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/**
* $Id$
* ***** BEGIN GPL/BL DUAL LICENSE BLOCK *****
*
* This program is free software; you can redistribute it and/or
* modify it under the terms of the GNU General Public License
* as published by the Free Software Foundation; either version 2
* of the License, or (at your option) any later version. The Blender
* Foundation also sells licenses for use in proprietary software under
* the Blender License. See http://www.blender.org/BL/ for information
* about this.
*
* 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 General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software Foundation,
* Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
*
* The Original Code is Copyright (C) 2001-2002 by NaN Holding BV.
* All rights reserved.
*
* The Original Code is: all of this file.
*
* Contributor(s): none yet.
*
* ***** END GPL/BL DUAL LICENSE BLOCK *****
*/
/*
*
* Template Numerical Toolkit (TNT): Linear Algebra Module
*
* Mathematical and Computational Sciences Division
* National Institute of Technology,
* Gaithersburg, MD USA
*
*
* This software was developed at the National Institute of Standards and
* Technology (NIST) by employees of the Federal Government in the course
* of their official duties. Pursuant to title 17 Section 105 of the
* United States Code, this software is not subject to copyright protection
* and is in the public domain. The Template Numerical Toolkit (TNT) is
* an experimental system. NIST assumes no responsibility whatsoever for
* its use by other parties, and makes no guarantees, expressed or implied,
* about its quality, reliability, or any other characteristic.
*
* BETA VERSION INCOMPLETE AND SUBJECT TO CHANGE
* see http://math.nist.gov/tnt for latest updates.
*
*/
#ifndef LU_H
#define LU_H
// Solve system of linear equations Ax = b.
//
// Typical usage:
//
// Matrix(double) A;
// Vector(Subscript) ipiv;
// Vector(double) b;
//
// 1) LU_Factor(A,ipiv);
// 2) LU_Solve(A,ipiv,b);
//
// Now b has the solution x. Note that both A and b
// are overwritten. If these values need to be preserved,
// one can make temporary copies, as in
//
// O) Matrix(double) T = A;
// 1) LU_Factor(T,ipiv);
// 1a) Vector(double) x=b;
// 2) LU_Solve(T,ipiv,x);
//
// See details below.
//
// for fabs()
//
#include <cmath>
// right-looking LU factorization algorithm (unblocked)
//
// Factors matrix A into lower and upper triangular matrices
// (L and U respectively) in solving the linear equation Ax=b.
//
//
// Args:
//
// A (input/output) Matrix(1:n, 1:n) In input, matrix to be
// factored. On output, overwritten with lower and
// upper triangular factors.
//
// indx (output) Vector(1:n) Pivot vector. Describes how
// the rows of A were reordered to increase
// numerical stability.
//
// Return value:
//
// int (0 if successful, 1 otherwise)
//
//
namespace TNT
{
template <class MaTRiX, class VecToRSubscript>
int LU_factor( MaTRiX &A, VecToRSubscript &indx)
{
assert(A.lbound() == 1); // currently for 1-offset
assert(indx.lbound() == 1); // vectors and matrices
Subscript M = A.num_rows();
Subscript N = A.num_cols();
if (M == 0 || N==0) return 0;
if (indx.dim() != M)
indx.newsize(M);
Subscript i=0,j=0,k=0;
Subscript jp=0;
typename MaTRiX::element_type t;
Subscript minMN = (M < N ? M : N) ; // min(M,N);
for (j=1; j<= minMN; j++)
{
// find pivot in column j and test for singularity.
jp = j;
t = fabs(A(j,j));
for (i=j+1; i<=M; i++)
if ( fabs(A(i,j)) > t)
{
jp = i;
t = fabs(A(i,j));
}
indx(j) = jp;
// jp now has the index of maximum element
// of column j, below the diagonal
if ( A(jp,j) == 0 )
return 1; // factorization failed because of zero pivot
if (jp != j) // swap rows j and jp
for (k=1; k<=N; k++)
{
t = A(j,k);
A(j,k) = A(jp,k);
A(jp,k) =t;
}
if (j<M) // compute elements j+1:M of jth column
{
// note A(j,j), was A(jp,p) previously which was
// guarranteed not to be zero (Label #1)
//
typename MaTRiX::element_type recp = 1.0 / A(j,j);
for (k=j+1; k<=M; k++)
A(k,j) *= recp;
}
if (j < minMN)
{
// rank-1 update to trailing submatrix: E = E - x*y;
//
// E is the region A(j+1:M, j+1:N)
// x is the column vector A(j+1:M,j)
// y is row vector A(j,j+1:N)
Subscript ii,jj;
for (ii=j+1; ii<=M; ii++)
for (jj=j+1; jj<=N; jj++)
A(ii,jj) -= A(ii,j)*A(j,jj);
}
}
return 0;
}
template <class MaTRiX, class VecToR, class VecToRSubscripts>
int LU_solve(const MaTRiX &A, const VecToRSubscripts &indx, VecToR &b)
{
assert(A.lbound() == 1); // currently for 1-offset
assert(indx.lbound() == 1); // vectors and matrices
assert(b.lbound() == 1);
Subscript i,ii=0,ip,j;
Subscript n = b.dim();
typename MaTRiX::element_type sum = 0.0;
for (i=1;i<=n;i++)
{
ip=indx(i);
sum=b(ip);
b(ip)=b(i);
if (ii)
for (j=ii;j<=i-1;j++)
sum -= A(i,j)*b(j);
else if (sum) ii=i;
b(i)=sum;
}
for (i=n;i>=1;i--)
{
sum=b(i);
for (j=i+1;j<=n;j++)
sum -= A(i,j)*b(j);
b(i)=sum/A(i,i);
}
return 0;
}
} // namespace TNT
#endif
// LU_H
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