Tridiagonal decomposition of a selfadjoint matrix. More...

#include <src/eigen/Eigen/src/Eigenvalues/Tridiagonalization.h>

Collaboration diagram for Eigen::Tridiagonalization< _MatrixType >:

Public Types
enum	{ Size = MatrixType::RowsAtCompileTime , SizeMinusOne = Size == Dynamic ? Dynamic : (Size > 1 ? Size - 1 : 1) , Options = MatrixType::Options , MaxSize = MatrixType::MaxRowsAtCompileTime , MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : (MaxSize > 1 ? MaxSize - 1 : 1) }

typedef _MatrixType	MatrixType
	Synonym for the template parameter `_MatrixType`.

typedef MatrixType::Scalar	Scalar

typedef NumTraits< Scalar >::Real	RealScalar

typedef Eigen::Index	Index

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

typedef internal::plain_col_type< MatrixType, RealScalar >::type	DiagonalType

typedef Matrix< RealScalar, SizeMinusOne, 1, Options &~RowMajor, MaxSizeMinusOne, 1 >	SubDiagonalType

typedef internal::remove_all< typenameMatrixType::RealReturnType >::type	MatrixTypeRealView

typedef internal::TridiagonalizationMatrixTReturnType< MatrixTypeRealView >	MatrixTReturnType

typedef internal::conditional< NumTraits< Scalar >::IsComplex, typenameinternal::add_const_on_value_type< typenameDiagonal< constMatrixType >::RealReturnType >::type, constDiagonal< constMatrixType > >::type	DiagonalReturnType

typedef internal::conditional< NumTraits< Scalar >::IsComplex, typenameinternal::add_const_on_value_type< typenameDiagonal< constMatrixType,-1 >::RealReturnType >::type, constDiagonal< constMatrixType,-1 > >::type	SubDiagonalReturnType

typedef HouseholderSequence< MatrixType, typename internal::remove_all< typename CoeffVectorType::ConjugateReturnType >::type >	HouseholderSequenceType
	Return type of matrixQ()

Public Member Functions
	Tridiagonalization (Index size=Size==Dynamic ? 2 :Size)
	Default constructor.

template<typename InputType >
	Tridiagonalization (const EigenBase< InputType > &matrix)
	Constructor; computes tridiagonal decomposition of given matrix.

template<typename InputType >
Tridiagonalization &	compute (const EigenBase< InputType > &matrix)
	Computes tridiagonal decomposition of given matrix.

CoeffVectorType	householderCoefficients () const
	Returns the Householder coefficients.

const MatrixType &	packedMatrix () const
	Returns the internal representation of the decomposition.

HouseholderSequenceType	matrixQ () const
	Returns the unitary matrix Q in the decomposition.

MatrixTReturnType	matrixT () const
	Returns an expression of the tridiagonal matrix T in the decomposition.

DiagonalReturnType	diagonal () const
	Returns the diagonal of the tridiagonal matrix T in the decomposition.

SubDiagonalReturnType	subDiagonal () const
	Returns the subdiagonal of the tridiagonal matrix T in the decomposition.

Protected Attributes
MatrixType	m_matrix

CoeffVectorType	m_hCoeffs

bool	m_isInitialized

Detailed Description

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

Tridiagonal decomposition of a selfadjoint matrix.

\eigenvalues_module

Template Parameters

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

This class performs a tridiagonal decomposition of a selfadjoint matrix $ A $ such that: $ A = Q T Q^* $ where $ Q $ is unitary and $ T $ a real symmetric tridiagonal matrix.

A tridiagonal matrix is a matrix which has nonzero elements only on the main diagonal and the first diagonal below and above it. The Hessenberg decomposition of a selfadjoint matrix is in fact a tridiagonal decomposition. This class is used in SelfAdjointEigenSolver to compute the eigenvalues and eigenvectors of a selfadjoint matrix.

Call the function compute() to compute the tridiagonal decomposition of a given matrix. Alternatively, you can use the Tridiagonalization(const MatrixType&) constructor which computes the tridiagonal Schur decomposition at construction time. Once the decomposition is computed, you can use the matrixQ() and matrixT() functions to retrieve the matrices Q and T in the decomposition.

The documentation of Tridiagonalization(const MatrixType&) contains an example of the typical use of this class.

