public static void data_to_r8poly(int ntab, double[] xtab, double[] ytab, ref double[] c)
//****************************************************************************80
//
// Purpose:
//
// DATA_TO_R8POLY computes the coefficients of a polynomial interpolating data.
//
// Discussion:
//
// Space can be saved by using a single array for both the C and
// YTAB parameters. In that case, the coefficients will
// overwrite the Y data without interfering with the computation.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 05 September 2004
//
// Author:
//
// John Burkardt
//
// Reference:
//
// Carl deBoor,
// A Practical Guide to Splines,
// Springer, 2001,
// ISBN: 0387953663,
// LC: QA1.A647.v27.
//
// Parameters:
//
// Input, int NTAB, the number of data points.
//
// Input, double XTAB[NTAB], YTAB[NTAB], the data values.
//
// Output, double C[NTAB], the coefficients of the polynomial that passes
// through the data (XTAB,YTAB). C(0) is the constant term.
//
{
if (!typeMethods.r8vec_distinct(ntab, xtab))
{
Console.WriteLine("");
Console.WriteLine("DATA_TO_R8POLY - Fatal error!");
Console.WriteLine(" Two entries of XTAB are equal.");
return;
}
data_to_dif(ntab, xtab, ytab, ref c);
Dif.dif_to_r8poly(ntab, xtab, c, ref c);
}
private static void dif_basis_deriv_test()
//****************************************************************************80
//
// Purpose:
//
// Dif.dif_basis_deriv_test tests Dif.dif_basis_deriv;
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 01 June 2013
//
// Author:
//
// John Burkardt
//
{
const int nd = 3;
Console.WriteLine("");
Console.WriteLine("Dif.dif_basis_deriv_test");
Console.WriteLine(" Dif.dif_basis_deriv() computes difference tables for");
Console.WriteLine(" the first derivative of each Lagrange basis.");
double[] xd = new double[nd];
double[] xdp = new double[nd - 1];
double[] ddp = new double[(nd - 1) * nd];
xd[0] = -2.0;
xd[1] = 1.0;
xd[2] = 5.0;
Dif.dif_basis_deriv(nd, xd, ref xdp, ref ddp);
//
// Because the difference tables were shifted to all 0 abscissas,
// they contain the polynomial coefficients.
//
typeMethods.r8mat_transpose_print(nd - 1, nd, ddp,
" Lagrange basis derivative polynomial coefficients:");
double[] c = new double[nd - 1];
Dif.dif_to_r8poly(nd - 1, xdp, ddp, ref c, diftabIndex: +0 * (nd - 1));
typeMethods.r8poly_print(nd - 1, c, " P1'=-(2x-6)/21");
Dif.dif_to_r8poly(nd - 1, xdp, ddp, ref c, diftabIndex: +1 * (nd - 1));
typeMethods.r8poly_print(nd - 1, c, " P2'=-(2x-3)/12");
Dif.dif_to_r8poly(nd - 1, xdp, ddp, ref c, diftabIndex: +2 * (nd - 1));
typeMethods.r8poly_print(nd - 1, c, " P3'=(2x+1)/28");
}
public static double[] hermite_interpolant_rule(int n, double a, double b, double[] x)
//****************************************************************************80
//
// Purpose:
//
// HERMITE_INTERPOLANT_RULE: quadrature rule for a Hermite interpolant.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 31 October 2011
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int N, the number of abscissas.
//
// Input, double A, B, the integration limits.
//
// Input, double X[N], the abscissas.
//
// Output, double HERMITE_INTERPOLANT_RULE[2*N], the quadrature
// coefficients, given as pairs for function and derivative values
// at each abscissa.
