public static void triangle()
//****************************************************************************80
//
// Purpose:
//
// MAIN is the main program for TRIANGLE.
//
// Discussion:
//
// This driver computes the interpolation of the Franke function
// on the triangle T(U,V,W) with vertices U=(U1,U2)=(0,0),
// V=(V1,V2)=(1,0) and W=(W1,W2)=(0,1) (unit triangle)
// at the first family of Padua points.
//
// The degree of interpolation is DEG = 60 and the number of target
// points is NTG = NTG1 ** 2 - NTG1 + 1, NTG1 = 100.
//
// The maps from the reference square [-1,1]^2 to the triangle
// are SIGMA1 and SIGMA2 with inverses ISIGM1 and ISIGM2.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 16 February 2014
//
// Author:
//
// Original FORTRAN77 version by Marco Caliari, Stefano De Marchi,
// Marco Vianello.
// This C version by John Burkardt.
//
// Reference:
//
// Marco Caliari, Stefano de Marchi, Marco Vianello,
// Algorithm 886:
// Padua2D: Lagrange Interpolation at Padua Points on Bivariate Domains,
// ACM Transactions on Mathematical Software,
// Volume 35, Number 3, October 2008, Article 21, 11 pages.
//
// Parameters:
//
// Local, int DEGMAX, the maximum degree of interpolation.
//
// Local, int NPDMAX, the maximum number of Padua points
// = (DEGMAX + 1) * (DEGMAX + 2) / 2.
//
// Local, int NTG1MX, the maximum value of the parameter determining
// the number of target points.
//
// Local, int NTGMAX, the maximum number of target points,
// dependent on NTG1MX.
//
// Local, int DEG, the degree of interpolation.
//
// Local, int NTG1, the parameter determining the number
// of target points.
//
// Local, int NPD, the number of Padua points = (DEG + 1) * (DEG + 2) / 2.
//
// Local, int NTG, the number of target points, dependent on NTG1.
//
// Local, double PD1(NPDMAX), the first coordinates of
// the Padua points.
//
// Local, double PD2(NPDMAX), the second coordinates of the
// Padua points.
//
// Local, double WPD(NPDMAX), the weights.
//
// Local, double FPD(NPDMAX), the function at the Padua points.
//
// Workspace, double RAUX1(DEGMAX+1)*(DEGMAX+2)).
//
// Workspace, double RAUX2(DEGMAX+1)*(DEGMAX+2)).
//
// Local, double C0(0:DEGMAX+1,0:DEGMAX+1), the coefficient matrix.
//
// Local, double TG1(NTGMAX), the first coordinates of the
// target points.
//
// Local, double TG2(NTGMAX), the second coordinates of the
// target points.
//
// Local, double INTFTG(NTGMAX), the values of the
// interpolated function.
//
// Local, double MAXERR, the maximum norm of the error at target
// points.
//
// Local, double ESTERR, the estimated error.
//
{
const int DEGMAX = 60;
const int NTG1MX = 100;
const int NPDMAX = (DEGMAX + 1) * (DEGMAX + 2) / 2;
const int NTGMAX = NTG1MX * NTG1MX - NTG1MX + 1;
double[] c0 = new double[(DEGMAX + 2) * (DEGMAX + 2)];
double esterr = 0;
double[] fpd = new double[NPDMAX];
int i;
double[] intftg = new double[NTGMAX];
double maxdev;
int npd = 0;
int ntg = 0;
List <string> output = new();
double[] pd1 = new double[NPDMAX];
double[] pd2 = new double[NPDMAX];
double[] raux1 = new double[(DEGMAX + 1) * (DEGMAX + 2)];
double[] raux2 = new double[(DEGMAX + 1) * (DEGMAX + 2)];
double[] tg1 = new double[NTGMAX];
double[] tg2 = new double[NTGMAX];
double[] wpd = new double[NPDMAX];
double x;
double y;
const double u1 = 0.0;
const double u2 = 0.0;
const double v1 = 1.0;
const double v2 = 0.0;
const double w1 = 0.0;
const double w2 = 1.0;
const int deg = 60;
const int ntg1 = 100;
Console.WriteLine("");
Console.WriteLine("TRIANGLE:");
Console.WriteLine(" Interpolation of the Franke function");
Console.WriteLine(" on the unit triangle T((0,0),(1,0),(0,1))");
Console.WriteLine(" at degree = " + deg + "");
//
// Build the first family of Padua points in the square [-1,1]^2
//
Padua.pdpts(deg, ref pd1, ref pd2, ref wpd, ref npd);
//
// Compute the Franke function at Padua points mapped to T(U,V,W).
