public static double sphere01_monomial_quadrature(int[] expon, int point_num, double[] xyz,
double[] w)
//****************************************************************************80
//
// Purpose:
//
// SPHERE01_MONOMIAL_QUADRATURE applies quadrature to a monomial in a sphere.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 05 July 2007
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int DIM_NUM, the spatial dimension.
//
// Input, int EXPON[DIM_NUM], the exponents.
//
// Input, int POINT_NUM, the number of points in the rule.
//
// Input, double XYZ[DIM_NUM*POINT_NUM], the quadrature points.
//
// Input, double W[POINT_NUM], the quadrature weights.
//
// Output, double SPHERE01_MONOMIAL_QUADRATURE, the quadrature error.
//
{
//
// Get the exact value of the integral.
//
double exact = Integrals.sphere01_monomial_integral(expon);
//
// Evaluate the monomial at the quadrature points.
//
double[] value = MonomialNS.Monomial.monomial_value(3, point_num, expon, xyz);
//
// Compute the weighted sum.
//
double quad = typeMethods.r8vec_dot(point_num, w, value);
//
// Error:
//
double quad_error = Math.Abs(quad - exact);
return(quad_error);
}
private static void test01()
//****************************************************************************80
//
// Purpose:
//
// TEST01 uses SPHERE01_SAMPLE to estimate monomial integrands.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 06 January 2014
//
// Author:
//
// John Burkardt
//
{
const int m = 3;
int test;
const int test_num = 20;
Console.WriteLine("");
Console.WriteLine("TEST01");
Console.WriteLine(" Estimate monomial integrands using Monte Carlo");
Console.WriteLine(" over the surface of the unit sphere in 3D.");
//
// Get sample points.
//
const int n = 8192;
int seed = 123456789;
double[] x = Integrals.sphere01_sample(n, ref seed);
Console.WriteLine("");
Console.WriteLine(" Number of sample points used is " + n + "");
//
// Randomly choose X,Y,Z exponents between (0,0,0) and (9,9,9).
//
Console.WriteLine("");
Console.WriteLine(" If any exponent is odd, the integral is zero.");
Console.WriteLine(" We will restrict this test to randomly chosen even exponents.");
Console.WriteLine("");
Console.WriteLine(" Ex Ey Ez MC-Estimate Exact Error");
Console.WriteLine("");
for (test = 1; test <= test_num; test++)
{
int[] e = UniformRNG.i4vec_uniform_ab_new(m, 0, 4, ref seed);
int i;
for (i = 0; i < m; i++)
{
e[i] *= 2;
}
double[] value = Monomial.monomial_value(m, n, e, x);
double result = Integrals.sphere01_area() * typeMethods.r8vec_sum(n, value) / n;
double exact = Integrals.sphere01_monomial_integral(e);
double error = Math.Abs(result - exact);
Console.WriteLine(" " + e[0].ToString().PadLeft(2)
+ " " + e[1].ToString().PadLeft(2)
+ " " + e[2].ToString().PadLeft(2)
+ " " + result.ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + exact.ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + error.ToString(CultureInfo.InvariantCulture).PadLeft(14) + "");
}
}
private static void test05(ref int[] e_save)
//****************************************************************************80
//
// Purpose:
//
// TEST05 tests SPHERE01_QUAD_ICOS2V.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 24 September 2010
//
// Author:
//
// John Burkardt
//
{
int[] e = new int[3];
int i;
int n = 0;
Console.WriteLine("");
Console.WriteLine("TEST05");
Console.WriteLine(" Approximate the integral of a function on the unit sphere.");
Console.WriteLine(" SPHERE01_QUAD_ICOS2V uses vertices of spherical triangles.");
Console.WriteLine("");
Console.WriteLine("FACTOR N QUAD EXACT ERROR");
for (i = 1; i <= 17; i++)
{
switch (i)
{
case 1:
e[0] = 0;
e[1] = 0;
e[2] = 0;
break;
case 2:
e[0] = 1;
e[1] = 0;
e[2] = 0;
break;
case 3:
e[0] = 0;
e[1] = 1;
e[2] = 0;
break;
case 4:
e[0] = 0;
e[1] = 0;
e[2] = 1;
break;
case 5:
e[0] = 2;
e[1] = 0;
e[2] = 0;
break;
case 6:
e[0] = 0;
e[1] = 2;
e[2] = 2;
break;
case 7:
e[0] = 2;
e[1] = 2;
e[2] = 2;
break;
case 8:
e[0] = 0;
e[1] = 2;
e[2] = 4;
break;
case 9:
e[0] = 0;
e[1] = 0;
e[2] = 6;
break;
case 10:
e[0] = 1;
e[1] = 2;
e[2] = 4;
break;
case 11:
e[0] = 2;
e[1] = 4;
e[2] = 2;
break;
case 12:
e[0] = 6;
e[1] = 2;