//
{
int i;
double[] y = new double[n];
double[] yp = new double[n];
int nd = 2 * n;
double[] c = new double[nd];
double[] w = new double[nd];
double[] xd = new double[nd];
double[] yd = new double[nd];
int ndp = 2 * n - 1;
double[] xdp = new double[ndp];
double[] ydp = new double[ndp];
for (i = 0; i < n; i++)
{
y[i] = 0.0;
yp[i] = 0.0;
}
int k = 0;
for (i = 0; i < n; i++)
{
y[i] = 1.0;
hermite_interpolant(n, x, y, yp, ref xd, ref yd, ref xdp, ref ydp);
Dif.dif_to_r8poly(nd, xd, yd, ref c);
double a_value = typeMethods.r8poly_ant_val(n, c, a);
double b_value = typeMethods.r8poly_ant_val(n, c, b);
w[k] = b_value - a_value;
y[i] = 0.0;
k += 1;
yp[i] = 1.0;
hermite_interpolant(n, x, y, yp, ref xd, ref yd, ref xdp, ref ydp);
Dif.dif_to_r8poly(nd, xd, yd, ref c);
a_value = typeMethods.r8poly_ant_val(n, c, a);
b_value = typeMethods.r8poly_ant_val(n, c, b);
w[k] = b_value - a_value;
yp[i] = 0.0;
k += 1;
}
return(w);
}
public static void hermite_demo(int n, double[] x, double[] y, double[] yp)
//****************************************************************************80
//
// Purpose:
//
// HERMITE_DEMO computes and prints Hermite interpolant information for data.
//
// Discussion:
//
// Given a set of Hermite data, this routine calls HERMITE_INTERPOLANT to
// determine and print the divided difference table, and then DIF_TO_R8POLY to
// determine and print the coefficients of the polynomial in standard form.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 31 October 2011
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int N, the number of data points.
//
// Input, double X[N], the abscissas.
//
// Input, double Y[N], YP[N], the function and derivative
// values at the abscissas.
//
{
int i;
Console.WriteLine("");
Console.WriteLine("HERMITE_DEMO");
Console.WriteLine(" Compute coefficients CD of the Hermite polynomial");
Console.WriteLine(" interpolant to given data (x,y,yp).");
Console.WriteLine("");
Console.WriteLine(" Data:");
Console.WriteLine(" X Y Y'");
Console.WriteLine("");
for (i = 0; i < n; i++)
{
Console.WriteLine(" " + i.ToString(CultureInfo.InvariantCulture).PadLeft(4)
+ " " + x[i].ToString(CultureInfo.InvariantCulture).PadLeft(10)
+ " " + y[i].ToString(CultureInfo.InvariantCulture).PadLeft(10)
+ " " + yp[i].ToString(CultureInfo.InvariantCulture).PadLeft(10) + "");
}
int nd = 2 * n;
double[] xd = new double[nd];
double[] yd = new double[nd];
int ndp = 2 * n - 1;
double[] xdp = new double[ndp];
double[] ydp = new double[ndp];
hermite_interpolant(n, x, y, yp, ref xd, ref yd, ref xdp, ref ydp);
Console.WriteLine("");
Console.WriteLine(" Difference table:");
Console.WriteLine(" XD YD");
Console.WriteLine("");
for (i = 0; i < nd; i++)
{
Console.WriteLine(" " + i.ToString(CultureInfo.InvariantCulture).PadLeft(4)
+ " " + xd[i].ToString(CultureInfo.InvariantCulture).PadLeft(10)
+ " " + yd[i].ToString(CultureInfo.InvariantCulture).PadLeft(10) + "");
}
Console.WriteLine("");
Console.WriteLine(" Difference table:");
Console.WriteLine(" XDP YDP");
Console.WriteLine("");
for (i = 0; i < nd - 1; i++)
{
Console.WriteLine(" " + i.ToString(CultureInfo.InvariantCulture).PadLeft(4)
+ " " + xdp[i].ToString(CultureInfo.InvariantCulture).PadLeft(10)
+ " " + ydp[i].ToString(CultureInfo.InvariantCulture).PadLeft(10) + "");
}
double[] cd = new double[2 * n];
Dif.dif_to_r8poly(nd, xd, yd, ref cd);
typeMethods.r8poly_print(nd - 1, cd, " Hermite interpolating polynomial:");
//
// Verify interpolation claim!
