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/* -*- c++ -*- (enables emacs c++ mode) *//*===========================================================================
Copyright (C) 2003-2015 Yves Renard 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. ===========================================================================*/
// This file is a modified version of lu.h from MTL.
// See http://osl.iu.edu/research/mtl/
// Following the corresponding Copyright notice.
//===========================================================================
//
// Copyright (c) 1998-2001, University of Notre Dame. All rights reserved.
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//
// * Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above copyright
// notice, this list of conditions and the following disclaimer in the
// documentation and/or other materials provided with the distribution.
// * Neither the name of the University of Notre Dame nor the
// names of its contributors may be used to endorse or promote products
// derived from this software without specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE TRUSTEES OF INDIANA UNIVERSITY AND
// CONTRIBUTORS ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING,
// BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
// FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE TRUSTEES
// OF INDIANA UNIVERSITY AND CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
// INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
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/**@file gmm_dense_lu.h
@author Andrew Lumsdaine, Jeremy G. Siek, Lie-Quan Lee, Y. Renard @date June 5, 2003. @brief LU factorizations and determinant computation for dense matrices.*/#ifndef GMM_DENSE_LU_H
#define GMM_DENSE_LU_H
#include "gmm_dense_Householder.h"
#include "gmm_opt.h"
namespace gmm {
/** LU Factorization of a general (dense) matrix (real or complex).
This is the outer product (a level-2 operation) form of the LU Factorization with pivoting algorithm . This is equivalent to LAPACK's dgetf2. Also see "Matrix Computations" 3rd Ed. by Golub and Van Loan section 3.2.5 and especially page 115. The pivot indices in ipvt are indexed starting from 1 so that this is compatible with LAPACK (Fortran). */ template <typename DenseMatrix, typename Pvector> size_type lu_factor(DenseMatrix& A, Pvector& ipvt) { typedef typename linalg_traits<DenseMatrix>::value_type T; typedef typename linalg_traits<Pvector>::value_type int_T; typedef typename number_traits<T>::magnitude_type R; size_type info(0), i, j, jp, M(mat_nrows(A)), N(mat_ncols(A)); size_type NN = std::min(M, N); std::vector<T> c(M), r(N); GMM_ASSERT2(ipvt.size()+1 >= NN, "IPVT too small"); for (i = 0; i+1 < NN; ++i) ipvt[i] = int_T(i); if (M || N) { for (j = 0; j+1 < NN; ++j) { R max = gmm::abs(A(j,j)); jp = j; for (i = j+1; i < M; ++i) /* find pivot. */ if (gmm::abs(A(i,j)) > max) { jp = i; max = gmm::abs(A(i,j)); } ipvt[j] = int_T(jp + 1); if (max == R(0)) { info = j + 1; break; } if (jp != j) for (i = 0; i < N; ++i) std::swap(A(jp, i), A(j, i)); for (i = j+1; i < M; ++i) { A(i, j) /= A(j,j); c[i-j-1] = -A(i, j); } for (i = j+1; i < N; ++i) r[i-j-1] = A(j, i); // avoid the copy ?
rank_one_update(sub_matrix(A, sub_interval(j+1, M-j-1), sub_interval(j+1, N-j-1)), c, conjugated(r)); } ipvt[j] = int_T(j + 1); } return info; } /** LU Solve : Solve equation Ax=b, given an LU factored matrix.*/ // Thanks to Valient Gough for this routine!
