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/*
* Computation of Legendre Gauss-Lobatto nodes and weights
*/
/*
* Copyright (c) 2005 Philippe Grandclment
*
* This file is part of LORENE.
*
* LORENE 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.
*
* LORENE 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 LORENE; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*
*/
/*
* $Id: legendre_nodes.C,v 1.2 2014/10/06 15:09:48 j_novak Exp $
* $Log: legendre_nodes.C,v $
* Revision 1.2 2014/10/06 15:09:48 j_novak
* Modified #include directives to use c++ syntax.
*
* Revision 1.1 2005/11/14 01:56:59 e_gourgoulhon
* First version
*
*
* $Header: /cvsroot/Lorene/School05/Monday/legendre_nodes.C,v 1.2 2014/10/06 15:09:48 j_novak Exp $
*
*/
#include <iostream>
using namespace std ;
#include<cmath>
#include<cstdlib>
void legendre_poly_der(int n, double& poly, double& pder, double& polym1,
double& pderm1, double& polym2, double& pderm2, double x) {
if (n==0) {
poly = 1 ;
pder = 0 ;
}
else
if (n==1) {
polym1 = 1 ;
pderm1 = 0 ;
poly = x ;
pder = 1 ;
}
else {
polym1 = 1 ;
pderm1 = 0 ;
poly = x ;
pder = 1 ;
for (int i=1 ; i<n ; i++) {
polym2 = polym1 ;
pderm2 = pderm1 ;
polym1 = poly ;
pderm1 = pder ;
poly = ((2*i+1)*x*polym1 - i*polym2)/(i+1) ;
pder = ((2*i+1)*polym1+(2*i+1)*x*pderm1-i*pderm2)/(i+1) ;
}
}
}
void legendre_nodes_weight_GL(int n, double* coloc, double* weight, double prec,
int itemax) {
double x_plus = 1 ;
double x_moins = -1 ;
double p_plus, dp_plus, p_plus_m1, dp_plus_m1, p_plus_m2, dp_plus_m2 ;
double p_moins, dp_moins, p_moins_m1, dp_moins_m1, p_moins_m2, dp_moins_m2 ;
double p, dp, p_m1, dp_m1, p_m2, dp_m2 ;
legendre_poly_der(n, p_plus, dp_plus, p_plus_m1, dp_plus_m1, p_plus_m2,
dp_plus_m2, x_plus) ;
legendre_poly_der(n, p_moins, dp_moins, p_moins_m1, dp_moins_m1,
p_moins_m2, dp_moins_m2, x_moins) ;
double det = p_plus_m1*p_moins_m2 - p_moins_m1*p_plus_m2 ;
double r_plus = -p_plus ;
double r_moins = -p_moins ;
double a = (r_plus*p_moins_m2 - r_moins*p_plus_m2)/det ;
double b = (r_moins*p_plus_m1 - r_plus*p_moins_m1)/det ;
coloc[n-1] = 1 ;
double dth = M_PI/(2*n+1) ;
double cd = cos (2*dth) ;
double sd = sin (2*dth) ;
double cs = cos(dth) ;
double ss = sin(dth) ;
int borne_sup = (n%2==0) ?
n/2 : (n+1)/2 ;
for (int j=1 ; j<borne_sup ; j++) {
double x = cs ;
bool loop = true ;
int ite = 0 ;
while (loop) {
legendre_poly_der(n, p, dp, p_m1, dp_m1, p_m2, dp_m2, x) ;
double poly = p + a*p_m1 + b*p_m2 ;
double pder = dp + a * dp_m1 + b*dp_m2 ;
double sum = 0 ;
for (int i=0 ; i<j ; i++)
sum += 1./(x-coloc[n-i-1]) ;
double increm = -poly/(pder-sum*poly) ;
x += increm ;
ite ++ ;
if ((fabs(increm) < prec) || (ite >itemax))
loop = false ;
}
if (ite > itemax) {
cout << "legendre_poly_der: too many iterations !..." << endl ;
abort() ;
}
coloc[n-j-1] = x ;
double auxi = cs*cd-ss*sd ;
ss = cs*sd+ss*cd ;
cs = auxi ;
}
if (n%2==1)
coloc[(n-1)/2] = 0 ;
// Copy of the symetric ones :
for (int i=0 ; i<borne_sup ; i++)
coloc[i] = - coloc[n-i-1] ;
for (int i=0 ; i<n ; i++) {
legendre_poly_der(n-1, p, dp, p_m1, dp_m1, p_m2, dp_m2, coloc[i]) ;
weight[i] = 2./(n-1)/n/p/p ;
}
}
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