//
double[] xv = new double[n];
double[] yv = new double[n];
double[] yvp = new double[n];
for (i = 0; i < n; i++)
{
xv[i] = x[i];
}
hermite_interpolant_value(nd, xd, yd, xdp, ydp, n, xv, ref yv, ref yvp);
Console.WriteLine("");
Console.WriteLine(" Data Versus Interpolant:");
Console.WriteLine(" X Y H YP HP");
Console.WriteLine("");
for (i = 0; i < n; i++)
{
Console.WriteLine(" " + i.ToString(CultureInfo.InvariantCulture).PadLeft(4)
+ " " + xv[i].ToString(CultureInfo.InvariantCulture).PadLeft(10)
+ " " + y[i].ToString(CultureInfo.InvariantCulture).PadLeft(10)
+ " " + yv[i].ToString(CultureInfo.InvariantCulture).PadLeft(10)
+ " " + yp[i].ToString(CultureInfo.InvariantCulture).PadLeft(10)
+ " " + yvp[i].ToString(CultureInfo.InvariantCulture).PadLeft(10) + "");
}
}
private static void dif_derivk_table_test()
//****************************************************************************80
//
// Purpose:
//
// Dif.dif_derivk_table_test tests Dif.dif_derivk_table;
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 01 June 2013
//
// Author:
//
// John Burkardt
//
{
int i;
int j;
Console.WriteLine("");
Console.WriteLine("Dif.dif_derivk_table_test");
Console.WriteLine(" Dif.dif_derivk_table() computes the K-th derivative");
Console.WriteLine(" for a divided difference table.");
//
// Set the 0 data points.
//
int n0 = 5;
double[] x0 = new double[n0];
double[] f0 = new double[n0];
for (i = 0; i < 5; i++)
{
x0[i] = i - 2;
}
//
// Set data for x^4/24+x^3/3+x^2/2+x+1
//
for (j = 0; j < n0; j++)
{
f0[j] = 1.0;
for (i = 4; 1 <= i; i--)
{
f0[j] = f0[j] * x0[j] / i + 1.0;
}
}
//
// Compute the difference table.
//
double[] d0 = new double[n0];
Data.data_to_dif(n0, x0, f0, ref d0);
Dif.dif_print(n0, x0, d0, " The divided difference polynomial P0:");
double[] c0 = new double[n0];
Dif.dif_to_r8poly(n0, x0, d0, ref c0);
typeMethods.r8poly_print(n0, c0, " Using DIF_TO_R8POLY");
//
// Compute the difference table for the K=1 derivative.
//
int k = 1;
int n1 = n0 - k;
double[] x1 = new double[n1];
double[] d1 = new double[n1];
Dif.dif_derivk_table(n0, x0, d0, k, x1, ref d1);
//
// Compute the difference table for the K=2 derivative.
//
k = 2;
int n2 = n0 - k;
double[] x2 = new double[n2];
double[] d2 = new double[n2];
Dif.dif_derivk_table(n0, x0, d0, k, x2, ref d2);
//
// Compute the difference table for the K=3 derivative.
//
k = 3;
int n3 = n0 - k;
double[] x3 = new double[n3];
double[] d3 = new double[n3];
Dif.dif_derivk_table(n0, x0, d0, k, x3, ref d3);
//
// Compute the difference table for the K=4 derivative.
//
k = 4;
int n4 = n0 - k;
double[] x4 = new double[n4];
double[] d4 = new double[n4];
Dif.dif_derivk_table(n0, x0, d0, k, x4, ref d4);
//
// Evaluate all 5 polynomials.
