Tridiagonalization< _MatrixType > Class Template Reference

Trigiagonal decomposition of a selfadjoint matrix. More...

#include <Tridiagonalization.h>

List of all members.

Public Types
enum	{ Size = MatrixType::RowsAtCompileTime, SizeMinusOne, PacketSize = ei_packet_traits<Scalar>::size }
typedef Matrix< Scalar, SizeMinusOne, 1 >	CoeffVectorType
typedef NestByValue < DiagonalCoeffs< MatrixType > >::RealReturnType	DiagonalReturnType
typedef Matrix< RealScalar, Size, 1 >	DiagonalType
typedef _MatrixType	MatrixType
typedef ei_packet_traits < Scalar >::type	Packet
typedef NumTraits< Scalar >::Real	RealScalar
typedef MatrixType::Scalar	Scalar
typedef NestByValue < DiagonalCoeffs< NestByValue < Block< MatrixType, SizeMinusOne, SizeMinusOne > > > >::RealReturnType	SubDiagonalReturnType
typedef Matrix< RealScalar, SizeMinusOne, 1 >	SubDiagonalType
Public Member Functions
void	compute (const MatrixType &matrix)
const DiagonalReturnType	diagonal (void) const
CoeffVectorType	householderCoefficients (void) const
MatrixType	matrixQ (void) const
MatrixType	matrixT (void) const
const MatrixType &	packedMatrix (void) const
const SubDiagonalReturnType	subDiagonal (void) const
	Tridiagonalization (const MatrixType &matrix)
	Tridiagonalization (int size=Size==Dynamic?2:Size)
Static Public Member Functions
static void	decomposeInPlace (MatrixType &mat, DiagonalType &diag, SubDiagonalType &subdiag, bool extractQ=true)
Protected Attributes
CoeffVectorType	m_hCoeffs
MatrixType	m_matrix
Static Private Member Functions
static void	_compute (MatrixType &matA, CoeffVectorType &hCoeffs)
static void	_decomposeInPlace3x3 (MatrixType &mat, DiagonalType &diag, SubDiagonalType &subdiag, bool extractQ=true)

---

Detailed Description

template<typename _MatrixType>
class Tridiagonalization< _MatrixType >

Trigiagonal decomposition of a selfadjoint matrix.

Parameters:

MatrixType

the type of the matrix of which we are performing the tridiagonalization

This class performs a tridiagonal decomposition of a selfadjoint matrix such that: where is unitary and a real symmetric tridiagonal matrix.

See also:

MatrixBase::tridiagonalize()

Definition at line 42 of file Tridiagonalization.h.

---

Member Typedef Documentation

template<typename _MatrixType >

typedef Matrix<Scalar, SizeMinusOne, 1> Tridiagonalization< _MatrixType >::CoeffVectorType

Definition at line 59 of file Tridiagonalization.h.

template<typename _MatrixType >

typedef NestByValue<DiagonalCoeffs<MatrixType> >::RealReturnType Tridiagonalization< _MatrixType >::DiagonalReturnType

Definition at line 63 of file Tridiagonalization.h.

template<typename _MatrixType >

typedef Matrix<RealScalar, Size, 1> Tridiagonalization< _MatrixType >::DiagonalType

Definition at line 60 of file Tridiagonalization.h.

template<typename _MatrixType >

typedef _MatrixType Tridiagonalization< _MatrixType >::MatrixType

Definition at line 46 of file Tridiagonalization.h.

template<typename _MatrixType >

typedef ei_packet_traits<Scalar>::type Tridiagonalization< _MatrixType >::Packet

Definition at line 49 of file Tridiagonalization.h.

template<typename _MatrixType >

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

Definition at line 48 of file Tridiagonalization.h.

template<typename _MatrixType >

typedef MatrixType::Scalar Tridiagonalization< _MatrixType >::Scalar

Definition at line 47 of file Tridiagonalization.h.

template<typename _MatrixType >

typedef NestByValue<DiagonalCoeffs< NestByValue<Block<MatrixType,SizeMinusOne,SizeMinusOne> > > >::RealReturnType Tridiagonalization< _MatrixType >::SubDiagonalReturnType

Definition at line 66 of file Tridiagonalization.h.

template<typename _MatrixType >

typedef Matrix<RealScalar, SizeMinusOne, 1> Tridiagonalization< _MatrixType >::SubDiagonalType

Definition at line 61 of file Tridiagonalization.h.

---

Member Enumeration Documentation

template<typename _MatrixType >

anonymous enum

Enumerator:

Size
SizeMinusOne
PacketSize

Definition at line 51 of file Tridiagonalization.h.