template <typename DenseMatrix, typename VectorB, typename VectorX, typename Pvector> void lu_solve(const DenseMatrix &LU, const Pvector& pvector, VectorX &x, const VectorB &b) { typedef typename linalg_traits<DenseMatrix>::value_type T; copy(b, x); for(size_type i = 0; i < pvector.size(); ++i) { size_type perm = pvector[i]-1; // permutations stored in 1's offset
if(i != perm) { T aux = x[i]; x[i] = x[perm]; x[perm] = aux; } } /* solve Ax = b -> LUx = b -> Ux = L^-1 b. */ lower_tri_solve(LU, x, true); upper_tri_solve(LU, x, false); }
template <typename DenseMatrix, typename VectorB, typename VectorX> void lu_solve(const DenseMatrix &A, VectorX &x, const VectorB &b) { typedef typename linalg_traits<DenseMatrix>::value_type T; dense_matrix<T> B(mat_nrows(A), mat_ncols(A)); std::vector<int> ipvt(mat_nrows(A)); gmm::copy(A, B); size_type info = lu_factor(B, ipvt); GMM_ASSERT1(!info, "Singular system, pivot = " << info); lu_solve(B, ipvt, x, b); } template <typename DenseMatrix, typename VectorB, typename VectorX, typename Pvector> void lu_solve_transposed(const DenseMatrix &LU, const Pvector& pvector, VectorX &x, const VectorB &b) { typedef typename linalg_traits<DenseMatrix>::value_type T; copy(b, x); lower_tri_solve(transposed(LU), x, false); upper_tri_solve(transposed(LU), x, true); for(size_type i = pvector.size(); i > 0; --i) { size_type perm = pvector[i-1]-1; // permutations stored in 1's offset
if(i-1 != perm) { T aux = x[i-1]; x[i-1] = x[perm]; x[perm] = aux; } }
}
///@cond DOXY_SHOW_ALL_FUNCTIONS
template <typename DenseMatrixLU, typename DenseMatrix, typename Pvector> void lu_inverse(const DenseMatrixLU& LU, const Pvector& pvector, DenseMatrix& AInv, col_major) { typedef typename linalg_traits<DenseMatrixLU>::value_type T; std::vector<T> tmp(pvector.size(), T(0)); std::vector<T> result(pvector.size()); for(size_type i = 0; i < pvector.size(); ++i) { tmp[i] = T(1); lu_solve(LU, pvector, result, tmp); copy(result, mat_col(AInv, i)); tmp[i] = T(0); } }
template <typename DenseMatrixLU, typename DenseMatrix, typename Pvector> void lu_inverse(const DenseMatrixLU& LU, const Pvector& pvector, DenseMatrix& AInv, row_major) { typedef typename linalg_traits<DenseMatrixLU>::value_type T; std::vector<T> tmp(pvector.size(), T(0)); std::vector<T> result(pvector.size()); for(size_type i = 0; i < pvector.size(); ++i) { tmp[i] = T(1); // to be optimized !!
// on peut sur le premier tri solve reduire le systeme
// et peut etre faire un solve sur une serie de vecteurs au lieu
// de vecteur a vecteur (accumulation directe de l'inverse dans la
// matrice au fur et a mesure du calcul ... -> evite la copie finale
lu_solve_transposed(LU, pvector, result, tmp); copy(result, mat_row(AInv, i)); tmp[i] = T(0); } } ///@endcond
/** Given an LU factored matrix, build the inverse of the matrix. */ template <typename DenseMatrixLU, typename DenseMatrix, typename Pvector> void lu_inverse(const DenseMatrixLU& LU, const Pvector& pvector, const DenseMatrix& AInv_) { DenseMatrix& AInv = const_cast<DenseMatrix&>(AInv_); lu_inverse(LU, pvector, AInv, typename principal_orientation_type<typename linalg_traits<DenseMatrix>::sub_orientation>::potype()); }
/** Given a dense matrix, build the inverse of the matrix, and
return the determinant */ template <typename DenseMatrix> typename linalg_traits<DenseMatrix>::value_type lu_inverse(const DenseMatrix& A_, bool doassert = true) { typedef typename linalg_traits<DenseMatrix>::value_type T; DenseMatrix& A = const_cast<DenseMatrix&>(A_); dense_matrix<T> B(mat_nrows(A), mat_ncols(A)); std::vector<int> ipvt(mat_nrows(A)); gmm::copy(A, B); size_type info = lu_factor(B, ipvt); if (doassert) GMM_ASSERT1(!info, "Non invertible matrix, pivot = "<<info); if (!info) lu_inverse(B, ipvt, A); return lu_det(B, ipvt); }
/** Compute the matrix determinant (via a LU factorization) */ template <typename DenseMatrixLU, typename Pvector> typename linalg_traits<DenseMatrixLU>::value_type lu_det(const DenseMatrixLU& LU, const Pvector &pvector) { typedef typename linalg_traits<DenseMatrixLU>::value_type T; T det(1); for (size_type j = 0; j < std::min(mat_nrows(LU), mat_ncols(LU)); ++j) det *= LU(j,j); for(size_type i = 0; i < pvector.size(); ++i) if (i != size_type(pvector[i]-1)) { det = -det; } return det; }
template <typename DenseMatrix> typename linalg_traits<DenseMatrix>::value_type lu_det(const DenseMatrix& A) { typedef typename linalg_traits<DenseMatrix>::value_type T; dense_matrix<T> B(mat_nrows(A), mat_ncols(A)); std::vector<int> ipvt(mat_nrows(A)); gmm::copy(A, B); lu_factor(B, ipvt); return lu_det(B, ipvt); }
}
#endif
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