Expose gauss_legendre_tbl publicly as gauss_legendre_roots

libsharp/sharp_legendre_roots.c

@@ -0,0 +1,67 @@
+/* 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
+  }