//
for (i = 0; i < npd; i++)
{
x = sigma1(pd1[i], pd2[i], u1, u2, v1, v2, w1, w2);
y = sigma2(pd1[i], pd2[i], u1, u2, v1, v2, w1, w2);
fpd[i] = Franke.franke(x, y);
}
//
// Write X, Y, F(X,Y) to a file.
//
string filename = "triangle_fpd.txt";
for (i = 0; i < npd; i++)
{
x = sigma1(pd1[i], pd2[i], u1, u2, v1, v2, w1, w2);
y = sigma2(pd1[i], pd2[i], u1, u2, v1, v2, w1, w2);
output.Add(x
+ " " + y
+ " " + fpd[i] + "");
}
File.WriteAllLines(filename, output);
Console.WriteLine("");
Console.WriteLine(" Wrote F(x,y) at Padua points in '" + filename + "'");
//
// Compute the matrix C0 of the coefficients in the bivariate
// orthonormal Chebyshev basis
//
Padua.padua2(deg, DEGMAX, npd, wpd, fpd, raux1, raux2, ref c0, ref esterr);
//
// Build the set of target points on T(U,V,W)
//
target(u1, u2, v1, v2, w1, w2, ntg1, NTGMAX, ref tg1, ref tg2, ref ntg);
//
// Evaluate the interpolant at the target points.
//
for (i = 0; i < ntg; i++)
{
x = isigm1(tg1[i], tg2[i], u1, u2, v1, v2, w1, w2);
y = isigm2(tg1[i], tg2[i], u1, u2, v1, v2, w1, w2);
intftg[i] = Padua.pd2val(deg, DEGMAX, c0, x, y);
}
//
// Write the function value at target points to a file.
//
filename = "triangle_ftg.txt";
for (i = 0; i < ntg; i++)
{
output.Add(tg1[i]
+ " " + tg2[i]
+ " " + Franke.franke(tg1[i], tg2[i]) + "");
}
File.WriteAllLines(filename, output);
Console.WriteLine(" Wrote F(x,y) at target points in '" + filename + "'");
//
// Write the interpolated function value at target points to a file.
//
filename = "triangle_itg.txt";
for (i = 0; i < ntg; i++)
{
output.Add(tg1[i]
+ " " + tg2[i]
+ " " + intftg[i] + "");
}
File.WriteAllLines(filename, output);
Console.WriteLine(" Wrote I(F)(x,y) at target points in '" + filename + "'");
//
// Compute the error relative to the max deviation from the mean.
//
double maxerr = 0.0;
double mean = 0.0;
double fmax = -typeMethods.r8_huge();
double fmin = +typeMethods.r8_huge();
for (i = 0; i < ntg; i++)
{
double fxy = Franke.franke(tg1[i], tg2[i]);
double ixy = intftg[i];
maxerr = Math.Max(maxerr, Math.Abs(fxy - ixy));
mean += fxy;
fmax = Math.Max(fmax, fxy);
fmin = Math.Min(fmin, fxy);
}
if (Math.Abs(fmax - fmin) <= double.Epsilon)
{
maxdev = 1.0;
}
else
{
mean /= ntg;
maxdev = Math.Max(fmax - mean, mean - fmin);
}
//
// Print error ratios.
//
Console.WriteLine("");
Console.WriteLine(" Estimated error: " + esterr / maxdev + "");
Console.WriteLine(" Actual error: " + maxerr / maxdev + "");
Console.WriteLine(" Expected error: " + 0.1226E-09 + "");
//
// Terminate.
//
Console.WriteLine("");
Console.WriteLine("TRIANGLE:");
Console.WriteLine(" Normal end of execution.");
Console.WriteLine("");
}
public static void ellipse()
//****************************************************************************80
//
// Purpose:
//
// MAIN is the main program for ELLIPSE.
