Reduces a square matrix to Hessenberg form by an orthogonal similarity transformation. More...

#include <HessenbergDecomposition.h>

Public Types
enum	{ Size = MatrixType::RowsAtCompileTime, SizeMinusOne = Size == Dynamic ? Dynamic : Size - 1, Options = MatrixType::Options, MaxSize = MatrixType::MaxRowsAtCompileTime, MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : MaxSize - 1 }
typedef _MatrixType	MatrixType
	Synonym for the template parameter `_MatrixType`.
typedef MatrixType::Scalar	Scalar
	Scalar type for matrices of type MatrixType.
typedef MatrixType::Index	Index
typedef Matrix< Scalar, SizeMinusOne, 1, Options &~RowMajor, MaxSizeMinusOne, 1 >	CoeffVectorType
	Type for vector of Householder coefficients.
typedef HouseholderSequence < MatrixType, CoeffVectorType > ::ConjugateReturnType	HouseholderSequenceType
	Return type of matrixQ().
typedef internal::HessenbergDecompositionMatrixHReturnType < MatrixType >	MatrixHReturnType
Public Member Functions
	HessenbergDecomposition (Index size=Size==Dynamic?2:Size)
	Default constructor; the decomposition will be computed later.
	HessenbergDecomposition (const MatrixType &matrix)
	Constructor; computes Hessenberg decomposition of given matrix.
HessenbergDecomposition &	compute (const MatrixType &matrix)
	Computes Hessenberg decomposition of given matrix.
const CoeffVectorType &	householderCoefficients () const
	Returns the Householder coefficients.
const MatrixType &	packedMatrix () const
	Returns the internal representation of the decomposition.
HouseholderSequenceType	matrixQ () const
	Reconstructs the orthogonal matrix Q in the decomposition.
MatrixHReturnType	matrixH () const
	Constructs the Hessenberg matrix H in the decomposition.
Protected Attributes
MatrixType	m_matrix
CoeffVectorType	m_hCoeffs
VectorType	m_temp
bool	m_isInitialized

Detailed Description

template<typename _MatrixType>
class HessenbergDecomposition< _MatrixType >

Reduces a square matrix to Hessenberg form by an orthogonal similarity transformation.

Template Parameters:

_MatrixType

the type of the matrix of which we are computing the Hessenberg decomposition

This class performs an Hessenberg decomposition of a matrix $ A $ . In the real case, the Hessenberg decomposition consists of an orthogonal matrix $ Q $ and a Hessenberg matrix $ H $ such that $ A = Q H Q^T $ . An orthogonal matrix is a matrix whose inverse equals its transpose ( $Q^{-1} = Q^T$ ). A Hessenberg matrix has zeros below the subdiagonal, so it is almost upper triangular. The Hessenberg decomposition of a complex matrix is $ A = Q H Q^* $ with $ Q $ unitary (that is, $Q^{-1} = Q^*$ ).

Call the function compute() to compute the Hessenberg decomposition of a given matrix. Alternatively, you can use the HessenbergDecomposition(const MatrixType&) constructor which computes the Hessenberg decomposition at construction time. Once the decomposition is computed, you can use the matrixH() and matrixQ() functions to construct the matrices H and Q in the decomposition.

The documentation for matrixH() contains an example of the typical use of this class.

See also:: class ComplexSchur, class Tridiagonalization, QR Module

Member Typedef Documentation

template<typename _MatrixType>

typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> HessenbergDecomposition< _MatrixType >::CoeffVectorType

Type for vector of Householder coefficients.

This is column vector with entries of type Scalar. The length of the vector is one less than the size of MatrixType, if it is a fixed-side type.

template<typename _MatrixType>

typedef HouseholderSequence<MatrixType,CoeffVectorType>::ConjugateReturnType HessenbergDecomposition< _MatrixType >::HouseholderSequenceType

Return type of matrixQ().

template<typename _MatrixType>

typedef MatrixType::Index HessenbergDecomposition< _MatrixType >::Index

template<typename _MatrixType>

typedef internal::HessenbergDecompositionMatrixHReturnType<MatrixType> HessenbergDecomposition< _MatrixType >::MatrixHReturnType

template<typename _MatrixType>

typedef _MatrixType HessenbergDecomposition< _MatrixType >::MatrixType

Synonym for the template parameter _MatrixType.

template<typename _MatrixType>

typedef MatrixType::Scalar HessenbergDecomposition< _MatrixType >::Scalar

Scalar type for matrices of type MatrixType.

