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/* -*- c++ -*- (enables emacs c++ mode) *//*===========================================================================
Copyright (C) 2003-2015 Yves Renard, Julien Pommier This file is a part of GETFEM++ Getfem++ 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 along with the GCC Runtime Library Exception either version 3.1 or (at your option) any later version. This program 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 and GCC Runtime Library Exception for more details. You should have received a copy of the GNU Lesser General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. As a special exception, you may use this file as it is a part of a free software library without restriction. Specifically, if other files instantiate templates or use macros or inline functions from this file, or you compile this file and link it with other files to produce an executable, this file does not by itself cause the resulting executable to be covered by the GNU Lesser General Public License. This exception does not however invalidate any other reasons why the executable file might be covered by the GNU Lesser General Public License. ===========================================================================*/
/**@file gmm_condition_number.h
@author Yves Renard <Yves.Renard@insa-lyon.fr>, Julien Pommier <Julien.Pommier@insa-toulouse.fr> @date August 27, 2003. @brief computation of the condition number of dense matrices.*/#ifndef GMM_CONDITION_NUMBER_H__
#define GMM_CONDITION_NUMBER_H__
#include "gmm_dense_qr.h"
namespace gmm {
/** computation of the condition number of dense matrices using SVD.
Uses symmetric_qr_algorithm => dense matrices only.
@param M a matrix. @param emin smallest (in magnitude) eigenvalue @param emax largest eigenvalue. */ template <typename MAT> typename number_traits<typename linalg_traits<MAT>::value_type>::magnitude_type condition_number(const MAT& M, typename number_traits<typename linalg_traits<MAT>::value_type>::magnitude_type& emin, typename number_traits<typename linalg_traits<MAT>::value_type>::magnitude_type& emax) { typedef typename linalg_traits<MAT>::value_type T; typedef typename number_traits<T>::magnitude_type R;
size_type m = mat_nrows(M), n = mat_ncols(M); emax = emin = R(0); std::vector<R> eig(m+n);
if (m+n == 0) return R(0); if (is_hermitian(M)) { eig.resize(m); gmm::symmetric_qr_algorithm(M, eig); } else { dense_matrix<T> B(m+n, m+n); // not very efficient ??
gmm::copy(conjugated(M), sub_matrix(B, sub_interval(m, n), sub_interval(0, m))); gmm::copy(M, sub_matrix(B, sub_interval(0, m), sub_interval(m, n))); gmm::symmetric_qr_algorithm(B, eig); } emin = emax = gmm::abs(eig[0]); for (size_type i = 1; i < eig.size(); ++i) { R e = gmm::abs(eig[i]); emin = std::min(emin, e); emax = std::max(emax, e); } // cout << "emin = " << emin << " emax = " << emax << endl;
if (emin == R(0)) return gmm::default_max(R()); return emax / emin; }
template <typename MAT> typename number_traits<typename linalg_traits<MAT>::value_type>::magnitude_type condition_number(const MAT& M) { typename number_traits<typename linalg_traits<MAT>::value_type>::magnitude_type emax, emin; return condition_number(M, emin, emax); }
template <typename MAT> typename number_traits<typename linalg_traits<MAT>::value_type>::magnitude_type Frobenius_condition_number_sqr(const MAT& M) { typedef typename linalg_traits<MAT>::value_type T; typedef typename number_traits<T>::magnitude_type R; size_type m = mat_nrows(M), n = mat_ncols(M); dense_matrix<T> B(std::min(m,n), std::min(m,n)); if (m < n) mult(M,gmm::conjugated(M),B); else mult(gmm::conjugated(M),M,B); R trB = abs(mat_trace(B)); lu_inverse(B); return trB*abs(mat_trace(B)); }
template <typename MAT> typename number_traits<typename linalg_traits<MAT>::value_type>::magnitude_type Frobenius_condition_number(const MAT& M) { return sqrt(Frobenius_condition_number_sqr(M)); }
/** estimation of the condition number (TO BE DONE...)
*/ template <typename MAT> typename number_traits<typename linalg_traits<MAT>::value_type>::magnitude_type condest(const MAT& M, typename number_traits<typename linalg_traits<MAT>::value_type>::magnitude_type& emin, typename number_traits<typename linalg_traits<MAT>::value_type>::magnitude_type& emax) { return condition_number(M, emin, emax); } template <typename MAT> typename number_traits<typename linalg_traits<MAT>::value_type>::magnitude_type condest(const MAT& M) { typename number_traits<typename linalg_traits<MAT>::value_type>::magnitude_type emax, emin; return condest(M, emin, emax); }}
#endif
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