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

Public Types
enum	{ Size , SizeMinusOne , Options , MaxSize , MaxSizeMinusOne }

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

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

typedef Eigen::Index	Index

typedef internal::HessenbergDecompositionMatrixHReturnType< MatrixType >	MatrixHReturnType

typedef MatrixType_	MatrixType
	Synonym for the template parameter `MatrixType_`. More...

typedef MatrixType::Scalar	Scalar
	Scalar type for matrices of type MatrixType. More...

Public Member Functions
template<typename InputType >
HessenbergDecomposition &	compute (const EigenBase< InputType > &matrix)
	Computes Hessenberg decomposition of given matrix. More...

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

	HessenbergDecomposition (Index size=Size==Dynamic ? 2 :Size)
	Default constructor; the decomposition will be computed later. More...

const CoeffVectorType &	householderCoefficients () const
	Returns the Householder coefficients. More...

MatrixHReturnType	matrixH () const
	Constructs the Hessenberg matrix H in the decomposition. More...

HouseholderSequenceType	matrixQ () const
	Reconstructs the orthogonal matrix Q in the decomposition. More...

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

Protected Attributes
CoeffVectorType	m_hCoeffs

bool	m_isInitialized

MatrixType	m_matrix

VectorType	m_temp

Private Types
typedef NumTraits< Scalar >::Real	RealScalar

typedef Matrix< Scalar, 1, Size, int(Options)\|int(RowMajor), 1, MaxSize >	VectorType

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.

This is defined in the Eigenvalues module.

#include <Eigen/Eigenvalues>

Eigenvalues

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

Definition at line 59 of file HessenbergDecomposition.h.

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.

Definition at line 84 of file HessenbergDecomposition.h.

◆ HouseholderSequenceType

template<typename MatrixType_ >

typedef HouseholderSequence<MatrixType,internal::remove_all_t<typename CoeffVectorType::ConjugateReturnType> > Eigen::HessenbergDecomposition< MatrixType_ >::HouseholderSequenceType

Return type of matrixQ()

Definition at line 87 of file HessenbergDecomposition.h.

◆ Index

template<typename MatrixType_ >

typedef Eigen::Index Eigen::HessenbergDecomposition< MatrixType_ >::Index

Deprecated:: since Eigen 3.3

Definition at line 76 of file HessenbergDecomposition.h.

◆ MatrixHReturnType

template<typename MatrixType_ >

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

Definition at line 89 of file HessenbergDecomposition.h.

◆ MatrixType

template<typename MatrixType_ >

typedef MatrixType_ Eigen::HessenbergDecomposition< MatrixType_ >::MatrixType

Synonym for the template parameter MatrixType_.

Definition at line 64 of file HessenbergDecomposition.h.

◆ RealScalar

template<typename MatrixType_ >

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

private

Definition at line 273 of file HessenbergDecomposition.h.

◆ Scalar

template<typename MatrixType_ >

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

Scalar type for matrices of type MatrixType.

Definition at line 75 of file HessenbergDecomposition.h.

◆ VectorType

template<typename MatrixType_ >

typedef Matrix<Scalar, 1, Size, int(Options) | int(RowMajor), 1, MaxSize> Eigen::HessenbergDecomposition< MatrixType_ >::VectorType

private

Definition at line 272 of file HessenbergDecomposition.h.

Member Enumeration Documentation

◆ anonymous enum

template<typename MatrixType_ >

anonymous enum

Enumerator
Size
SizeMinusOne
Options
MaxSize
MaxSizeMinusOne

Definition at line 66 of file HessenbergDecomposition.h.

          {
       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.

Definition at line 102 of file HessenbergDecomposition.h.

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

◆ 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.

Definition at line 121 of file HessenbergDecomposition.h.

       : 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;
     }

Member Function Documentation

◆ _compute()

template<typename MatrixType >

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

staticprivate

Definition at line 296 of file HessenbergDecomposition.h.

 {
   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), numext::conj(h), &temp.coeffRef(0));
   }
 }

◆ 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:

MatrixXcf A = MatrixXcf::Random(4,4);
HessenbergDecomposition<MatrixXcf> hd(4);
hd.compute(A);
cout << "The matrix H in the decomposition of A is:" << endl << hd.matrixH() << endl;
hd.compute(2*A); // re-use hd to compute and store decomposition of 2A
cout << "The matrix H in the decomposition of 2A is:" << endl << hd.matrixH() << endl;

Output:

The matrix H in the decomposition of A is:
    (-0.211,0.68)     (0.346,0.216)  (-0.688,0.00979)    (0.0451,0.584)
        (-1.45,0) (-0.0574,-0.0123)    (-0.196,0.385)     (0.395,0.389)
            (0,0)          (1.68,0)   (-0.397,-0.552)    (0.156,-0.241)
            (0,0)             (0,0)          (1.56,0)    (0.876,-0.423)
The matrix H in the decomposition of 2A is:
   (-0.422,1.36)    (0.691,0.431)   (-1.38,0.0196)    (0.0902,1.17)
       (-2.91,0) (-0.115,-0.0246)    (-0.392,0.77)    (0.791,0.777)
           (0,0)         (3.36,0)    (-0.795,-1.1)   (0.311,-0.482)
           (0,0)            (0,0)         (3.12,0)    (1.75,-0.846)

Definition at line 154 of file HessenbergDecomposition.h.

