Eigen  5.0.1
 
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Eigen::HessenbergDecomposition< MatrixType_ > Class Template Reference
Dense linear problems and decompositions » Reference » Eigenvalues module

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

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>
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

Public Types

using CoeffVectorType
 Type for vector of Householder coefficients.
 
using HouseholderSequenceType
 Return type of matrixQ()
 
using Index
 
using MatrixType
 Synonym for the template parameter MatrixType_.
 
using Scalar
 Scalar type for matrices of type MatrixType.
 

Public Member Functions

template<typename InputType>
HessenbergDecomposition & compute (const EigenBase< InputType > &matrix)
 Computes Hessenberg decomposition of given matrix.
 
template<typename InputType>
 HessenbergDecomposition (const EigenBase< InputType > &matrix)
 Constructor; computes Hessenberg decomposition of given matrix.
 
template<typename InputType>
 HessenbergDecomposition (EigenBase< InputType > &matrix)
 Constructor for inplace decomposition .
 
 HessenbergDecomposition (Index size=Size==Dynamic ? 2 :Size)
 Default constructor; the decomposition will be computed later.
 
const CoeffVectorType & householderCoefficients () const
 Returns the Householder coefficients.
 
MatrixHReturnType matrixH () const
 Constructs the Hessenberg matrix H in the decomposition.
 
HouseholderSequenceType matrixQ () const
 Reconstructs the orthogonal matrix Q in the decomposition.
 
const MatrixType & packedMatrix () const
 Returns the internal representation of the decomposition.
 

Member Typedef Documentation

◆ CoeffVectorType

template<typename MatrixType_>
using 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-size type.

◆ Index

template<typename MatrixType_>
using Eigen::HessenbergDecomposition< MatrixType_ >::Index

Constructor & Destructor Documentation

◆ HessenbergDecomposition() [1/3]

template<typename MatrixType_>
Eigen::HessenbergDecomposition< MatrixType_ >::HessenbergDecomposition ( Index size = Size == Dynamic ? 2 : Size)
inlineexplicit

Default constructor; the decomposition will be computed later.

Parameters
[in]sizeThe 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.

◆ HessenbergDecomposition() [2/3]

template<typename MatrixType_>
template<typename InputType>
Eigen::HessenbergDecomposition< MatrixType_ >::HessenbergDecomposition ( const EigenBase< InputType > & matrix)
inlineexplicit

Constructor; computes Hessenberg decomposition of given matrix.

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

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

See also
matrixH() for an example.

◆ HessenbergDecomposition() [3/3]

template<typename MatrixType_>
template<typename InputType>
Eigen::HessenbergDecomposition< MatrixType_ >::HessenbergDecomposition ( EigenBase< InputType > & matrix)
inlineexplicit

Constructor for inplace decomposition .

Parameters
[in,out]matrixSquare matrix whose Hessenberg decomposition is to be computed.

When MatrixType is a Ref<>, the decomposition is computed within the memory of matrix, which then holds the packed representation returned by packedMatrix(). Otherwise this constructor behaves like HessenbergDecomposition(const EigenBase<InputType>&).

Member Function Documentation

◆ 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]matrixSquare 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 the allocated data in the HessenbergDecomposition object.

Example:

// SPDX-FileCopyrightText: The Eigen Authors
// SPDX-License-Identifier: MPL-2.0
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); // reuse hd to compute and store decomposition of 2A
cout << "The matrix H in the decomposition of 2A is:" << endl << hd.matrixH() << endl;
Eigen::HessenbergDecomposition::HessenbergDecomposition
HessenbergDecomposition(Index size=Size==Dynamic ? 2 :Size)
Default constructor; the decomposition will be computed later.
Definition HessenbergDecomposition.h:107
Eigen::MatrixXcf
Matrix< std::complex< float >, Dynamic, Dynamic > MatrixXcf
Dynamic×Dynamic matrix of type std::complex<float>.
Definition Matrix.h:490

Output: 

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

◆ 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

◆ 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:

// SPDX-FileCopyrightText: The Eigen Authors
// SPDX-License-Identifier: MPL-2.0
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;
Eigen::MatrixXf
Matrix< float, Dynamic, Dynamic > MatrixXf
Dynamic×Dynamic matrix of type float.
Definition Matrix.h:488
Eigen::Matrix4f
Matrix< float, 4, 4 > Matrix4f
4×4 matrix of type float.
Definition Matrix.h:488

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()

◆ 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

◆ 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:

// SPDX-FileCopyrightText: The Eigen Authors
// SPDX-License-Identifier: MPL-2.0
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;
Eigen::Vector3d
Matrix< double, 3, 1 > Vector3d
3×1 vector of type double.
Definition Matrix.h:489
Eigen::Matrix4d
Matrix< double, 4, 4 > Matrix4d
4×4 matrix of type double.
Definition Matrix.h:489

Output: 

Here is a random 4x4 matrix:
-0.211  0.108  0.435 -0.198
 0.597  0.258  0.214 -0.782
-0.605 0.0268 -0.514 -0.563
 0.536  0.832  0.608  0.678
The packed matrix M is:
 -0.211   0.303  -0.199   0.329
     -1  0.0133    0.15  -0.975
 -0.378  -0.629   0.473   0.826
  0.335  -0.826  -0.742 -0.0647
The upper Hessenberg part corresponds to the matrix H, which is:
 -0.211   0.303  -0.199   0.329
     -1  0.0133    0.15  -0.975
      0  -0.629   0.473   0.826
      0       0  -0.742 -0.0647
The vector of Householder coefficients is:
1.59
1.19
   0
See also
householderCoefficients()
The documentation for this class was generated from the following file: