description

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

Performs a complex Schur decomposition of a real or complex square matrix.

\eigenvalues_module

Template Parameters

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

Given a real or complex square matrix A, this class computes the Schur decomposition: $ A = U T U^*$ where U is a unitary complex matrix, and T is a complex upper triangular matrix. The diagonal of the matrix T corresponds to the eigenvalues of the matrix A.

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

Note: This code is inspired from Jampack

See also: class RealSchur, class EigenSolver, class ComplexEigenSolver

Definition at line 51 of file ComplexSchur.h.

Member Typedef Documentation

◆ ComplexMatrixType

template<typename _MatrixType >

typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> Eigen::ComplexSchur< _MatrixType >::ComplexMatrixType

Type for the matrices in the Schur decomposition.

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

Definition at line 81 of file ComplexSchur.h.

◆ ComplexScalar

template<typename _MatrixType >

typedef std::complex<RealScalar> Eigen::ComplexSchur< _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 74 of file ComplexSchur.h.

◆ Index

template<typename _MatrixType >

typedef Eigen::Index Eigen::ComplexSchur< _MatrixType >::Index

Deprecated:: since Eigen 3.3

Definition at line 66 of file ComplexSchur.h.

◆ MatrixType

template<typename _MatrixType >

typedef _MatrixType Eigen::ComplexSchur< _MatrixType >::MatrixType

Definition at line 54 of file ComplexSchur.h.

◆ RealScalar

template<typename _MatrixType >

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

Definition at line 65 of file ComplexSchur.h.

◆ Scalar

template<typename _MatrixType >

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

Scalar type for matrices of type _MatrixType.

Definition at line 64 of file ComplexSchur.h.

Member Enumeration Documentation

◆ anonymous enum

template<typename _MatrixType >

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 55 of file ComplexSchur.h.

Constructor & Destructor Documentation

◆ ComplexSchur() [1/2]

template<typename _MatrixType >

Eigen::ComplexSchur< _MatrixType >::ComplexSchur ( 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 94 of file ComplexSchur.h.

◆ ComplexSchur() [2/2]

template<typename _MatrixType >

template<typename InputType >

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

inlineexplicit

Constructor; 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.

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

See also: matrixT() and matrixU() for examples.

Definition at line 113 of file ComplexSchur.h.

Member Function Documentation

◆ compute() [1/2]

template<typename _MatrixType >

template<typename InputType >

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

Definition at line 322 of file ComplexSchur.h.

◆ compute() [2/2]

template<typename _MatrixType >

template<typename InputType >

ComplexSchur& Eigen::ComplexSchur< _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 QR iterations with a single shift. The cost of computing the Schur decomposition depends on the number of iterations; as a rough guide, it may be taken on the number of iterations; as a rough guide, it may be taken to be $25n^3$ complex flops, or $10n^3$ complex flops if computeU is false.

Example:

MatrixXcf A = MatrixXcf::Random(4,4);
ComplexSchur<MatrixXcf> schur(4);
schur.compute(A);
cout << "The matrix T in the decomposition of A is:" << endl << schur.matrixT() << endl;
schur.compute(A.inverse());
cout << "The matrix T in the decomposition of A^(-1) is:" << endl << schur.matrixT() << endl;

Output:

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

◆ computeFromHessenberg() [1/2]

template<typename _MatrixType >

template<typename HessMatrixType , typename OrthMatrixType >

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

Definition at line 344 of file ComplexSchur.h.

◆ computeFromHessenberg() [2/2]

template<typename _MatrixType >

template<typename HessMatrixType , typename OrthMatrixType >

ComplexSchur& Eigen::ComplexSchur< _MatrixType >::computeFromHessenberg	(	const HessMatrixType &	matrixH,
		const OrthMatrixType &	matrixQ,
		bool	computeU = `true`
	)

Compute Schur decomposition from a given Hessenberg matrix.

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)

◆ computeShift()

template<typename MatrixType >

ComplexSchur< MatrixType >::ComplexScalar Eigen::ComplexSchur< MatrixType >::computeShift	(	Index	iu,
		Index	iter
	)

private

Compute the shift in the current QR iteration.

Definition at line 281 of file ComplexSchur.h.

◆ getMaxIterations()

template<typename _MatrixType >

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

inline

Returns the maximum number of iterations.

Definition at line 235 of file ComplexSchur.h.

◆ info()

template<typename _MatrixType >

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

inline

Reports whether previous computation was successful.

Returns: Success if computation was successful, NoConvergence otherwise.

Definition at line 217 of file ComplexSchur.h.

◆ matrixT()

template<typename _MatrixType >

const ComplexMatrixType& Eigen::ComplexSchur< _MatrixType >::matrixT ( ) const

inline

Returns the triangular matrix in the Schur decomposition.

Returns: A const reference to the matrix T.

It is assumed that either the constructor ComplexSchur(const MatrixType& matrix, bool computeU) or the member function compute(const MatrixType& matrix, bool computeU) has been called before to compute the Schur decomposition of a matrix.

