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sphere-quad.cpp
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sphere-quad.cpp
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/* Copyright (C) 2005-2015 Massachusetts Institute of Technology.
*
* This program 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.
*
* This program 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 this program; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*/
/* This file is compiled into a program sphere_quad that is used to
generate the file sphere-quad.h, which is a table of quadrature
points and weights for integrating on spheres in 1d/2d/3d. */
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define SHIFT3(x,y,z) {double SHIFT3_dummy = z; z = y; y = x; x = SHIFT3_dummy;}
#define CHECK(condition, message) do { \
if (!(condition)) { \
fprintf(stderr, "CHECK failure on line %d of " __FILE__ ": " \
message "\n", __LINE__); exit(EXIT_FAILURE); \
} \
} while (0)
/* Compute quadrature points and weights for integrating on the
unit sphere. x, y, z, and weight should be arrays of num_sq_pts
values to hold the coordinates and weights of the quadrature points
on output. Currently, num_sq_pts = 12, 50, and 72 are supported. */
void spherical_quadrature_points(double *x, double *y, double *z,
double *weight, int num_sq_pts)
{
int i,j,k,l, n = 0;
double x0, y0, z0, w;
if (num_sq_pts == 50) {
/* Computes quadrature points and weights for 50-point 11th degree
integration formula on a unit sphere. This particular quadrature
formula has the advantage, for our purposes, of preserving the
symmetry group of an octahedron (i.e. simple cubic symmetry, with
respect to the Cartesian xyz axes). References:
A. D. McLaren, "Optimal Numerical Integration on a Sphere,"
Math. Comp. 17, pp. 361-383 (1963).
Also in: Arthur H. Stroud, "Approximate Calculation of Multiple
Integrals" (Prentice Hall, 1971) (formula number U3:11-1).
This code was written with the help of example code by
John Burkardt:
http://www.psc.edu/~burkardt/src_pt/stroud/stroud.html */
x0 = 1; y0 = z0 = 0;
w = 9216 / 725760.0;
for (i = 0; i < 2; ++i) {
x0 = -x0;
for (j = 0; j < 3; ++j) {
SHIFT3(x0,y0,z0);
x[n] = x0; y[n] = y0; z[n] = z0; weight[n++] = w;
}
}
x0 = y0 = sqrt(0.5); z0 = 0;
w = 16384 / 725760.0;
for (i = 0; i < 2; ++i) {
x0 = -x0;
for (j = 0; j < 2; ++j) {
y0 = -y0;
for (k = 0; k < 3; ++k) {
SHIFT3(x0,y0,z0);
x[n] = x0; y[n] = y0; z[n] = z0; weight[n++] = w;
}
}
}
x0 = y0 = z0 = sqrt(1.0 / 3.0);
w = 15309 / 725760.0;
for (i = 0; i < 2; ++i) {
x0 = -x0;
for (j = 0; j < 2; ++j) {
y0 = -y0;
for (k = 0; k < 2; ++k) {
z0 = -z0;
x[n] = x0; y[n] = y0; z[n] = z0; weight[n++] = w;
}
}
}
x0 = y0 = sqrt(1.0 / 11.0); z0 = 3 * x0;
w = 14641 / 725760.0;
for (i = 0; i < 2; ++i) {
x0 = -x0;
for (j = 0; j < 2; ++j) {
y0 = -y0;
for (k = 0; k < 2; ++k) {
z0 = -z0;
for (l = 0; l < 3; ++l) {
SHIFT3(x0,y0,z0);
x[n] = x0; y[n] = y0; z[n] = z0; weight[n++] = w;
}
}
}
}
}
else if (num_sq_pts == 72 || num_sq_pts == 12) {
/* As above (same references), but with a 72-point 14th degree
formula, this time with the symmetry group of an icosohedron.