Member Enumeration Documentation

template<typename _MatrixType>

anonymous enum

Enumerator:

Size
SizeMinusOne
Options
MaxSize
MaxSizeMinusOne

Constructor & Destructor Documentation

template<typename _MatrixType>

HessenbergDecomposition< _MatrixType >::HessenbergDecomposition ( Index size = Size==Dynamic ? 2 : Size ) [inline]

Default constructor; the decomposition will be computed later.

Parameters:

[in] size The size of the matrix whose Hessenberg 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.

template<typename _MatrixType>

HessenbergDecomposition< _MatrixType >::HessenbergDecomposition ( const MatrixType & matrix ) [inline]

Constructor; computes Hessenberg decomposition of given matrix.

Parameters:

[in] matrix Square matrix whose Hessenberg decomposition is to be computed.

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

See also:: matrixH() for an example.

Member Function Documentation

template<typename _MatrixType>

HessenbergDecomposition& HessenbergDecomposition< _MatrixType >::compute ( const MatrixType & matrix ) [inline]

Computes Hessenberg decomposition of given matrix.

Parameters:

[in] matrix Square matrix whose Hessenberg decomposition is to be computed.

Returns:: Reference to *this

The Hessenberg decomposition is computed by bringing the columns of the matrix successively in the required form using Householder reflections (see, e.g., Algorithm 7.4.2 in Golub & Van Loan, Matrix Computations). The cost is $ 10n^3/3 $ flops, where $ n $ denotes the size of the given matrix.

This method reuses of the allocated data in the HessenbergDecomposition object.

Example:

Output:

template<typename _MatrixType>

const CoeffVectorType& HessenbergDecomposition< _MatrixType >::householderCoefficients ( ) const [inline]

Returns the Householder coefficients.

Returns:: a const reference to the vector of Householder coefficients

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

The Householder coefficients allow the reconstruction of the matrix $ Q $ in the Hessenberg decomposition from the packed data.

See also:: packedMatrix(), Householder module

template<typename _MatrixType>

MatrixHReturnType HessenbergDecomposition< _MatrixType >::matrixH ( ) const [inline]

Constructs the Hessenberg matrix H in the decomposition.

Returns:: expression object representing the matrix H

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

The object returned by this function constructs the Hessenberg matrix H when it is assigned to a matrix or otherwise evaluated. The matrix H is constructed from the packed matrix as returned by packedMatrix(): The upper part (including the subdiagonal) of the packed matrix contains the matrix H. It may sometimes be better to directly use the packed matrix instead of constructing the matrix H.

Example:

Output:

See also:: matrixQ(), packedMatrix()

template<typename _MatrixType>

HouseholderSequenceType HessenbergDecomposition< _MatrixType >::matrixQ ( void ) const [inline]

Reconstructs the orthogonal matrix Q in the decomposition.

Returns:: object representing the matrix Q

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

This function returns a light-weight object of template class HouseholderSequence. You can either apply it directly to a matrix or you can convert it to a matrix of type MatrixType.

See also:: matrixH() for an example, class HouseholderSequence

template<typename _MatrixType>

const MatrixType& HessenbergDecomposition< _MatrixType >::packedMatrix ( ) const [inline]

Returns the internal representation of the decomposition.

Returns:: a const reference to a matrix with the internal representation of the decomposition.

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

The returned matrix contains the following information:

the upper part and lower sub-diagonal represent the Hessenberg matrix H
the rest of the lower part contains the Householder vectors that, combined with Householder coefficients returned by householderCoefficients(), allows to reconstruct the matrix Q as $Q = H_{N-1} \ldots H_1 H_0$ . Here, the matrices are the Householder transformations where is the th Householder coefficient and is the Householder vector defined by $v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T$ with M the matrix returned by this function.