scattered_2d_interp
Computes a smooth bivariate interpolant to scattered data that is locally a quintic polynomial in two variables.
Synopsis
#include <imsl.h>
float *imsl_f_scattered_2d_interp (int ndata, float xydata[], float fdata[], int nx_out, int ny_out, float x_out[], float y_out[], …, 0)
The type double function is imsl_d_scattered_2d_interp.
Required Arguments
int ndata (Input)
Number of data points.
float xydata[] (Input)
Array with ndata*2 components containing the data points for the interpolation problem. The i-th data point (xi, yi) is stored consecutively in the 2i and 2i + 1 positions of xydata.
float fdata[] (Input)
Array of size ndata containing the values to be interpolated.
int nx_out (Input)
Number of data points in the x direction for the output grid.
int ny_out (Input)
Number of data points in the y direction for the output grid.
float x_out[] (Input)
Array of length nx_out specifying the x values for the output grid. It must be strictly increasing.
float y_out[] (Input)
Array of length ny_out specifying the y values for the output grid. It must be strictly increasing.s
Return Value
A pointer to the nx_out × ny_out grid of values of the interpolant. If no answer can be computed, then NULL is returned. To release this space, use imsl_free.
Synopsis with Optional Arguments
#include <imsl.h>
float *imsl_f_scattered_2d_interp (int ndata, float xydata[], float fdata[], int nx_out, int ny_out, float x_out[], float y_out[],
IMSL_RETURN_USER, float surface[],
IMSL_SUR_COL_DIM, int surface_col_dim,
0)
Optional Arguments
IMSL_RETURN_USER, float surface[] (Output)
This option allows the user to provide his own space for the result. In this case, the answer will be returned in surface.
IMSL_SUR_COL_DIM, int surface_col_dim (Input)
This option requires the user to provide the column dimension of the two-dimensional array surface.
Default: surface_col_dim = ny_out
Description
The function imsl_f_scattered_2d_interp computes a C1 interpolant to scattered data in the plane. Given the data points
in R3 where n = ndata, imsl_f_scattered_2d_interp returns the values of the interpolant s on the user-specified grid. The computation of s is as follows: First the Delaunay triangulation of the points
is computed. On each triangle T in this triangulation, s has the form
Thus, s is a bivariate quintic polynomial on each triangle of the triangulation. In addition, we have
and s is continuously differentiable across the boundaries of neighboring triangles. These conditions do not exhaust the freedom implied by the above representation. This additional freedom is exploited in an attempt to produce an interpolant that is faithful to the global shape properties implied by the data. For more information on this procedure, refer to the article by Akima (1978). The output grid is specified by the two integer variables nx_out and ny_out that represent the number of grid points in the first (second) variable and by two real vectors that represent the first (second) coordinates of the grid.
Examples
Example 1
In this example, the interpolant to the linear function (3 + 7x + 2y) is computed from 20 data points equally spaced on the circle of radius 3. Then the values are printed on a 3 ×3 grid.
