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

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

Inheritance diagram for Eigen::HessenbergDecomposition< _MatrixType >:

Collaboration diagram for Eigen::HessenbergDecomposition< _MatrixType >:

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 Eigen::Index	Index

typedef Matrix< Scalar, SizeMinusOne, 1, Options &~RowMajor, MaxSizeMinusOne, 1 >	CoeffVectorType
	Type for vector of Householder coefficients.

typedef HouseholderSequence< MatrixType, typename internal::remove_all< typename CoeffVectorType::ConjugateReturnType >::type >	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.

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

template<typename InputType >
HessenbergDecomposition &	compute (const EigenBase< InputType > &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

Private Types
typedef Matrix< Scalar, 1, Size, Options\|RowMajor, 1, MaxSize >	VectorType

typedef NumTraits< Scalar >::Real	RealScalar

Static Private Member Functions
static void	_compute (MatrixType &matA, CoeffVectorType &hCoeffs, VectorType &temp)

Detailed Description

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

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

\eigenvalues_module

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

◆ CoeffVectorType

template<typename _MatrixType >

typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> Eigen::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.

◆ HouseholderSequenceType

template<typename _MatrixType >

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

Return type of matrixQ()

◆ Index

template<typename _MatrixType >

typedef Eigen::Index Eigen::HessenbergDecomposition< _MatrixType >::Index

◆ MatrixHReturnType

template<typename _MatrixType >

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

◆ MatrixType

template<typename _MatrixType >

typedef _MatrixType Eigen::HessenbergDecomposition< _MatrixType >::MatrixType

Synonym for the template parameter _MatrixType.

◆ RealScalar

template<typename _MatrixType >

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

private

◆ Scalar

template<typename _MatrixType >

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

Scalar type for matrices of type MatrixType.

◆ VectorType

template<typename _MatrixType >

typedef Matrix<Scalar, 1, Size, Options | RowMajor, 1, MaxSize> Eigen::HessenbergDecomposition< _MatrixType >::VectorType

private

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,
      Options = MatrixType::Options,
      MaxSize = MatrixType::MaxRowsAtCompileTime,
      MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : MaxSize - 1
    };

Constructor & Destructor Documentation

◆ HessenbergDecomposition() [1/2]

template<typename _MatrixType >

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

inlineexplicit

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.

                                                                    : Size)
      : m_matrix(size,size),
        m_temp(size),
        m_isInitialized(false)
    {
      if(size>1)
        m_hCoeffs.resize(size-1);
    }

References Eigen::HessenbergDecomposition< _MatrixType >::m_hCoeffs, and Eigen::PlainObjectBase< Derived >::resize().

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◆ HessenbergDecomposition() [2/2]

template<typename _MatrixType >

template<typename InputType >

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

inlineexplicit

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.

      : m_matrix(matrix.derived()),
        m_temp(matrix.rows()),
        m_isInitialized(false)
    {
      if(matrix.rows()<2)
      {
        m_isInitialized = true;
        return;
      }
      m_hCoeffs.resize(matrix.rows()-1,1);
      _compute(m_matrix, m_hCoeffs, m_temp);
      m_isInitialized = true;
    }

References Eigen::HessenbergDecomposition< _MatrixType >::_compute(), Eigen::HessenbergDecomposition< _MatrixType >::m_hCoeffs, Eigen::HessenbergDecomposition< _MatrixType >::m_isInitialized, Eigen::HessenbergDecomposition< _MatrixType >::m_matrix, Eigen::HessenbergDecomposition< _MatrixType >::m_temp, Eigen::PlainObjectBase< Derived >::resize(), and Eigen::EigenBase< Derived >::rows().

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Member Function Documentation

◆ _compute()

template<typename MatrixType >

void Eigen::HessenbergDecomposition< MatrixType >::_compute	(	MatrixType &	matA,
		CoeffVectorType &	hCoeffs,
		VectorType &	temp
	)

staticprivate

{
  eigen_assert(matA.rows()==matA.cols());
  Index n = matA.rows();
  temp.resize(n);
  for (Index i = 0; i<n-1; ++i)
  {
    // let's consider the vector v = i-th column starting at position i+1
    Index remainingSize = n-i-1;
    RealScalar beta;
    Scalar h;
    matA.col(i).tail(remainingSize).makeHouseholderInPlace(h, beta);
    matA.col(i).coeffRef(i+1) = beta;
    hCoeffs.coeffRef(i) = h;
 
    // Apply similarity transformation to remaining columns,
    // i.e., compute A = H A H'
 
    // A = H A
    matA.bottomRightCorner(remainingSize, remainingSize)
        .applyHouseholderOnTheLeft(matA.col(i).tail(remainingSize-1), h, &temp.coeffRef(0));
 
    // A = A H'
    matA.rightCols(remainingSize)
        .applyHouseholderOnTheRight(matA.col(i).tail(remainingSize-1).conjugate(), numext::conj(h), &temp.coeffRef(0));
  }
}

References Eigen::Matrix< _Scalar, _Rows, _Cols, _Options, _MaxRows, _MaxCols >::coeffRef(), eigen_assert, and Eigen::PlainObjectBase< Derived >::resize().

Referenced by Eigen::HessenbergDecomposition< _MatrixType >::HessenbergDecomposition(), and Eigen::HessenbergDecomposition< _MatrixType >::compute().