---

Constructor & Destructor Documentation

template<typename _MatrixType >

Tridiagonalization< _MatrixType >::Tridiagonalization

(

int

size = Size==Dynamic ? 2 : Size

)

[inline]

This constructor initializes a Tridiagonalization object for further use with Tridiagonalization::compute()

Definition at line 71 of file Tridiagonalization.h.

template<typename _MatrixType >

Tridiagonalization< _MatrixType >::Tridiagonalization

(

const MatrixType &

matrix

)

[inline]

Definition at line 75 of file Tridiagonalization.h.

---

Member Function Documentation

template<typename MatrixType >

void Tridiagonalization< MatrixType >::_compute	(	MatrixType &	matA,
		CoeffVectorType &	hCoeffs
	)			[inline, static, private]

Definition at line 191 of file Tridiagonalization.h.

template<typename MatrixType >

void Tridiagonalization< MatrixType >::_decomposeInPlace3x3	(	MatrixType &	mat,
		DiagonalType &	diag,
		SubDiagonalType &	subdiag,
		bool	extractQ = true
	)			[inline, static, private]

Definition at line 390 of file Tridiagonalization.h.

template<typename _MatrixType >

void Tridiagonalization< _MatrixType >::compute

(

const MatrixType &

matrix

)

[inline]

Computes or re-compute the tridiagonalization for the matrix matrix.

This method allows to re-use the allocated data.

Definition at line 86 of file Tridiagonalization.h.

template<typename MatrixType >

void Tridiagonalization< MatrixType >::decomposeInPlace	(	MatrixType &	mat,
		DiagonalType &	diag,
		SubDiagonalType &	subdiag,
		bool	extractQ = true
	)			[inline, static]

Performs a full decomposition in place

Definition at line 367 of file Tridiagonalization.h.

template<typename MatrixType >

const Tridiagonalization< MatrixType >::DiagonalReturnType Tridiagonalization< MatrixType >::diagonal

(

void

)

const [inline]

Returns:

an expression of the diagonal vector

Definition at line 139 of file Tridiagonalization.h.

template<typename _MatrixType >

CoeffVectorType Tridiagonalization< _MatrixType >::householderCoefficients

(

void

)

const [inline]

Returns:

the householder coefficients allowing to reconstruct the matrix Q from the packed data.

See also:

packedMatrix()

Definition at line 98 of file Tridiagonalization.h.

template<typename MatrixType >

Tridiagonalization< MatrixType >::MatrixType Tridiagonalization< MatrixType >::matrixQ

(

void

)

const [inline]

reconstructs and returns the matrix Q

Definition at line 347 of file Tridiagonalization.h.

template<typename MatrixType >

Tridiagonalization< MatrixType >::MatrixType Tridiagonalization< MatrixType >::matrixT

(

void

)

const [inline]

constructs and returns the tridiagonal matrix T. Note that the matrix T is equivalent to the diagonal and sub-diagonal of the packed matrix. Therefore, it might be often sufficient to directly use the packed matrix, or the vector expressions returned by diagonal() and subDiagonal() instead of creating a new matrix.

Definition at line 161 of file Tridiagonalization.h.

template<typename _MatrixType >

const MatrixType& Tridiagonalization< _MatrixType >::packedMatrix

(

void

)

const [inline]

Returns:

the internal result of the decomposition.

The returned matrix contains the following information:

• the strict upper part is equal to the input matrix A
• the diagonal and lower sub-diagonal represent the tridiagonal symmetric matrix (real).
• 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 follow: Q = H_{N-1} ... H_1 H_0 where the matrices H are the Householder transformations: H_i = (I - h_i * v_i * v_i') where h_i == householderCoefficients()[i] and v_i is a Householder vector: v_i = [ 0, ..., 0, 1, M(i+2,i), ..., M(N-1,i) ]

See LAPACK for further details on this packed storage.

Definition at line 116 of file Tridiagonalization.h.

template<typename MatrixType >

const Tridiagonalization< MatrixType >::SubDiagonalReturnType Tridiagonalization< MatrixType >::subDiagonal

(

void

)

const [inline]

Returns:

an expression of the sub-diagonal vector

Definition at line 147 of file Tridiagonalization.h.

---

Member Data Documentation

template<typename _MatrixType >

CoeffVectorType Tridiagonalization< _MatrixType >::m_hCoeffs [protected]

Definition at line 133 of file Tridiagonalization.h.

template<typename _MatrixType >

MatrixType Tridiagonalization< _MatrixType >::m_matrix [protected]

Definition at line 132 of file Tridiagonalization.h.

---

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

• Tridiagonalization.h