Reduces a square matrix to Hessenberg form by an orthogonal similarity transformation
_MatrixType | the type of the matrix of which we are computing the Hessenberg decomposition |
This class performs an Hessenberg decomposition of a matrix . In the real case, the Hessenberg decomposition consists of an orthogonal matrix
and a Hessenberg matrix
such that
. An orthogonal matrix is a matrix whose inverse equals its transpose (
). A Hessenberg matrix has zeros below the subdiagonal, so it is almost upper triangular. The Hessenberg decomposition of a complex matrix is
with
unitary (that is,
).
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.
Definition at line 70 of file HessenbergDecomposition.h.
#include <src/Eigenvalues/HessenbergDecomposition.h>
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 MatrixType::Index | Index |
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 internal::HessenbergDecompositionMatrixHReturnType < MatrixType > | MatrixHReturnType |
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. | |
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) |
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.
typedef HouseholderSequence<MatrixType,CoeffVectorType>::ConjugateReturnType HessenbergDecomposition< _MatrixType >::HouseholderSequenceType |
Return type of matrixQ().
Definition at line 98 of file HessenbergDecomposition.h.
typedef MatrixType::Index HessenbergDecomposition< _MatrixType >::Index |
Definition at line 87 of file HessenbergDecomposition.h.
typedef internal::HessenbergDecompositionMatrixHReturnType<MatrixType> HessenbergDecomposition< _MatrixType >::MatrixHReturnType |
Definition at line 100 of file HessenbergDecomposition.h.
typedef _MatrixType HessenbergDecomposition< _MatrixType >::MatrixType |
Synonym for the template parameter _MatrixType
.
Definition at line 75 of file HessenbergDecomposition.h.
typedef NumTraits<Scalar>::Real HessenbergDecomposition< _MatrixType >::RealScalar [private] |
Definition at line 282 of file HessenbergDecomposition.h.
typedef MatrixType::Scalar HessenbergDecomposition< _MatrixType >::Scalar |
Scalar type for matrices of type MatrixType.
Definition at line 86 of file HessenbergDecomposition.h.
typedef Matrix<Scalar, 1, Size, Options | RowMajor, 1, MaxSize> HessenbergDecomposition< _MatrixType >::VectorType [private] |
Definition at line 281 of file HessenbergDecomposition.h.
anonymous enum |
Definition at line 77 of file HessenbergDecomposition.h.
HessenbergDecomposition< _MatrixType >::HessenbergDecomposition | ( | Index | size = Size==Dynamic ? 2 : Size |
) | [inline] |
Default constructor; the decomposition will be computed later.
[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.
Definition at line 113 of file HessenbergDecomposition.h.
HessenbergDecomposition< _MatrixType >::HessenbergDecomposition | ( | const MatrixType & | matrix | ) | [inline] |
Constructor; computes Hessenberg decomposition of given matrix.
[in] | matrix | Square matrix whose Hessenberg decomposition is to be computed. |
This constructor calls compute() to compute the Hessenberg decomposition.
Definition at line 131 of file HessenbergDecomposition.h.
static void HessenbergDecomposition< _MatrixType >::_compute | ( | MatrixType & | matA, | |
CoeffVectorType & | hCoeffs, | |||
VectorType & | temp | |||
) | [static, private] |
HessenbergDecomposition& HessenbergDecomposition< _MatrixType >::compute | ( | const MatrixType & | matrix | ) | [inline] |
Computes Hessenberg decomposition of given matrix.
[in] | matrix | Square matrix whose Hessenberg decomposition is to be computed. |
*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 flops, where
denotes the size of the given matrix.
This method reuses of the allocated data in the HessenbergDecomposition object.
Example:
Output:
Definition at line 163 of file HessenbergDecomposition.h.
Referenced by RealSchur< MatrixType >::compute(), internal::complex_schur_reduce_to_hessenberg< MatrixType, false >::run(), and internal::complex_schur_reduce_to_hessenberg< MatrixType, IsComplex >::run().
const CoeffVectorType& HessenbergDecomposition< _MatrixType >::householderCoefficients | ( | ) | const [inline] |
Returns the Householder coefficients.
The Householder coefficients allow the reconstruction of the matrix in the Hessenberg decomposition from the packed data.
Definition at line 190 of file HessenbergDecomposition.h.
MatrixHReturnType HessenbergDecomposition< _MatrixType >::matrixH | ( | ) | const [inline] |
Constructs the Hessenberg matrix H in the decomposition.
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:
Definition at line 273 of file HessenbergDecomposition.h.
Referenced by RealSchur< MatrixType >::compute(), internal::complex_schur_reduce_to_hessenberg< MatrixType, false >::run(), and internal::complex_schur_reduce_to_hessenberg< MatrixType, IsComplex >::run().
HouseholderSequenceType HessenbergDecomposition< _MatrixType >::matrixQ | ( | ) | const [inline] |
Reconstructs the orthogonal matrix Q in the decomposition.
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.
Definition at line 245 of file HessenbergDecomposition.h.
Referenced by RealSchur< MatrixType >::compute(), internal::complex_schur_reduce_to_hessenberg< MatrixType, false >::run(), and internal::complex_schur_reduce_to_hessenberg< MatrixType, IsComplex >::run().
const MatrixType& HessenbergDecomposition< _MatrixType >::packedMatrix | ( | ) | const [inline] |
Returns the internal representation of the decomposition.
The returned matrix contains the following information:
See LAPACK for further details on this packed storage.
Example:
Output:
Definition at line 225 of file HessenbergDecomposition.h.
Referenced by internal::HessenbergDecompositionMatrixHReturnType< MatrixType >::cols(), internal::HessenbergDecompositionMatrixHReturnType< MatrixType >::evalTo(), and internal::HessenbergDecompositionMatrixHReturnType< MatrixType >::rows().
CoeffVectorType HessenbergDecomposition< _MatrixType >::m_hCoeffs [protected] |
Definition at line 287 of file HessenbergDecomposition.h.
Referenced by HessenbergDecomposition< MatrixType >::compute(), HessenbergDecomposition< MatrixType >::HessenbergDecomposition(), HessenbergDecomposition< MatrixType >::householderCoefficients(), and HessenbergDecomposition< MatrixType >::matrixQ().
bool HessenbergDecomposition< _MatrixType >::m_isInitialized [protected] |
Definition at line 289 of file HessenbergDecomposition.h.
Referenced by HessenbergDecomposition< MatrixType >::compute(), HessenbergDecomposition< MatrixType >::HessenbergDecomposition(), HessenbergDecomposition< MatrixType >::householderCoefficients(), HessenbergDecomposition< MatrixType >::matrixH(), HessenbergDecomposition< MatrixType >::matrixQ(), and HessenbergDecomposition< MatrixType >::packedMatrix().
MatrixType HessenbergDecomposition< _MatrixType >::m_matrix [protected] |
VectorType HessenbergDecomposition< _MatrixType >::m_temp [protected] |
Definition at line 288 of file HessenbergDecomposition.h.
Referenced by HessenbergDecomposition< MatrixType >::compute(), and HessenbergDecomposition< MatrixType >::HessenbergDecomposition().
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