//
// Discussion:
//
// This driver computes the interpolation of the Franke function
// on the ellipse E((C1,C2),ALPHA,BETA) = E((0.5,0.5),0.5,0.5)
// at the first family of Padua points.
//
// The ellipse has the equation:
//
// ( ( X - C1 ) / ALPHA )^2 + ( ( Y - C2 ) / BETA )^2 = 1
//
// The degree of interpolation DEG = 60 and the number of target
// points is NTG = NTG1 ^ 2 - 2 * NTG1 + 2, NTG1 = 100.
//
// The maps from the reference square [-1,1]^2 to the current domain
// are SIGMA1 and SIGMA2 with inverses ISIGM1 and ISIGM2.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 16 February 2014
//
// Author:
//
// Original FORTRAN77 version by Marco Caliari, Stefano De Marchi,
// Marco Vianello.
// C++ version by John Burkardt.
//
// Reference:
//
// Marco Caliari, Stefano de Marchi, Marco Vianello,
// Algorithm 886:
// Padua2D: Lagrange Interpolation at Padua Points on Bivariate Domains,
// ACM Transactions on Mathematical Software,
// Volume 35, Number 3, October 2008, Article 21, 11 pages.
//
// Local Parameters:
//
// Local, int DEGMAX, the maximum degree of interpolation.
//
// Local, int NPDMAX, the maximum number of Padua points
// = (DEGMAX + 1) * (DEGMAX + 2) / 2.
//
// Local, int NTG1MX, the maximum value of the parameter determining
// the number of target points.
//
// Local, int NTGMAX, the maximum number of target points,
// dependent on NTG1MX.
//
// Local, int DEG, the degree of interpolation.
//
// Local, int NTG1, the parameter determining the number of target points.
//
// Local, int NPD, the number of Padua points = (DEG + 1) * (DEG + 2) / 2.
//
// Local, int NTG, the number of target points, dependent on NTG1.
//
// Local, double PD1[NPDMAX], the first coordinates of
// the Padua points.
//
// Local, double PD2[NPDMAX], the second coordinates of the
// Padua points.
//
// Local, double WPD[NPDMAX], the weights.
//
// Local, double FPD[NPDMAX], the function at the Padua points.
//
// Workspace, double RAUX1[(DEGMAX+1)*(DEGMAX+2)].
//
// Workspace, double RAUX2[(DEGMAX+1)*(DEGMAX+2)].
//
// Local, double C0[(0:DEGMAX+1)*(0:DEGMAX+1)], the coefficient matrix.
//
// Local, double TG1[NTGMAX], the first coordinates of the
// target points.
//
// Local, double TG2[NTGMAX], the second coordinates of the
// target points.
//
// Local, double INTFTG[NTGMAX], the values of the
// interpolated function.
//
// Local, double MAXERR, the maximum norm of the error at target
// points.
//
// Local, double ESTERR, the estimated error.
//
{
const int DEGMAX = 60;
const int NTG1MX = 100;
const int NPDMAX = (DEGMAX + 1) * (DEGMAX + 2) / 2;
const int NTGMAX = NTG1MX * NTG1MX - 2 * NTG1MX + 2;
double[] c0 = new double[(DEGMAX + 2) * (DEGMAX + 2)];
double esterr = 0;
double[] fpd = new double[NPDMAX];
int i;
double[] intftg = new double[NTGMAX];
double maxdev;
int npd = 0;
int ntg = 0;
List <string> output = new();
double[] pd1 = new double[NPDMAX];
double[] pd2 = new double[NPDMAX];
double[] raux1 = new double[(DEGMAX + 1) * (DEGMAX + 2)];
double[] raux2 = new double[(DEGMAX + 1) * (DEGMAX + 2)];
double[] tg1 = new double[NTGMAX];
double[] tg2 = new double[NTGMAX];
double[] wpd = new double[NPDMAX];
double x;
double y;
const double alpha = 0.5;
const double beta = 0.5;
const double c1 = 0.5;
const double c2 = 0.5;
const int deg = 60;
const int ntg1 = 100;
Console.WriteLine("");
Console.WriteLine("ELLIPSE:");
Console.WriteLine(" Interpolation of the Franke function");
Console.WriteLine(" on the disk with center = (0.5,0.5) and radius = 0.5");
Console.WriteLine(" of degree = " + deg + "");
//
// Build the first family of Padua points in the square [-1,1]^2.
