Reduces a square matrix to Hessenberg form by an orthogonal similarity transformation. More...
#include <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 Matrix< Scalar, SizeMinusOne, 1, Options &~RowMajor, MaxSizeMinusOne, 1 > | CoeffVectorType |
Type for vector of Householder coefficients. More... | |
typedef HouseholderSequence< MatrixType, typename internal::remove_all< typename CoeffVectorType::ConjugateReturnType >::type > | HouseholderSequenceType |
Return type of matrixQ() More... | |
typedef MatrixType::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 | |
HessenbergDecomposition & | compute (const MatrixType &matrix) |
Computes Hessenberg decomposition of given matrix. More... | |
HessenbergDecomposition (Index size=Size==Dynamic?2:Size) | |
Default constructor; the decomposition will be computed later. More... | |
HessenbergDecomposition (const MatrixType &matrix) | |
Constructor; computes Hessenberg decomposition of given matrix. 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, Options|RowMajor, 1, MaxSize > | VectorType |
Static Private Member Functions | |
static void | _compute (MatrixType &matA, CoeffVectorType &hCoeffs, VectorType &temp) |
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 57 of file HessenbergDecomposition.h.
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 82 of file HessenbergDecomposition.h.
typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename CoeffVectorType::ConjugateReturnType>::type> Eigen::HessenbergDecomposition< _MatrixType >::HouseholderSequenceType |
Return type of matrixQ()
Definition at line 85 of file HessenbergDecomposition.h.
typedef MatrixType::Index Eigen::HessenbergDecomposition< _MatrixType >::Index |
Definition at line 74 of file HessenbergDecomposition.h.
typedef internal::HessenbergDecompositionMatrixHReturnType<MatrixType> Eigen::HessenbergDecomposition< _MatrixType >::MatrixHReturnType |
Definition at line 87 of file HessenbergDecomposition.h.
typedef _MatrixType Eigen::HessenbergDecomposition< _MatrixType >::MatrixType |
Synonym for the template parameter _MatrixType
.
Definition at line 62 of file HessenbergDecomposition.h.
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Definition at line 269 of file HessenbergDecomposition.h.
typedef MatrixType::Scalar Eigen::HessenbergDecomposition< _MatrixType >::Scalar |
Scalar type for matrices of type MatrixType.
Definition at line 73 of file HessenbergDecomposition.h.
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private |
Definition at line 268 of file HessenbergDecomposition.h.
anonymous enum |
Enumerator | |
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Size | |
SizeMinusOne | |
Options | |
MaxSize | |
MaxSizeMinusOne |
Definition at line 64 of file HessenbergDecomposition.h.
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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 100 of file HessenbergDecomposition.h.
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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 118 of file HessenbergDecomposition.h.
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staticprivate |
Definition at line 292 of file HessenbergDecomposition.h.
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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:
Definition at line 150 of file HessenbergDecomposition.h.
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Returns the Householder coefficients.
The Householder coefficients allow the reconstruction of the matrix in the Hessenberg decomposition from the packed data.
Definition at line 177 of file HessenbergDecomposition.h.
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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:
Definition at line 260 of file HessenbergDecomposition.h.
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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 232 of file HessenbergDecomposition.h.
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Returns the internal representation of the decomposition.
The returned matrix contains the following information:
See LAPACK for further details on this packed storage.
Example:
Definition at line 212 of file HessenbergDecomposition.h.
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Definition at line 274 of file HessenbergDecomposition.h.
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Definition at line 276 of file HessenbergDecomposition.h.
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Definition at line 273 of file HessenbergDecomposition.h.
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Definition at line 275 of file HessenbergDecomposition.h.