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*
*
* \brief Algorithm for Triangular-Quadratic-Decomposition
*
*
*
*
* \author O. Krause
* \date 2011
*
*
* \par Copyright 1995-2015 Shark Development Team
*
* <BR><HR>
* This file is part of Shark.
* <http://image.diku.dk/shark/>
*
* Shark is free software: you can redistribute it and/or modify
* it under the terms of the GNU Lesser General Public License as published
* by the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* Shark is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with Shark. If not, see <http://www.gnu.org/licenses/>.
*
*/
#ifndef SHARK_LINALG_RQ_H
#define SHARK_LINALG_RQ_H
#include <shark/LinAlg/Base.h>
namespace shark{ namespace blas{
/**
* \ingroup shark_globals
*
* @{
*/
/*!
* \brief Determines the RQ Decomposition of the matrix A using pivoting
* returning the housholder transformation instead of Q.
*
* The pivoting RQ-Decomposition finds an orthonormal matrix Q and a lower Triangular matrix R
* as well as a permutation matrix P such that PA = R*Q.
* Since Q is the multiplication of all householder transformations,
* It is quite expensive to compute. Often, Q is only an intermediate step in computations which can be
* carried out more efficiently using the Householder Transformations themselves.
*
* The Matrix format of the householder transform is that the transformations are stored as
* upper triangular matrix. The first transformation being in the first row and so on.
*/
template<class MatrixT,class Mat>
std::size_t pivotingRQ
(
blas::matrix_expression<MatrixT> const& matrixA,
blas::matrix_container<Mat>& matrixR,
blas::matrix_container<Mat>& matrixQ,
blas::permutation_matrix& permutation
);
/*!
* \brief Determines the RQ Decomposition of the matrix A using pivoting
*
* The pivoting RQ-Decomposition finds an orthonormal matrix Q and a lower Triangular matrix R
* as well as a permuation matrix P such that PA = R*Q.
* This function is better known as the QR-Decomposition
* of a transposed matrix B^T = A and B = QR.
*
* This version of the algorithm is based on householder transformations. since it uses pivoting it can
* be used to determine the rank of a matrix. The i-th applied householder decomposition has the form
* H_i=I-v_iv_i^T and the v_i are stored separately.
* A mxn natrix needs at most k=min{m,n} transformations.
* Also as RQ=AH_1,...,H_k Q=H_k,...,H_1
*/
template<class MatrixT,class MatrixU>
std::size_t pivotingRQHouseholder
(
blas::matrix_expression<MatrixT> const& matrixA,
blas::matrix_container<MatrixU>& matrixR,
blas::matrix_container<MatrixU>& householderV,
blas::permutation_matrix& permutation
);
/** @}*/
}}
#include "Impl/pivotingRQ.inl"
#endif
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