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.
typedef std::complex< RealScalar >	ComplexScalar
	Complex scalar type for `_MatrixType`.
typedef MatrixType::Index	Index
typedef _MatrixType	MatrixType
typedef NumTraits< Scalar >::Real	RealScalar
typedef MatrixType::Scalar	Scalar
	Scalar type for matrices of type `_MatrixType`.
Public Member Functions
	ComplexSchur (Index size=RowsAtCompileTime==Dynamic?1:RowsAtCompileTime)
	Default constructor.
	ComplexSchur (const MatrixType &matrix, bool computeU=true)
	Constructor; computes Schur decomposition of given matrix.
ComplexSchur &	compute (const MatrixType &matrix, bool computeU=true)
	Computes Schur decomposition of given matrix.
ComputationInfo	info () const
	Reports whether previous computation was successful.
const ComplexMatrixType &	matrixT () const
	Returns the triangular matrix in the Schur decomposition.
const ComplexMatrixType &	matrixU () const
	Returns the unitary matrix in the Schur decomposition.
Static Public Attributes
static const int	m_maxIterations = 30
	Maximum number of iterations.
Protected Attributes
HessenbergDecomposition < MatrixType >	m_hess
ComputationInfo	m_info
bool	m_isInitialized
ComplexMatrixType	m_matT
ComplexMatrixType	m_matU
bool	m_matUisUptodate
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 ComplexSchur< _MatrixType >

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

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 65 of file ComplexSchur.h.

Member Typedef Documentation

template<typename _MatrixType>

typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> 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 95 of file ComplexSchur.h.

template<typename _MatrixType>

typedef std::complex<RealScalar> 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 88 of file ComplexSchur.h.

template<typename _MatrixType>

typedef MatrixType::Index ComplexSchur< _MatrixType >::Index

Definition at line 80 of file ComplexSchur.h.

template<typename _MatrixType>

typedef _MatrixType ComplexSchur< _MatrixType >::MatrixType

Definition at line 68 of file ComplexSchur.h.

template<typename _MatrixType>

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

Definition at line 79 of file ComplexSchur.h.

template<typename _MatrixType>

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

Scalar type for matrices of type _MatrixType.

Definition at line 78 of file ComplexSchur.h.

Member Enumeration Documentation

template<typename _MatrixType>

anonymous enum

Enumerator:

RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 69 of file ComplexSchur.h.

Constructor & Destructor Documentation

template<typename _MatrixType>

ComplexSchur< _MatrixType >::ComplexSchur ( Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime ) [inline]

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 108 of file ComplexSchur.h.

template<typename _MatrixType>

ComplexSchur< _MatrixType >::ComplexSchur	(	const MatrixType &	matrix,
		bool	computeU = `true`
	)		`[inline]`

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 125 of file ComplexSchur.h.

Member Function Documentation

template<typename _MatrixType>

ComplexSchur& ComplexSchur< _MatrixType >::compute	(	const MatrixType &	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:

template<typename _MatrixType>

ComplexScalar ComplexSchur< _MatrixType >::computeShift	(	Index	iu,
		Index	iter
	)		`[private]`

template<typename _MatrixType>

ComputationInfo ComplexSchur< _MatrixType >::info ( ) const [inline]

Reports whether previous computation was successful.

Returns:: Success if computation was succesful, NoConvergence otherwise.

Definition at line 204 of file ComplexSchur.h.

template<typename _MatrixType>

const ComplexMatrixType& 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 173 of file ComplexSchur.h.

template<typename _MatrixType>

const ComplexMatrixType& 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 149 of file ComplexSchur.h.

template<typename _MatrixType>

void ComplexSchur< _MatrixType >::reduceToTriangularForm ( bool computeU ) [private]

template<typename _MatrixType>

bool ComplexSchur< _MatrixType >::subdiagonalEntryIsNeglegible ( Index i ) [private]

Friends And Related Function Documentation

template<typename _MatrixType>

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

Definition at line 227 of file ComplexSchur.h.

Member Data Documentation

template<typename _MatrixType>

HessenbergDecomposition<MatrixType> ComplexSchur< _MatrixType >::m_hess [protected]

Definition at line 218 of file ComplexSchur.h.

template<typename _MatrixType>

ComputationInfo ComplexSchur< _MatrixType >::m_info [protected]

Definition at line 219 of file ComplexSchur.h.

template<typename _MatrixType>

bool ComplexSchur< _MatrixType >::m_isInitialized [protected]

Definition at line 220 of file ComplexSchur.h.

template<typename _MatrixType>

ComplexMatrixType ComplexSchur< _MatrixType >::m_matT [protected]

Definition at line 217 of file ComplexSchur.h.

template<typename _MatrixType>

ComplexMatrixType ComplexSchur< _MatrixType >::m_matU [protected]

Definition at line 217 of file ComplexSchur.h.

template<typename _MatrixType>

bool ComplexSchur< _MatrixType >::m_matUisUptodate [protected]

Definition at line 221 of file ComplexSchur.h.

template<typename _MatrixType>

const int ComplexSchur< _MatrixType >::m_maxIterations = 30 [static]

Maximum number of iterations.

Maximum number of iterations allowed for an eigenvalue to converge.

Definition at line 214 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 ComplexSchur< _MatrixType >

Member Typedef Documentation

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Friends And Related Function Documentation

Member Data Documentation

template<typename _MatrixType>
class ComplexSchur< _MatrixType >