     {
       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;
     }

◆ 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

Definition at line 181 of file HessenbergDecomposition.h.

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

◆ 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:

Matrix4f A = MatrixXf::Random(4,4);
cout << "Here is a random 4x4 matrix:" << endl << A << endl;
HessenbergDecomposition<MatrixXf> hessOfA(A);
MatrixXf H = hessOfA.matrixH();
cout << "The Hessenberg matrix H is:" << endl << H << endl;
MatrixXf Q = hessOfA.matrixQ();
cout << "The orthogonal matrix Q is:" << endl << Q << endl;
cout << "Q H Q^T is:" << endl << Q * H * Q.transpose() << endl;

Output:

Here is a random 4x4 matrix:
   0.68   0.823  -0.444   -0.27
 -0.211  -0.605   0.108  0.0268
  0.566   -0.33 -0.0452   0.904
  0.597   0.536   0.258   0.832
The Hessenberg matrix H is:
   0.68  -0.691  -0.645   0.235
  0.849   0.836  -0.419   0.794
      0  -0.469  -0.547 -0.0731
      0       0  -0.559  -0.107
The orthogonal matrix Q is:
      1       0       0       0
      0  -0.249  -0.958   0.144
      0   0.667  -0.277  -0.692
      0   0.703 -0.0761   0.707
Q H Q^T is:
   0.68   0.823  -0.444   -0.27
 -0.211  -0.605   0.108  0.0268
  0.566   -0.33 -0.0452   0.904
  0.597   0.536   0.258   0.832

See also: matrixQ(), packedMatrix()

Definition at line 264 of file HessenbergDecomposition.h.

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

◆ 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

Definition at line 236 of file HessenbergDecomposition.h.

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

◆ 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 \( H_i \) are the Householder transformations \( H_i = (I - h_i v_i v_i^T) \) where \( h_i \) is the \( i \)th Householder coefficient and \( v_i \) 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:

Matrix4d A = Matrix4d::Random(4,4);
cout << "Here is a random 4x4 matrix:" << endl << A << endl;
HessenbergDecomposition<Matrix4d> hessOfA(A);
Matrix4d pm = hessOfA.packedMatrix();
cout << "The packed matrix M is:" << endl << pm << endl;
cout << "The upper Hessenberg part corresponds to the matrix H, which is:" 
     << endl << hessOfA.matrixH() << endl;
Vector3d hc = hessOfA.householderCoefficients();
cout << "The vector of Householder coefficients is:" << endl << hc << endl;

Output:

Here is a random 4x4 matrix:
   0.68   0.823  -0.444   -0.27
 -0.211  -0.605   0.108  0.0268
  0.566   -0.33 -0.0452   0.904
  0.597   0.536   0.258   0.832
The packed matrix M is:
   0.68  -0.691  -0.645   0.235
  0.849   0.836  -0.419   0.794
 -0.534  -0.469  -0.547 -0.0731
 -0.563   0.344  -0.559  -0.107
The upper Hessenberg part corresponds to the matrix H, which is:
   0.68  -0.691  -0.645   0.235
  0.849   0.836  -0.419   0.794
      0  -0.469  -0.547 -0.0731
      0       0  -0.559  -0.107
The vector of Householder coefficients is:
1.25
1.79
   0

See also: householderCoefficients()

Definition at line 216 of file HessenbergDecomposition.h.

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

Member Data Documentation

◆ m_hCoeffs

template<typename MatrixType_ >

CoeffVectorType Eigen::HessenbergDecomposition< MatrixType_ >::m_hCoeffs

protected

Definition at line 278 of file HessenbergDecomposition.h.

◆ m_isInitialized

template<typename MatrixType_ >

bool Eigen::HessenbergDecomposition< MatrixType_ >::m_isInitialized

protected

Definition at line 280 of file HessenbergDecomposition.h.

◆ m_matrix

template<typename MatrixType_ >

MatrixType Eigen::HessenbergDecomposition< MatrixType_ >::m_matrix

protected

Definition at line 277 of file HessenbergDecomposition.h.

◆ m_temp

template<typename MatrixType_ >

VectorType Eigen::HessenbergDecomposition< MatrixType_ >::m_temp

protected

Definition at line 279 of file HessenbergDecomposition.h.

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

HessenbergDecomposition.h

Public Types

Public Member Functions

Protected Attributes

Private Types

Static Private Member Functions

Detailed Description

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

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

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