See also: class HessenbergDecomposition, class SelfAdjointEigenSolver

Member Typedef Documentation

◆ CoeffVectorType

template<typename _MatrixType >

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

◆ DiagonalReturnType

template<typename _MatrixType >

typedef internal::conditional<NumTraits<Scalar>::IsComplex,typenameinternal::add_const_on_value_type<typenameDiagonal<constMatrixType>::RealReturnType>::type,constDiagonal<constMatrixType>>::type Eigen::Tridiagonalization< _MatrixType >::DiagonalReturnType

◆ DiagonalType

template<typename _MatrixType >

typedef internal::plain_col_type<MatrixType,RealScalar>::type Eigen::Tridiagonalization< _MatrixType >::DiagonalType

◆ HouseholderSequenceType

template<typename _MatrixType >

typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename CoeffVectorType::ConjugateReturnType>::type> Eigen::Tridiagonalization< _MatrixType >::HouseholderSequenceType

Return type of matrixQ()

◆ Index

template<typename _MatrixType >

typedef Eigen::Index Eigen::Tridiagonalization< _MatrixType >::Index

◆ MatrixTReturnType

template<typename _MatrixType >

typedef internal::TridiagonalizationMatrixTReturnType<MatrixTypeRealView> Eigen::Tridiagonalization< _MatrixType >::MatrixTReturnType

◆ MatrixType

template<typename _MatrixType >

typedef _MatrixType Eigen::Tridiagonalization< _MatrixType >::MatrixType

Synonym for the template parameter _MatrixType.

◆ MatrixTypeRealView

template<typename _MatrixType >

typedef internal::remove_all<typenameMatrixType::RealReturnType>::type Eigen::Tridiagonalization< _MatrixType >::MatrixTypeRealView

◆ RealScalar

template<typename _MatrixType >

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

◆ Scalar

template<typename _MatrixType >

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

◆ SubDiagonalReturnType

template<typename _MatrixType >

typedef internal::conditional<NumTraits<Scalar>::IsComplex,typenameinternal::add_const_on_value_type<typenameDiagonal<constMatrixType,-1>::RealReturnType>::type,constDiagonal<constMatrixType,-1>>::type Eigen::Tridiagonalization< _MatrixType >::SubDiagonalReturnType

◆ SubDiagonalType

template<typename _MatrixType >

typedef Matrix<RealScalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> Eigen::Tridiagonalization< _MatrixType >::SubDiagonalType

Member Enumeration Documentation

◆ anonymous enum

template<typename _MatrixType >

anonymous enum

Enumerator
Size
SizeMinusOne
Options
MaxSize
MaxSizeMinusOne

         {
      Size = MatrixType::RowsAtCompileTime,
      SizeMinusOne = Size == Dynamic ? Dynamic : (Size > 1 ? Size - 1 : 1),
      Options = MatrixType::Options,
      MaxSize = MatrixType::MaxRowsAtCompileTime,
      MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : (MaxSize > 1 ? MaxSize - 1 : 1)
    };

Constructor & Destructor Documentation

◆ Tridiagonalization() [1/2]

template<typename _MatrixType >

Eigen::Tridiagonalization< _MatrixType >::Tridiagonalization ( Index size = Size==Dynamic ? 2 : Size )

inlineexplicit

Default constructor.

Parameters

[in] size Positive integer, size of the matrix whose tridiagonal 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.

                                                               : Size)
      : m_matrix(size,size),
        m_hCoeffs(size > 1 ? size-1 : 1),
        m_isInitialized(false)
    {}

◆ Tridiagonalization() [2/2]

template<typename _MatrixType >

template<typename InputType >

Eigen::Tridiagonalization< _MatrixType >::Tridiagonalization ( const EigenBase< InputType > & matrix )

inlineexplicit

Constructor; computes tridiagonal decomposition of given matrix.

Parameters

[in] matrix Selfadjoint matrix whose tridiagonal decomposition is to be computed.

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

Example:

Output:

      : m_matrix(matrix.derived()),
        m_hCoeffs(matrix.cols() > 1 ? matrix.cols()-1 : 1),
        m_isInitialized(false)
    {
      internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
      m_isInitialized = true;
    }

References Eigen::Tridiagonalization< _MatrixType >::m_hCoeffs, Eigen::Tridiagonalization< _MatrixType >::m_isInitialized, Eigen::Tridiagonalization< _MatrixType >::m_matrix, and Eigen::internal::tridiagonalization_inplace().