//
Console.WriteLine("");
Console.WriteLine(" Evaluate difference tables for the function P0");
Console.WriteLine(" and its first four derivatives, P1...P4.");
Console.WriteLine("");
Console.WriteLine(" X P0 P1 P2 P3 P4");
Console.WriteLine("");
for (i = 0; i <= 10; i++)
{
double x = i / 5.0;
double y0 = Dif.dif_val(n0, x0, d0, x);
double y1 = Dif.dif_val(n1, x1, d1, x);
double y2 = Dif.dif_val(n2, x2, d2, x);
double y3 = Dif.dif_val(n3, x3, d3, x);
double y4 = Dif.dif_val(n4, x4, d4, x);
Console.WriteLine(" " + x.ToString(CultureInfo.InvariantCulture).PadLeft(8)
+ " " + y0.ToString(CultureInfo.InvariantCulture).PadLeft(8)
+ " " + y1.ToString(CultureInfo.InvariantCulture).PadLeft(8)
+ " " + y2.ToString(CultureInfo.InvariantCulture).PadLeft(8)
+ " " + y3.ToString(CultureInfo.InvariantCulture).PadLeft(8)
+ " " + y4.ToString(CultureInfo.InvariantCulture).PadLeft(8) + "");
}
}
private static void test18()
//****************************************************************************80
//
// Purpose:
//
// TEST18 tests ROOTS_TO_DIF and DIF_TO_R8POLY.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 18 January 2007
//
// Author:
//
// John Burkardt
//
{
const int MAXROOTS = 4;
double[] c = new double[MAXROOTS + 1];
double[] diftab = new double[MAXROOTS + 1];
int ntab = 0;
double[] roots = new double[MAXROOTS];
double[] xtab = new double[MAXROOTS + 1];
Console.WriteLine("");
Console.WriteLine("TEST18");
Console.WriteLine(" ROOTS_TO_DIF computes a divided difference");
Console.WriteLine(" polynomial with the given roots;");
Console.WriteLine(" DIF_TO_R8POLY converts it to a standard form polynomial.");
Console.WriteLine("");
int nroots = 1;
roots[0] = 3.0;
typeMethods.r8vec_print(nroots, roots, " The roots:");
Roots.roots_to_dif(nroots, roots, ref ntab, ref xtab, ref diftab);
Dif.dif_to_r8poly(ntab, xtab, diftab, ref c);
typeMethods.r8poly_print(ntab, c, " The polynomial:");
nroots = 2;
roots[0] = 3.0;
roots[1] = 1.0;
typeMethods.r8vec_print(nroots, roots, " The roots:");
Roots.roots_to_dif(nroots, roots, ref ntab, ref xtab, ref diftab);
Dif.dif_to_r8poly(ntab, xtab, diftab, ref c);
typeMethods.r8poly_print(ntab, c, " The polynomial:");
nroots = 3;
roots[0] = 3.0;
roots[1] = 1.0;
roots[2] = 2.0;
typeMethods.r8vec_print(nroots, roots, " The roots:");
Roots.roots_to_dif(nroots, roots, ref ntab, ref xtab, ref diftab);
Dif.dif_to_r8poly(ntab, xtab, diftab, ref c);
typeMethods.r8poly_print(ntab, c, " The polynomial:");
nroots = 4;
roots[0] = 3.0;
roots[1] = 1.0;
roots[2] = 2.0;
roots[3] = 4.0;
typeMethods.r8vec_print(nroots, roots, " The roots:");
Roots.roots_to_dif(nroots, roots, ref ntab, ref xtab, ref diftab);
Dif.dif_to_r8poly(ntab, xtab, diftab, ref c);
typeMethods.r8poly_print(ntab, c, " The polynomial:");
}
private static void test05()
//****************************************************************************80
//
// Purpose:
//
// TEST05 tests DATA_TO_DIF_DISPLAY, DIF_PRINT, DIF_SHIFT_ZERO, DIF_TO_R8POLY;
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 18 January 2007
//
// Author:
//
// John Burkardt
//
{
const int MAXTAB = 10;
double[] c = new double[MAXTAB];
double[] diftab1 = new double[MAXTAB];
double[] diftab2 = new double[MAXTAB];
int i;
double[] xtab1 = new double[MAXTAB];
double[] xtab2 = new double[MAXTAB];
double[] ytab1 = new double[MAXTAB];
double[] ytab2 = new double[MAXTAB];
Console.WriteLine("");
Console.WriteLine("TEST05");
Console.WriteLine(" DIF_TO_R8POLY converts a difference table to a polynomial;");
Console.WriteLine(" DIF_SHIFT_ZERO shifts a divided difference table to 0;");
Console.WriteLine("");
Console.WriteLine(" These are equivalent operations");
Console.WriteLine("");
//
// Set XTAB, YTAB to X, F(X)
//
const int ntab = 4;
for (i = 0; i < ntab; i++)
{
double x = i + 1;
xtab1[i] = x;
xtab2[i] = x;
ytab1[i] = -4.0 + x * (3.0 + x * (-2.0 + x));
ytab2[i] = ytab1[i];
}
//
// Compute and display the finite difference table.