(Stroud formula number U3:14-1.) Alternatively, just use
the 12-point 5th degree formula consisting of the vertices
of a regular icosohedron. */
/* first, the vertices of an icosohedron: */
x0 = sqrt(0.5 - sqrt(0.05));
y0 = sqrt(0.5 + sqrt(0.05));
z0 = 0;
if (num_sq_pts == 72)
w = 125 / 10080.0;
else
w = 1 / 12.0;
for (i = 0; i < 2; ++i) {
x0 = -x0;
for (j = 0; j < 2; ++j) {
y0 = -y0;
for (k = 0; k < 3; ++k) {
SHIFT3(x0,y0,z0);
x[n] = x0; y[n] = y0; z[n] = z0; weight[n++] = w;
}
}
}
if (num_sq_pts == 72) {
/* it would be nice, for completeness, to have more
digits here: */
double coords[3][5] = {
{ -0.151108275, 0.315838353, 0.346307112, -0.101808787, -0.409228403 },
{ 0.155240600, 0.257049387, 0.666277790, 0.817386065, 0.501547712 },
{ 0.976251323, 0.913330032, 0.660412970, 0.567022920, 0.762221757 }
};
w = 143 / 10080.0;
for (l = 0; l < 5; ++l) {
x0 = coords[0][l]; y0 = coords[1][l]; z0 = coords[2][l];
for (i = 0; i < 3; ++i) {
double dummy = x0;
x0 = z0;
z0 = -y0;
y0 = -dummy;
for (j = 0; j < 3; ++j) {
SHIFT3(x0,y0,z0);
x[n] = x0; y[n] = y0; z[n] = z0; weight[n++] = w;
}
y0 = -y0;
z0 = -z0;
x[n] = x0; y[n] = y0; z[n] = z0; weight[n++] = w;
}
}
}
}
else
CHECK(0, "spherical_quadrature_points: passed unknown # points!");
CHECK(n == num_sq_pts,
"bug in spherical_quadrature_points: wrong number of points!");
}
#define NQUAD3 50 /* use 50-point quadrature formula by default */
/**********************************************************************/
#define K_PI 3.141592653589793238462643383279502884197
#define NQUAD2 12
/**********************************************************************/
double sqr(double x) { return x * x; }
double dist2(double x1, double y1, double z1,
double x2, double y2, double z2)
{
return sqr(x1-x2) + sqr(y1-y2) + sqr(z1-z2);
}
double min2(double a, double b) { return a < b ? a : b; }
/* sort the array to maximize the spacing of each point with the
previous points */
void sort_by_distance(int n, double x[], double y[], double z[], double w[])
{
for (int i = 1; i < n; ++i) {
double d2max = 0;
double d2maxsum = 0;
int jmax = i;
for (int j = i; j < n; ++j) {
double d2min = 1e20, d2sum = 0;
for (int k = 0; k < i; ++k) {
double d2 = float(dist2(x[k],y[k],z[k], x[j],y[j],z[j]));
d2min = min2(d2min, d2);
d2sum += d2;
}
if (d2min > d2max ||
(d2min == d2max && d2sum > d2maxsum)) {
d2max = d2min;
d2maxsum = d2sum;
jmax = j;
}
}
double xi = x[i], yi = y[i], zi = z[i], wi = w[i];
x[i] = x[jmax]; y[i] = y[jmax]; z[i] = z[jmax]; w[i] = w[jmax];
x[jmax] = xi; y[jmax] = yi; z[jmax] = zi; w[jmax] = wi;
}
}
int main(void)
{
int i;
double x2[NQUAD2], y2[NQUAD2], z2[NQUAD2], w2[NQUAD2];
double x3[NQUAD3], y3[NQUAD3], z3[NQUAD3], w3[NQUAD3];
printf(
"/* For 1d, 2d, and 3d, quadrature points and weights on a unit sphere.\n"
" There are num_sphere_quad[dim-1] points i, with the i-th point at\n"
" (x,y,z) = (sphere_quad[dim-1][i][ 0, 1, 2 ]), and with a quadrature\n"
" weight sphere_quad[dim-1][i][3]. */\n\n");
printf("static const int num_sphere_quad[3] = { %d, %d, %d };\n\n",
2, NQUAD2, NQUAD3);
printf("static const double sphere_quad[3][%d][4] = {\n", NQUAD3);
printf(" { {0,0,1,0.5}, {0,0,-1,0.5} },\n");
for (i = 0; i < NQUAD2; ++i) {
x2[i] = cos(2*i * K_PI / NQUAD2);
y2[i] = sin(2*i * K_PI / NQUAD2);
z2[i] = 0.0;
w2[i] = 1.0 / NQUAD2;
}
sort_by_distance(NQUAD2,x2,y2,z2,w2);
printf(" {\n");
for (i = 0; i < NQUAD2; ++i) {
printf(" { %0.20g, %0.20g, %0.20g, %0.20g },\n",
x2[i], y2[i], z2[i], w2[i]);
}
printf(" },\n");
printf(" {\n");
spherical_quadrature_points(x3,y3,z3, w3, NQUAD3);
sort_by_distance(NQUAD3,x3,y3,z3,w3);
for (i = 0; i < NQUAD3; ++i) {
printf(" { %0.20g, %0.20g, %0.20g, %0.20g },\n",
x3[i], y3[i], z3[i], w3[i]);
}
printf(" }\n");
printf("};\n");
return 0;
}