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Member Function Documentation

◆ compute()

template<typename _MatrixType >

template<typename InputType >

Tridiagonalization & Eigen::Tridiagonalization< _MatrixType >::compute ( const EigenBase< InputType > & matrix )

inline

Computes tridiagonal decomposition of given matrix.

Parameters

[in] matrix Selfadjoint matrix whose tridiagonal decomposition is to be computed.

Returns: Reference to *this

The tridiagonal decomposition is computed by bringing the columns of the matrix successively in the required form using Householder reflections. The cost is $ 4n^3/3 $ flops, where $ n $ denotes the size of the given matrix.

This method reuses of the allocated data in the Tridiagonalization object, if the size of the matrix does not change.

Example:

Output:

    {
      m_matrix = matrix.derived();
      m_hCoeffs.resize(matrix.rows()-1, 1);
      internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
      m_isInitialized = true;
      return *this;
    }

References Eigen::EigenBase< Derived >::derived(), Eigen::Tridiagonalization< _MatrixType >::m_hCoeffs, Eigen::Tridiagonalization< _MatrixType >::m_isInitialized, Eigen::Tridiagonalization< _MatrixType >::m_matrix, Eigen::PlainObjectBase< Derived >::resize(), Eigen::EigenBase< Derived >::rows(), and Eigen::internal::tridiagonalization_inplace().

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◆ diagonal()

template<typename MatrixType >

Tridiagonalization< MatrixType >::DiagonalReturnType Eigen::Tridiagonalization< MatrixType >::diagonal

Returns the diagonal of the tridiagonal matrix T in the decomposition.

Returns: expression representing the diagonal of T

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

Example:

Output:

See also: matrixT(), subDiagonal()

{
  eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
  return m_matrix.diagonal().real();
}

References eigen_assert, Eigen::Tridiagonalization< _MatrixType >::m_isInitialized, and Eigen::Tridiagonalization< _MatrixType >::m_matrix.

Referenced by Eigen::Tridiagonalization< _MatrixType >::subDiagonal().

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◆ householderCoefficients()

template<typename _MatrixType >

CoeffVectorType Eigen::Tridiagonalization< _MatrixType >::householderCoefficients ( ) const

inline

Returns the Householder coefficients.

Returns: a const reference to the vector of Householder coefficients

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

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

Example:

Output:

See also: packedMatrix(), Householder module

    {
      eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
      return m_hCoeffs;
    }

References eigen_assert, Eigen::Tridiagonalization< _MatrixType >::m_hCoeffs, and Eigen::Tridiagonalization< _MatrixType >::m_isInitialized.

◆ matrixQ()

template<typename _MatrixType >

HouseholderSequenceType Eigen::Tridiagonalization< _MatrixType >::matrixQ ( ) const

inline

Returns the unitary matrix Q in the decomposition.

Returns: object representing the matrix Q

Precondition: Either the constructor Tridiagonalization(const MatrixType&) or the member function compute(const MatrixType&) has been called before to compute the tridiagonal 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: Tridiagonalization(const MatrixType&) for an example, matrixT(), class HouseholderSequence

    {
      eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
      return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate())
             .setLength(m_matrix.rows() - 1)
             .setShift(1);
    }

References eigen_assert, Eigen::Tridiagonalization< _MatrixType >::m_hCoeffs, Eigen::Tridiagonalization< _MatrixType >::m_isInitialized, and Eigen::Tridiagonalization< _MatrixType >::m_matrix.

◆ matrixT()

template<typename _MatrixType >

MatrixTReturnType Eigen::Tridiagonalization< _MatrixType >::matrixT ( ) const

inline

Returns an expression of the tridiagonal matrix T in the decomposition.

Returns: expression object representing the matrix T

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

Currently, this function can be used to extract the matrix T from internal data and copy it to a dense matrix object. In most cases, it may be sufficient to directly use the packed matrix or the vector expressions returned by diagonal() and subDiagonal() instead of creating a new dense copy matrix with this function.

See also: Tridiagonalization(const MatrixType&) for an example, matrixQ(), packedMatrix(), diagonal(), subDiagonal()

    {
      eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
      return MatrixTReturnType(m_matrix.real());
    }

References eigen_assert, Eigen::Tridiagonalization< _MatrixType >::m_isInitialized, and Eigen::Tridiagonalization< _MatrixType >::m_matrix.