//
Data.data_to_dif_display(ntab, xtab1, ytab1, ref diftab1);
Data.data_to_dif_display(ntab, xtab2, ytab2, ref diftab2);
//
// Examine the corresponding polynomial.
//
Dif.dif_print(ntab, xtab1, diftab1, " The divided difference table:");
//
// Shift to zero using DIF_SHIFT_ZERO.
//
Dif.dif_shift_zero(ntab, ref xtab1, ref diftab1);
typeMethods.r8poly_print(ntab, diftab1, " The polynomial using DIF_SHIFT_ZERO:");
//
// Shift to zero using DIF_TO_R8POLY.
//
Dif.dif_to_r8poly(ntab, xtab2, diftab2, ref c);
typeMethods.r8poly_print(ntab, c, " The polynomial using DIF_TO_R8POLY:");
}
private static void dif_basis_derivk_test()
//****************************************************************************80
//
// Purpose:
//
// Dif.dif_basis_derivk_test tests DIF_BASIS_DERIVK;
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 04 June 2013
//
// Author:
//
// John Burkardt
//
{
// ReSharper disable once JoinDeclarationAndInitializer
double[] ddp;
const int k = 2;
const int nd = 5;
Console.WriteLine("");
Console.WriteLine("Dif.dif_basis_derivk_test");
Console.WriteLine(" DIF_BASIS_DERIVK computes difference tables for");
Console.WriteLine(" the K-th derivative of each Lagrange basis.");
double[] xd = new double[nd];
double[] xdp = new double[nd - k];
ddp = new double[(nd - k) * nd];
xd[0] = 1.0;
xd[1] = 2.0;
xd[2] = 3.0;
xd[3] = 4.0;
xd[4] = 5.0;
Dif.dif_basis_derivk(nd, xd, k, ref xdp, ref ddp);
//
// Because the difference tables were shifted to all 0 abscissas,
// they contain the polynomial coefficients.
//
typeMethods.r8mat_transpose_print(nd - k, nd, ddp,
" Lagrange basis K-th derivative polynomial coefficients:");
double[] c = new double[nd - k];
Dif.dif_to_r8poly(nd - k, xdp, ddp, ref c, diftabIndex: +0 * (nd - k));
typeMethods.r8poly_print(nd - k, c, " P1''=(12x^2-84x+142)/24");
Dif.dif_to_r8poly(nd - k, xdp, ddp, ref c, diftabIndex: +1 * (nd - k));
typeMethods.r8poly_print(nd - k, c, " P2''=-2x^2+13x-59/3");
Dif.dif_to_r8poly(nd - k, xdp, ddp, ref c, diftabIndex: +2 * (nd - k));
typeMethods.r8poly_print(nd - k, c, " P3''=3x^2-18x+49/2");
Dif.dif_to_r8poly(nd - k, xdp, ddp, ref c, diftabIndex: +3 * (nd - k));
typeMethods.r8poly_print(nd - k, c, " P4''=-2x^2+11x-41/3");
Dif.dif_to_r8poly(nd - k, xdp, ddp, ref c, diftabIndex: +4 * (nd - k));
typeMethods.r8poly_print(nd - k, c, " P5''=(6x^2-30x+35)/12");
}