//
Padua.pdpts(deg, ref pd1, ref pd2, ref wpd, ref npd);
//
// Compute the Franke function at Padua points mapped to the region.
//
for (i = 0; i < npd; i++)
{
x = sigma1(pd1[i], pd2[i], c1, c2, alpha, beta);
y = sigma2(pd1[i], pd2[i], c1, c2, alpha, beta);
fpd[i] = Franke.franke(x, y);
}
//
// Write X, Y, F(X,Y) to a file.
//
string filename = "ellipse_fpd.txt";
for (i = 0; i < npd; i++)
{
x = sigma1(pd1[i], pd2[i], c1, c2, alpha, beta);
y = sigma2(pd1[i], pd2[i], c1, c2, alpha, beta);
output.Add(x
+ " " + y
+ " " + fpd[i] + "");
}
File.WriteAllLines(filename, output);
Console.WriteLine("");
Console.WriteLine(" Wrote F(x,y) at Padua points in '" + filename + "'");
//
// Compute the matrix C0 of the coefficients in the bivariate
// orthonormal Chebyshev basis.
//
Padua.padua2(deg, DEGMAX, npd, wpd, fpd, raux1, raux2, ref c0, ref esterr);
//
// Evaluate the target points in the region.
//
target(c1, c2, alpha, beta, ntg1, NTGMAX, ref tg1, ref tg2, ref ntg);
//
// Evaluate the interpolant at the target points.
//
for (i = 0; i < ntg; i++)
{
x = isigm1(tg1[i], tg2[i], c1, c2, alpha, beta);
y = isigm2(tg1[i], tg2[i], c1, c2, alpha, beta);
intftg[i] = Padua.pd2val(deg, DEGMAX, c0, x, y);
}
//
// Write the function value at target points to a file.
//
output.Clear();
filename = "ellipse_ftg.txt";
for (i = 0; i < ntg; i++)
{
output.Add(tg1[i]
+ " " + tg2[i]
+ " " + Franke.franke(tg1[i], tg2[i]) + "");
}
File.WriteAllLines(filename, output);
Console.WriteLine(" Wrote F(x,y) at target points in '" + filename + "'");
//
// Write the interpolated function value at target points to a file.
//
output.Clear();
filename = "ellipse_itg.txt";
for (i = 0; i < ntg; i++)
{
output.Add(tg1[i]
+ " " + tg2[i]
+ " " + intftg[i] + "");
}
File.WriteAllLines(filename, output);
Console.WriteLine(" Wrote I(F)(x,y) at target points in '" + filename + "'");
//
// Compute the error relative to the max deviation from the mean.
//
double maxerr = 0.0;
double mean = 0.0;
double fmax = -typeMethods.r8_huge();
double fmin = +typeMethods.r8_huge();
for (i = 0; i < ntg; i++)
{
double fxy = Franke.franke(tg1[i], tg2[i]);
double ixy = intftg[i];
maxerr = Math.Max(maxerr, Math.Abs(fxy - ixy));
mean += fxy;
fmax = Math.Max(fmax, fxy);
fmin = Math.Min(fmin, fxy);
}
if (Math.Abs(fmax - fmin) <= double.Epsilon)
{
maxdev = 1.0;
}
else
{
mean /= ntg;
maxdev = Math.Max(fmax - mean, mean - fmin);
}
//
// Print error ratios.
//
Console.WriteLine("");
Console.WriteLine(" Estimated error: " + esterr / maxdev + "");
Console.WriteLine(" Actual error: " + maxerr / maxdev + "");
Console.WriteLine(" Expected error: " + 0.1769E-09 + "");
//
// Terminate.
//
Console.WriteLine("");
Console.WriteLine("ELLIPSE:");
Console.WriteLine(" Normal end of execution.");
Console.WriteLine("");
}