◆ packedMatrix()

template<typename _MatrixType >

const MatrixType & Eigen::Tridiagonalization< _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 Tridiagonalization(const MatrixType&) or the member function compute(const MatrixType&) has been called before to compute the tridiagonal decomposition of a matrix.

The returned matrix contains the following information:

the strict upper triangular part is equal to the input matrix A.
the diagonal and lower sub-diagonal represent the real tridiagonal symmetric matrix T.
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.

See LAPACK for further details on this packed storage.

Example:

Output:

See also: householderCoefficients()

    {
      eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
      return m_matrix;
    }

References eigen_assert, Eigen::Tridiagonalization< _MatrixType >::m_isInitialized, and Eigen::Tridiagonalization< _MatrixType >::m_matrix.

◆ subDiagonal()

template<typename MatrixType >

Tridiagonalization< MatrixType >::SubDiagonalReturnType Eigen::Tridiagonalization< MatrixType >::subDiagonal

Returns the subdiagonal of the tridiagonal matrix T in the decomposition.

Returns: expression representing the subdiagonal of T

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

See also: diagonal() for an example, matrixT()

{
  eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
  return m_matrix.template diagonal<-1>().real();
}

References Eigen::Tridiagonalization< _MatrixType >::diagonal(), eigen_assert, Eigen::Tridiagonalization< _MatrixType >::m_isInitialized, Eigen::Tridiagonalization< _MatrixType >::m_matrix, and real().

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Member Data Documentation

◆ m_hCoeffs

template<typename _MatrixType >

CoeffVectorType Eigen::Tridiagonalization< _MatrixType >::m_hCoeffs

protected

Referenced by Eigen::Tridiagonalization< _MatrixType >::Tridiagonalization(), Eigen::Tridiagonalization< _MatrixType >::compute(), Eigen::Tridiagonalization< _MatrixType >::householderCoefficients(), and Eigen::Tridiagonalization< _MatrixType >::matrixQ().

◆ m_isInitialized

template<typename _MatrixType >

bool Eigen::Tridiagonalization< _MatrixType >::m_isInitialized

protected

Referenced by Eigen::Tridiagonalization< _MatrixType >::Tridiagonalization(), Eigen::Tridiagonalization< _MatrixType >::compute(), Eigen::Tridiagonalization< _MatrixType >::diagonal(), Eigen::Tridiagonalization< _MatrixType >::householderCoefficients(), Eigen::Tridiagonalization< _MatrixType >::matrixQ(), Eigen::Tridiagonalization< _MatrixType >::matrixT(), Eigen::Tridiagonalization< _MatrixType >::packedMatrix(), and Eigen::Tridiagonalization< _MatrixType >::subDiagonal().

◆ m_matrix

template<typename _MatrixType >

MatrixType Eigen::Tridiagonalization< _MatrixType >::m_matrix

protected

Referenced by Eigen::Tridiagonalization< _MatrixType >::Tridiagonalization(), Eigen::Tridiagonalization< _MatrixType >::compute(), Eigen::Tridiagonalization< _MatrixType >::diagonal(), Eigen::Tridiagonalization< _MatrixType >::matrixQ(), Eigen::Tridiagonalization< _MatrixType >::matrixT(), Eigen::Tridiagonalization< _MatrixType >::packedMatrix(), and Eigen::Tridiagonalization< _MatrixType >::subDiagonal().

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

src/eigen/Eigen/src/Eigenvalues/Tridiagonalization.h

Public Types

Public Member Functions

Protected Attributes

Detailed Description

Member Typedef Documentation

◆ CoeffVectorType

◆ DiagonalReturnType

◆ DiagonalType

◆ HouseholderSequenceType

◆ Index

◆ MatrixTReturnType

◆ MatrixType

◆ MatrixTypeRealView

◆ RealScalar

◆ Scalar

◆ SubDiagonalReturnType

◆ SubDiagonalType

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ Tridiagonalization() [1/2]

◆ Tridiagonalization() [2/2]

Member Function Documentation

◆ compute()

◆ diagonal()

◆ householderCoefficients()

◆ matrixQ()

◆ matrixT()

◆ packedMatrix()

◆ subDiagonal()

Member Data Documentation

◆ m_hCoeffs

◆ m_isInitialized

◆ m_matrix