e[2] = 0;
break;
case 13:
e[0] = 0;
e[1] = 0;
e[2] = 8;
break;
case 14:
e[0] = 6;
e[1] = 0;
e[2] = 4;
break;
case 15:
e[0] = 4;
e[1] = 6;
e[2] = 2;
break;
case 16:
e[0] = 2;
e[1] = 4;
e[2] = 8;
break;
case 17:
e[0] = 16;
e[1] = 0;
e[2] = 0;
break;
}
polyterm_exponent("SET", ref e_save, ref e);
polyterm_exponent("PRINT", ref e_save, ref e);
int factor = 1;
int factor_log;
for (factor_log = 0; factor_log <= 5; factor_log++)
{
double result = Quad.sphere01_quad_icos2v(factor, polyterm_value_3d, ref n);
double exact = Integrals.sphere01_monomial_integral(e);
double error = Math.Abs(exact - result);
Console.WriteLine(" " + factor.ToString().PadLeft(4)
+ " " + n.ToString().PadLeft(8)
+ " " + result.ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + exact.ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + error.ToString(CultureInfo.InvariantCulture).PadLeft(14) + "");
factor *= 2;
}
}
}
private static void test01(ref int[] e_save)
//****************************************************************************80
//
// Purpose:
//
// TEST01 tests SPHERE01_QUAD_LL*.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 25 September 2010
//
// Author:
//
// John Burkardt
//
{
int[] e = new int[3];
double h = 0;
int i;
int n_llc = 0;
int n_llm = 0;
int n_llv = 0;
int seed = 123456789;
Console.WriteLine("");
Console.WriteLine("TEST01");
Console.WriteLine(" Approximate the integral of a function on the unit sphere.");
Console.WriteLine("");
Console.WriteLine(" SPHERE01_QUAD_MC uses a Monte Carlo method.");
Console.WriteLine(" SPHERE01_QUAD_LLC uses centroids of spherical triangles.");
Console.WriteLine(" SPHERE01_QUAD_LLM uses midsides of spherical triangles.");
Console.WriteLine(" SPHERE01_QUAD_LLV uses vertices of spherical triangles.");
Console.WriteLine("");
Console.WriteLine(" H QUAD_MC QUAD_LLC QUAD_LLM QUAD_LLV EXACT");
for (i = 0; i <= 17; i++)
{
switch (i)
{
case 0:
e[0] = 0;
e[1] = 0;
e[2] = 0;
break;
case 1:
e[0] = 0;
e[1] = 0;
e[2] = 0;
break;
case 2:
e[0] = 1;
e[1] = 0;
e[2] = 0;
break;
case 3:
e[0] = 0;
e[1] = 1;
e[2] = 0;
break;
case 4:
e[0] = 0;
e[1] = 0;
e[2] = 1;
break;
case 5:
e[0] = 2;
e[1] = 0;
e[2] = 0;
break;
case 6:
e[0] = 0;
e[1] = 2;
e[2] = 2;
break;
case 7:
e[0] = 2;
e[1] = 2;
e[2] = 2;
break;
case 8:
e[0] = 0;
e[1] = 2;
e[2] = 4;
break;
case 9:
e[0] = 0;
e[1] = 0;
e[2] = 6;
break;
case 10:
e[0] = 1;
e[1] = 2;
e[2] = 4;
break;
case 11:
e[0] = 2;
e[1] = 4;
e[2] = 2;
break;
case 12:
e[0] = 6;
e[1] = 2;
e[2] = 0;
break;
case 13:
e[0] = 0;
e[1] = 0;
e[2] = 8;
break;
case 14:
e[0] = 6;
e[1] = 0;
e[2] = 4;
break;
case 15:
e[0] = 4;
e[1] = 6;
e[2] = 2;
break;
case 16:
e[0] = 2;
e[1] = 4;
e[2] = 8;
break;
case 17:
e[0] = 16;
e[1] = 0;
e[2] = 0;
break;
}
polyterm_exponent("SET", ref e_save, ref e);
switch (i)
{
case 0:
Console.WriteLine("");
Console.WriteLine("Point counts per method:");
break;
default:
polyterm_exponent("PRINT", ref e_save, ref e);
break;
}
int h_test;
for (h_test = 1; h_test <= 3; h_test++)
{
h = h_test switch
{
1 => 1.0,
2 => 0.1,
3 => 0.01,
_ => h
};
int n_mc = Quad.sphere01_quad_mc_size(h);
double result_mc = Quad.sphere01_quad_mc(polyterm_value_3d, h, ref seed, n_mc);
double result_llc = Quad.sphere01_quad_llc(polyterm_value_3d, h, ref n_llc);
double result_llm = Quad.sphere01_quad_llm(polyterm_value_3d, h, ref n_llm);
double result_llv = Quad.sphere01_quad_llv(polyterm_value_3d, h, ref n_llv);
double exact = Integrals.sphere01_monomial_integral(e);
switch (i)
{
case 0:
Console.WriteLine(" " + h.ToString(CultureInfo.InvariantCulture).PadLeft(12)
+ " " + n_mc.ToString().PadLeft(12)
+ " " + n_llc.ToString().PadLeft(12)
+ " " + n_llm.ToString().PadLeft(12)
+ " " + n_llv.ToString().PadLeft(12) + "");
break;
default:
Console.WriteLine(" " + h.ToString(CultureInfo.InvariantCulture).PadLeft(12)
+ " " + result_mc.ToString(CultureInfo.InvariantCulture).PadLeft(12)
+ " " + result_llc.ToString(CultureInfo.InvariantCulture).PadLeft(12)
+ " " + result_llm.ToString(CultureInfo.InvariantCulture).PadLeft(12)
+ " " + result_llv.ToString(CultureInfo.InvariantCulture).PadLeft(12)
+ " " + exact.ToString(CultureInfo.InvariantCulture).PadLeft(12) + "");
break;
}
}
}
}
