Public Types | Public Member Functions | Protected Attributes | Private Types | Static Private Member Functions
HessenbergDecomposition< _MatrixType > Class Template Reference

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

#include <HessenbergDecomposition.h>

List of all members.

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 Matrix< Scalar,
SizeMinusOne, 1, Options
&~RowMajor, MaxSizeMinusOne, 1 > 		CoeffVectorType
 Type for vector of Householder coefficients.
typedef HouseholderSequence
< MatrixType, CoeffVectorType >
::ConjugateReturnType 		HouseholderSequenceType
 Return type of matrixQ()
typedef MatrixType::Index Index
typedef
internal::HessenbergDecompositionMatrixHReturnType
< MatrixType > 		MatrixHReturnType
typedef _MatrixType MatrixType
 Synonym for the template parameter _MatrixType.
typedef MatrixType::Scalar Scalar
 Scalar type for matrices of type MatrixType.

Public Member Functions

HessenbergDecomposition & compute (const MatrixType &matrix)
 Computes Hessenberg decomposition of given matrix.
 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.
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.

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, Options|RowMajor,
1, MaxSize > 		VectorType

Static Private Member Functions

static void _compute (MatrixType &matA, CoeffVectorType &hCoeffs, VectorType &temp)

Detailed Description

template<typename _MatrixType>
class 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 70 of file HessenbergDecomposition.h.

Member Typedef Documentation

template<typename _MatrixType>
typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> 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 95 of file HessenbergDecomposition.h.

template<typename _MatrixType>
typedef HouseholderSequence<MatrixType,CoeffVectorType>::ConjugateReturnType HessenbergDecomposition< _MatrixType >::HouseholderSequenceType

Return type of matrixQ()

Definition at line 98 of file HessenbergDecomposition.h.

template<typename _MatrixType>
typedef MatrixType::Index HessenbergDecomposition< _MatrixType >::Index

Definition at line 87 of file HessenbergDecomposition.h.

template<typename _MatrixType>
typedef internal::HessenbergDecompositionMatrixHReturnType<MatrixType> HessenbergDecomposition< _MatrixType >::MatrixHReturnType

Definition at line 100 of file HessenbergDecomposition.h.

template<typename _MatrixType>
typedef _MatrixType HessenbergDecomposition< _MatrixType >::MatrixType

Synonym for the template parameter _MatrixType.

Definition at line 75 of file HessenbergDecomposition.h.

template<typename _MatrixType>
typedef NumTraits<Scalar>::Real HessenbergDecomposition< _MatrixType >::RealScalar [private]

Definition at line 282 of file HessenbergDecomposition.h.

template<typename _MatrixType>
typedef MatrixType::Scalar HessenbergDecomposition< _MatrixType >::Scalar

Scalar type for matrices of type MatrixType.

Definition at line 86 of file HessenbergDecomposition.h.

template<typename _MatrixType>
typedef Matrix<Scalar, 1, Size, Options | RowMajor, 1, MaxSize> HessenbergDecomposition< _MatrixType >::VectorType [private]

Definition at line 281 of file HessenbergDecomposition.h.

Member Enumeration Documentation

template<typename _MatrixType>
anonymous enum
Enumerator:
Size 
SizeMinusOne 
Options 
MaxSize 
MaxSizeMinusOne 

Definition at line 77 of file HessenbergDecomposition.h.

Constructor & Destructor Documentation

template<typename _MatrixType>
HessenbergDecomposition< _MatrixType >::HessenbergDecomposition ( Index  size = Size==Dynamic ? 2 : Size) [inline]

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 113 of file HessenbergDecomposition.h.

template<typename _MatrixType>
HessenbergDecomposition< _MatrixType >::HessenbergDecomposition ( const MatrixType &  matrix) [inline]

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 131 of file HessenbergDecomposition.h.

Member Function Documentation

template<typename _MatrixType>
static void HessenbergDecomposition< _MatrixType >::_compute ( MatrixType &  matA,
CoeffVectorType &  hCoeffs,
VectorType &  temp 
) [static, private]
template<typename _MatrixType>
HessenbergDecomposition& HessenbergDecomposition< _MatrixType >::compute ( const MatrixType &  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 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: 

 

Definition at line 163 of file HessenbergDecomposition.h.













template<typename _MatrixType> 

      

        

          const CoeffVectorType& 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 190 of file HessenbergDecomposition.h.













template<typename _MatrixType> 

      

        

          MatrixHReturnType 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: 


See also:
matrixQ(), packedMatrix() 



Definition at line 273 of file HessenbergDecomposition.h.













template<typename _MatrixType> 

      

        

          HouseholderSequenceType HessenbergDecomposition< _MatrixType >::matrixQ 
          (
          void 
          )
           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 245 of file HessenbergDecomposition.h.













template<typename _MatrixType> 

      

        

          const MatrixType& 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: 


See also:
householderCoefficients() 



Definition at line 225 of file HessenbergDecomposition.h.







Member Data Documentation








template<typename _MatrixType> 

      

        

          CoeffVectorType HessenbergDecomposition< _MatrixType >::m_hCoeffs [protected]
        

      







Definition at line 287 of file HessenbergDecomposition.h.













template<typename _MatrixType> 

      

        

          bool HessenbergDecomposition< _MatrixType >::m_isInitialized [protected]
        

      







Definition at line 289 of file HessenbergDecomposition.h.













template<typename _MatrixType> 

      

        

          MatrixType HessenbergDecomposition< _MatrixType >::m_matrix [protected]
        

      







Definition at line 286 of file HessenbergDecomposition.h.













template<typename _MatrixType> 

      

        

          VectorType HessenbergDecomposition< _MatrixType >::m_temp [protected]
        

      







Definition at line 288 of file HessenbergDecomposition.h.







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







libicr

Author(s): Robert Krug

autogenerated on Mon Jan 6 2014 11:34:21