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HessenbergDecomposition< _MatrixType > Class Template Reference
Public Types | Public Member Functions

HessenbergDecomposition< _MatrixType > Class Template Reference
[Eigenvalues module]

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#include <HessenbergDecomposition.h>

Public Types

typedef _MatrixType MatrixType
 Synonym for the template parameter _MatrixType.
typedef MatrixType::Scalar Scalar
 Scalar type for matrices of type MatrixType.
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()

Public Member Functions

 HessenbergDecomposition (Index size=Size==Dynamic?2:Size)
 Default constructor; the decomposition will be computed later.
 HessenbergDecomposition (const MatrixType &matrix)
 Constructor; computes Hessenberg decomposition of given matrix.
HessenbergDecomposition & compute (const MatrixType &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.

Detailed Description

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

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

Template Parameters:
_MatrixTypethe 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 57 of file HessenbergDecomposition.h.

Member Typedef Documentation

typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> 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 82 of file HessenbergDecomposition.h.

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

Return type of matrixQ()

Definition at line 85 of file HessenbergDecomposition.h.

typedef _MatrixType MatrixType

Synonym for the template parameter _MatrixType.

Definition at line 62 of file HessenbergDecomposition.h.

typedef MatrixType::Scalar Scalar

Scalar type for matrices of type MatrixType.

Definition at line 73 of file HessenbergDecomposition.h.

Constructor & Destructor Documentation

HessenbergDecomposition ( Index  size = Size==Dynamic ? 2 : Size )

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.

Definition at line 100 of file HessenbergDecomposition.h.

HessenbergDecomposition ( const MatrixType &  matrix )

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.

Definition at line 118 of file HessenbergDecomposition.h.

Member Function Documentation

HessenbergDecomposition& compute ( const MatrixType &  matrix )

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 of the allocated data in the HessenbergDecomposition object.

Example: 

 Output: 


 

Definition at line 150 of file HessenbergDecomposition.h.











      

        

          const CoeffVectorType& householderCoefficients 
          (
           )
           const
        

      







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 177 of file HessenbergDecomposition.h.











      

        

          MatrixHReturnType matrixH 
          (
           )
           const
        

      







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() 



Definition at line 260 of file HessenbergDecomposition.h.











      

        

          HouseholderSequenceType matrixQ 
          (
           )
           const
        

      







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 232 of file HessenbergDecomposition.h.











      

        

          const MatrixType& packedMatrix 
          (
           )
           const
        

      







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: 


 Output: 


See also:
householderCoefficients() 



Definition at line 212 of file HessenbergDecomposition.h.









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