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// GetDP - Copyright (C) 1997-2018 P. Dular and C. Geuzaine, University of Liege
//
// See the LICENSE.txt file for license information. Please report all
// issues on https://gitlab.onelab.info/getdp/getdp/issues
//
// Contributor(s):
// Ruth Sabariego
//
#include <math.h>
#include <stdio.h>
#include "Message.h"
#define THESIGN(a) ((a)>=0 ? 1 : -1)
#define THEABS(a) ((a)>=0 ? a : -a)
#define ONE_OVER_FOUR_PI 7.9577471545947668E-02
double Factorial(double n)
{
/* FACTORIAL(n) is the product of all the integers from 1 to n */
double F ;
if ( n < 0 ){
Message::Error("Factorial(n): n must be a positive integer") ;
return 0;
}
if ( n == 0 ) return 1. ;
if ( n <= 2 ) return n ;
F = n ;
while ( n > 2 ){ n-- ; F *= n ; }
return F;
}
double BinomialCoef(double n, double m)
{
/* Binomial Coefficients: (n m) Computes de number of ways of
choosing m objects from a collection of n distinct objects */
int i ;
double B = 1. ;
if (m==0 || n==m) return 1. ;
for(i = (int)n ; i > m ; i--)
B *= (double)i/(double)(i-m);
return B;
}
double Legendre(int l, int m, double x)
{
/* Computes the associated Legendre polynomial P_l^m(x). Here the
degree l and the order m are the integers satisfying -l<=m<=l,
while x lies in the range -1<=x<=1 */
double fact, pll=0., pmm, pmmp1, somx2, Cte ;
int i, ll;
if ( THEABS(m) > l || fabs(x) > 1.){
Message::Error("Bad arguments for Legendre: P_l^m(x) with -l<=m<=l (int),"
" -1<=x<=1 l = %d m = %d x = %.8g", l, m, x);
return 0.;
}
Cte = (m > 0) ? 1. : Factorial((double)(l-THEABS(m))) /
Factorial((double)(l+THEABS(m))) * pow(-1.,(double)THEABS(m)) ;
m = THEABS(m) ;
pmm = 1. ;
if (m > 0) {
somx2 = sqrt((1.-x)*(1.+x)) ;
fact = 1. ;
for (i=1;i<=m;i++){
pmm *= -fact*somx2 ;
fact += 2. ;
}
}
if (l==m){
return Cte*pmm ;
}
else {
pmmp1 = x * (2*m+1)*pmm ;
if (l==(m+1)){
return Cte*pmmp1 ;
}
else {
for (ll=(m+2);ll<=l;ll++) {
pll = (x*(2*ll-1)*pmmp1-(ll+m-1)*pmm)/(ll-m) ;
pmm = pmmp1 ;
pmmp1 = pll ;
}
return Cte*pll ;
}
}
}
void LegendreRecursive(int l, int m, double x, double P[])
{
/* Computes recursively a (l+1)-terms sequence of the associated
Legendre polynomial P_l^m(x).
l and m are the integers satisfying 0<=m<=l
x lies in the range -1<=x<=1
l = maximum order considered, m = invariable */
int il ;
double Pl_m, Plm1_m ;
P[0] = Plm1_m = Legendre(0, m, x) ;
P[1] = Pl_m = Legendre(1, m, x) ;
if (l >=2)
for(il = 1 ; il < l ; il ++){
P[il+1] = (2*il+1)*x*Pl_m/(il-m+1) + (il+m)*Plm1_m/(m-il-1) ;
Plm1_m = Pl_m ;
Pl_m = P[il+1];
}
}
void LegendreRecursiveM(int l, double x, double P[])
{
/* Computes recursively a (l+1)-terms sequence of the associated
Legendre polynomial P_l^m(x).
x lies in the range -1<=x<=1, l = invariable, -l<=m<=l */
int m ;
double Pl_m, Plm1_m ;
if (fabs(x) == 1.)
for(m = -l ; m <= l ; m ++)
P[l+m] = (m==0) ? pow(THESIGN(x),(double)l) : 0. ;
else{
if (l==0){
P[0] = Legendre(0, 0, x) ;
return;
}
P[0] = Plm1_m = Legendre(l, -l, x) ;
P[1] = Pl_m = Legendre(l, -l+1, x) ;
if (l >= 1)
for(m = -l+1 ; m < l ; m ++){
P[l+m+1] = -2*m*x*Pl_m/sqrt(1-x*x) + (m*(m-1)-l*(l+1))*Plm1_m ;
Plm1_m = Pl_m ;
Pl_m = P[l+m+1];
}
else return ;
}
}
double dLegendre (int l, int m, double x)
{
/* Computes the derivative of the associated Legendre polynomial
P_l^m(x) */
double dpl;
if ( THEABS(m) > l || fabs(x) > 1.){
Message::Error("Bad arguments for dLegendre: -l<=m<=l (integers), -1<=x<=1."
" Current values: l %d m %d x %.8g", l, m, x) ;
return 0.;
}
if (fabs(x)==1.) dpl = 0.;
else
dpl = ((l+m)*(l-m+1)*sqrt(1-x*x)*((THEABS((m-1))>l) ? 0. :
Legendre(l, m-1, x)) + m*x* Legendre (l,m,x))/(1-x*x);
return dpl;
}
double dLegendreFinDif (int l, int m, double x)
{
/* Computes the derivative of the associated Legendre polynomial
P_l^m(x) using Finite Differences: f'(x) = (f(x+\delta
x)-f(x-\delta x))/(2 \delta) */
double dpl, delta = 1e-6;
if ( THEABS(m) > l || fabs(x) > 1.){
Message::Error("Bad arguments for dLegendreFinDif: -l<=m<=l (integers), "
"-1<=x<=1. Current values: l %d m %d x %.8g", l, m, x );
return 0.;
}
dpl = (Legendre (l, m, x+delta) - Legendre (l, m, x-delta))/(2*delta);
return dpl;
}
void SphericalHarmonics(int l, int m, double Theta, double Phi, double Yl_m[])
{
int am ;
double cn, Pl_m, F, cRe ;
cn = sqrt((2*l+1)*ONE_OVER_FOUR_PI) ; /* Normalization Factor */
am = THESIGN(m)*m ;
F= sqrt(Factorial((double)(l-am))/ Factorial((double)(l+am))) ;
Pl_m = Legendre(l, am, cos(Theta));
cRe = cn * F * Pl_m ;
Yl_m[0] = cRe*cos(m*Phi) ;
Yl_m[1] = cRe*sin(m*Phi) ;
}
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