#include <imsl.h>
#include <stdio.h>
#include <math.h>
#define NDATA 20
#define OUTDATA 3
/* Define function */
#define F(x,y) (float)(3.+7.*x+2.*y)
#define SURF(I,J) surf[(J) +(I)*OUTDATA]
int main()
{
int i, j;
float fdata[NDATA], xydata[2*NDATA], *surf;
float x, y, z, x_out[OUTDATA], y_out[OUTDATA], pi;
pi = imsl_f_constant("pi", 0);
/* Set up output grid */
for (i = 0; i < OUTDATA; i++) {
x_out[i] = y_out[i] = (float) i / ((float) (OUTDATA - 1));
}
for (i = 0; i < 2*NDATA; i += 2) {
xydata[i] = 3.*cos(pi*i/NDATA);
xydata[i+1] = 3.*sin(pi*i/NDATA);
fdata[i/2] = F(xydata[i], xydata[i+1]);
}
/* Compute scattered data interpolant */
surf = imsl_f_scattered_2d_interp (NDATA, xydata, fdata, OUTDATA,
OUTDATA, x_out, y_out, 0);
/* Print results */
printf(" x y F(x, y) Interpolant Error\n");
for (i = 0; i < OUTDATA; i++) {
for (j = 0; j < OUTDATA; j++) {
x = x_out[i];
y = y_out[j];
z = SURF(i,j);
printf(" %6.3f %6.3f %10.3f %10.3f %10.4f\n",
x, y, F(x,y), z, fabs(F(x,y)-z));
}
}
}
Output
x y F(x, y) Interpolant Error
0.000 0.000 3.000 3.000 0.0000
0.000 0.500 4.000 4.000 0.0000
0.000 1.000 5.000 5.000 0.0000
0.500 0.000 6.500 6.500 0.0000
0.500 0.500 7.500 7.500 0.0000
0.500 1.000 8.500 8.500 0.0000
1.000 0.000 10.000 10.000 0.0000
1.000 0.500 11.000 11.000 0.0000
1.000 1.000 12.000 12.000 0.0000
Example 2
Recall that in the first example, the interpolant to the linear function 3 + 7x + 2y is computed from 20 data points equally spaced on the circle of radius 3. We then print the values on a 3 × 3 grid. This example used the optional arguments to indicate that the answer is stored noncontiguously in a two-dimensional array surf with column dimension equal to 11.
#include <imsl.h>
#include <stdio.h>
#include <math.h>
#define NDATA 20
#define OUTDATA 3
#define COLDIM 11
/* Define function */
#define F(x,y) (float)(3.+7.*x+2.*y)
int main()
{
int i, j;
float fdata[NDATA], xydata[2*NDATA];
float surf[OUTDATA][COLDIM];
float x, y, z, x_out[OUTDATA], y_out[OUTDATA], pi;
pi = imsl_f_constant("pi", 0);
/* Set up output grid */
for (i = 0; i < OUTDATA; i++) {
x_out[i] = y_out[i] = (float) i / ((float) (OUTDATA - 1));
}
for (i = 0; i < 2*NDATA; i += 2) {
xydata[i] = 3.*cos(pi*i/NDATA);
xydata[i+1] = 3.*sin(pi*i/NDATA);
fdata[i/2] = F(xydata[i], xydata[i+1]);
}
/* Compute scattered data interpolant */
imsl_f_scattered_2d_interp (NDATA, xydata, fdata, OUTDATA,
OUTDATA, x_out, y_out,
IMSL_RETURN_USER, surf,
IMSL_SUR_COL_DIM, COLDIM,
0);
/* Print results */
printf(" x y F(x, y) Interpolant Error\n");
for (i = 0; i < OUTDATA; i++) {
for (j = 0; j < OUTDATA; j++) {
x = x_out[i];
y = y_out[j];
z = surf[i][j];
printf(" %6.3f %6.3f %10.3f %10.3f %10.4f\n",
x, y, F(x,y), z, fabs(F(x,y)-z));
}
}
}
Output
x y F(x, y) Interpolant Error
0.000 0.000 3.000 3.000 0.0000
0.000 0.500 4.000 4.000 0.0000
0.000 1.000 5.000 5.000 0.0000
0.500 0.000 6.500 6.500 0.0000
0.500 0.500 7.500 7.500 0.0000
0.500 1.000 8.500 8.500 0.0000
1.000 0.000 10.000 10.000 0.0000
1.000 0.500 11.000 11.000 0.0000
1.000 1.000 12.000 12.000 0.0000
Fatal Errors
IMSL_DUPLICATE_XYDATA_VALUES | The two-dimensional data values must be distinct. |
IMSL_XOUT_NOT_STRICTLY_INCRSING | The vector x_out must be strictly increasing. |
IMSL_YOUT_NOT_STRICTLY_INCRSING | The vector y_out must be strictly increasing. |