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