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◆ compute()

template<typename _MatrixType >

template<typename InputType >

HessenbergDecomposition & Eigen::HessenbergDecomposition< _MatrixType >::compute ( const EigenBase< InputType > & 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:

    {
      m_matrix = matrix.derived();
      if(matrix.rows()<2)
      {
        m_isInitialized = true;
        return *this;
      }
      m_hCoeffs.resize(matrix.rows()-1,1);
      _compute(m_matrix, m_hCoeffs, m_temp);
      m_isInitialized = true;
      return *this;
    }

References Eigen::HessenbergDecomposition< _MatrixType >::_compute(), Eigen::EigenBase< Derived >::derived(), Eigen::HessenbergDecomposition< _MatrixType >::m_hCoeffs, Eigen::HessenbergDecomposition< _MatrixType >::m_isInitialized, Eigen::HessenbergDecomposition< _MatrixType >::m_matrix, Eigen::HessenbergDecomposition< _MatrixType >::m_temp, Eigen::PlainObjectBase< Derived >::resize(), and Eigen::EigenBase< Derived >::rows().

Referenced by Eigen::internal::complex_schur_reduce_to_hessenberg< MatrixType, IsComplex >::run(), and Eigen::internal::complex_schur_reduce_to_hessenberg< MatrixType, false >::run().

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◆ householderCoefficients()

template<typename _MatrixType >

const CoeffVectorType & Eigen::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

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

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

◆ matrixH()

template<typename _MatrixType >

MatrixHReturnType Eigen::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()

    {
      eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized.");
      return MatrixHReturnType(*this);
    }

References eigen_assert, and Eigen::HessenbergDecomposition< _MatrixType >::m_isInitialized.

Referenced by Eigen::internal::complex_schur_reduce_to_hessenberg< MatrixType, IsComplex >::run(), and Eigen::internal::complex_schur_reduce_to_hessenberg< MatrixType, false >::run().

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◆ matrixQ()

template<typename _MatrixType >

HouseholderSequenceType Eigen::HessenbergDecomposition< _MatrixType >::matrixQ ( ) 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

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

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

Referenced by Eigen::internal::complex_schur_reduce_to_hessenberg< MatrixType, IsComplex >::run(), and Eigen::internal::complex_schur_reduce_to_hessenberg< MatrixType, false >::run().

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◆ packedMatrix()

template<typename _MatrixType >

const MatrixType & Eigen::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.

See LAPACK for further details on this packed storage.

Example:

Output:

See also: householderCoefficients()

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

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

Referenced by Eigen::internal::HessenbergDecompositionMatrixHReturnType< MatrixType >::cols(), Eigen::internal::HessenbergDecompositionMatrixHReturnType< MatrixType >::evalTo(), and Eigen::internal::HessenbergDecompositionMatrixHReturnType< MatrixType >::rows().

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Member Data Documentation

◆ m_hCoeffs

template<typename _MatrixType >

CoeffVectorType Eigen::HessenbergDecomposition< _MatrixType >::m_hCoeffs

protected

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

◆ m_isInitialized

template<typename _MatrixType >

bool Eigen::HessenbergDecomposition< _MatrixType >::m_isInitialized

protected

Referenced by Eigen::HessenbergDecomposition< _MatrixType >::HessenbergDecomposition(), Eigen::HessenbergDecomposition< _MatrixType >::compute(), Eigen::HessenbergDecomposition< _MatrixType >::householderCoefficients(), Eigen::HessenbergDecomposition< _MatrixType >::matrixH(), Eigen::HessenbergDecomposition< _MatrixType >::matrixQ(), and Eigen::HessenbergDecomposition< _MatrixType >::packedMatrix().

◆ m_matrix

template<typename _MatrixType >

MatrixType Eigen::HessenbergDecomposition< _MatrixType >::m_matrix

protected

Referenced by Eigen::HessenbergDecomposition< _MatrixType >::HessenbergDecomposition(), Eigen::HessenbergDecomposition< _MatrixType >::compute(), Eigen::HessenbergDecomposition< _MatrixType >::matrixQ(), and Eigen::HessenbergDecomposition< _MatrixType >::packedMatrix().

◆ m_temp

template<typename _MatrixType >

VectorType Eigen::HessenbergDecomposition< _MatrixType >::m_temp

protected

Referenced by Eigen::HessenbergDecomposition< _MatrixType >::HessenbergDecomposition(), and Eigen::HessenbergDecomposition< _MatrixType >::compute().

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

src/eigen/Eigen/src/Eigenvalues/HessenbergDecomposition.h

Public Types

Public Member Functions

Protected Attributes

Private Types

Static Private Member Functions

Detailed Description

Member Typedef Documentation

◆ CoeffVectorType

◆ HouseholderSequenceType

◆ Index

◆ MatrixHReturnType

◆ MatrixType

◆ RealScalar

◆ Scalar

◆ VectorType

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ HessenbergDecomposition() [1/2]

◆ HessenbergDecomposition() [2/2]

Member Function Documentation

◆ _compute()

◆ compute()

◆ householderCoefficients()

◆ matrixH()

◆ matrixQ()

◆ packedMatrix()

Member Data Documentation

◆ m_hCoeffs

◆ m_isInitialized

◆ m_matrix

◆ m_temp