cosmotool/external/sharp/libsharp/sharp_legendre_roots.c

68 lines
1.6 KiB
C

/* Function adapted from GNU GSL file glfixed.c
Original author: Pavel Holoborodko (http://www.holoborodko.com)
Adjustments by M. Reinecke
- adjusted interface (keep epsilon internal, return full number of points)
- removed precomputed tables
- tweaked Newton iteration to obtain higher accuracy */
#include <math.h>
#include "sharp_legendre_roots.h"
#include "c_utils.h"
static inline double one_minus_x2 (double x)
{ return (fabs(x)>0.1) ? (1.+x)*(1.-x) : 1.-x*x; }
void sharp_legendre_roots(int n, double *x, double *w)
{
const double pi = 3.141592653589793238462643383279502884197;
const double eps = 3e-14;
int m = (n+1)>>1;
double t0 = 1 - (1-1./n) / (8.*n*n);
double t1 = 1./(4.*n+2.);
#pragma omp parallel
{
int i;
#pragma omp for schedule(dynamic,100)
for (i=1; i<=m; ++i)
{
double x0 = cos(pi * ((i<<2)-1) * t1) * t0;
int dobreak=0;
int j=0;
double dpdx;
while(1)
{
double P_1 = 1.0;
double P0 = x0;
double dx, x1;
for (int k=2; k<=n; k++)
{
double P_2 = P_1;
P_1 = P0;
// P0 = ((2*k-1)*x0*P_1-(k-1)*P_2)/k;
P0 = x0*P_1 + (k-1.)/k * (x0*P_1-P_2);
}
dpdx = (P_1 - x0*P0) * n / one_minus_x2(x0);
/* Newton step */
x1 = x0 - P0/dpdx;
dx = x0-x1;
x0 = x1;
if (dobreak) break;
if (fabs(dx)<=eps) dobreak=1;
UTIL_ASSERT(++j<100,"convergence problem");
}
x[i-1] = -x0;
x[n-i] = x0;
w[i-1] = w[n-i] = 2. / (one_minus_x2(x0) * dpdx * dpdx);
}
} // end of parallel region
}