Note that this function returns a plain square matrix. If you want to reference only the upper triangular part, use:

schur.matrixT().triangularView<Upper>()

Example:

MatrixXcf A = MatrixXcf::Random(4,4);
cout << "Here is a random 4x4 matrix, A:" << endl << A << endl << endl;
ComplexSchur<MatrixXcf> schurOfA(A, false); // false means do not compute U
cout << "The triangular matrix T is:" << endl << schurOfA.matrixT() << endl;

Output:

Definition at line 162 of file ComplexSchur.h.

◆ matrixU()

template<typename _MatrixType >

const ComplexMatrixType& Eigen::ComplexSchur< _MatrixType >::matrixU ( ) const

inline

Returns the unitary matrix in the Schur decomposition.

Returns: A const reference to the matrix U.

It is assumed that either the constructor ComplexSchur(const MatrixType& matrix, bool computeU) or the member function compute(const MatrixType& matrix, bool computeU) has been called before to compute the Schur decomposition of a matrix, and that computeU was set to true (the default value).

Example:

MatrixXcf A = MatrixXcf::Random(4,4);
cout << "Here is a random 4x4 matrix, A:" << endl << A << endl << endl;
ComplexSchur<MatrixXcf> schurOfA(A);
cout << "The unitary matrix U is:" << endl << schurOfA.matrixU() << endl;

Output:

Definition at line 138 of file ComplexSchur.h.

◆ reduceToTriangularForm()

template<typename MatrixType >

void Eigen::ComplexSchur< MatrixType >::reduceToTriangularForm ( bool computeU )

private

Definition at line 390 of file ComplexSchur.h.

◆ setMaxIterations()

template<typename _MatrixType >

ComplexSchur& Eigen::ComplexSchur< _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 228 of file ComplexSchur.h.

◆ subdiagonalEntryIsNeglegible()

template<typename MatrixType >

bool Eigen::ComplexSchur< MatrixType >::subdiagonalEntryIsNeglegible ( Index i )

inlineprivate

If m_matT(i+1,i) is neglegible in floating point arithmetic compared to m_matT(i,i) and m_matT(j,j), then set it to zero and return true, else return false.

Definition at line 266 of file ComplexSchur.h.

Friends And Related Function Documentation

◆ internal::complex_schur_reduce_to_hessenberg< MatrixType, NumTraits< Scalar >::IsComplex >

template<typename _MatrixType >

friend struct internal::complex_schur_reduce_to_hessenberg< MatrixType, NumTraits< Scalar >::IsComplex >

friend

Definition at line 259 of file ComplexSchur.h.

Member Data Documentation

◆ m_hess

template<typename _MatrixType >

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

protected

Definition at line 249 of file ComplexSchur.h.

◆ m_info

template<typename _MatrixType >

ComputationInfo Eigen::ComplexSchur< _MatrixType >::m_info

protected

Definition at line 250 of file ComplexSchur.h.

◆ m_isInitialized

template<typename _MatrixType >

bool Eigen::ComplexSchur< _MatrixType >::m_isInitialized

protected

Definition at line 251 of file ComplexSchur.h.

◆ m_matT

template<typename _MatrixType >

ComplexMatrixType Eigen::ComplexSchur< _MatrixType >::m_matT

protected

Definition at line 248 of file ComplexSchur.h.

◆ m_matU

template<typename _MatrixType >

ComplexMatrixType Eigen::ComplexSchur< _MatrixType >::m_matU

protected

Definition at line 248 of file ComplexSchur.h.

◆ m_matUisUptodate

template<typename _MatrixType >

bool Eigen::ComplexSchur< _MatrixType >::m_matUisUptodate

protected

Definition at line 252 of file ComplexSchur.h.

◆ m_maxIterationsPerRow

template<typename _MatrixType >

const int Eigen::ComplexSchur< _MatrixType >::m_maxIterationsPerRow = 30

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 30.

Definition at line 245 of file ComplexSchur.h.

◆ m_maxIters

template<typename _MatrixType >

Index Eigen::ComplexSchur< _MatrixType >::m_maxIters

protected

Definition at line 253 of file ComplexSchur.h.

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

ComplexSchur.h

Public Types

Public Member Functions

Static Public Attributes

Protected Attributes

Private Member Functions

Friends

Detailed Description

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

Member Typedef Documentation

◆ ComplexMatrixType

◆ ComplexScalar

◆ Index

◆ MatrixType

◆ RealScalar

◆ Scalar

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ ComplexSchur() [1/2]

◆ ComplexSchur() [2/2]

Member Function Documentation

◆ compute() [1/2]

◆ compute() [2/2]

◆ computeFromHessenberg() [1/2]

◆ computeFromHessenberg() [2/2]

◆ computeShift()

◆ getMaxIterations()

◆ info()

◆ matrixT()

◆ matrixU()

◆ reduceToTriangularForm()

◆ setMaxIterations()

◆ subdiagonalEntryIsNeglegible()

Friends And Related Function Documentation

◆ internal::complex_schur_reduce_to_hessenberg< MatrixType, NumTraits< Scalar >::IsComplex >

Member Data Documentation

◆ m_hess

◆ m_info

◆ m_isInitialized

◆ m_matT

◆ m_matU

◆ m_matUisUptodate

◆ m_maxIterationsPerRow

◆ m_maxIters

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