spline_2d_value
Computes the value of a tensor-product spline or the value of one of its partial derivatives.
Synopsis
#include <imsl.h>
float imsl_f_spline_2d_value (float x, float y, Imsl_f_spline *sp, …, 0)
The type double function is imsl_d_spline_2d_value.
Required Arguments
float x (Input)
float y (Input)
The (x, y) coordinates of the evaluation point for the tensor-product spline.
Imsl_f_spline *sp (Input)
Pointer to the structure that represents the spline.
Return Value
The value of a tensor-product spline or one of its derivatives at the point (x, y).
Synopsis with Optional Arguments
#include <imsl.h>
float imsl_f_spline_2d_value (float x, float y, Imsl_f_spline *sp,
IMSL_DERIV, int x_partial, int y_partial,
IMSL_GRID, int nx, float *xvec, int ny, float *yvec, float **value,
IMSL_GRID_USER, int nx, float *xvec, int ny, float *yvec, float value_user[],
0)
Optional Arguments
IMSL_DERIV, int x_partial, int y_partial (Input)
Let p = x_partial and q = y_partial, and let s be the spline that is represented by the structure *sp, then this option produces the (p, q)-th derivative of s at (x, y), s(p,q) (x, y).
Default: x_partial = y_partial = 0
IMSL_GRID, int nx, float *xvec, int ny, float *yvec, float **value (Input/Output)
The argument xvec is the array of length nx containing the X coordinates at which the spline is to be evaluated. The argument yvec is the array of length ny containing the Y coordinates at which the spline is to be evaluated. The value of the spline on the nx by ny grid is returned in value.
IMSL_GRID_USER, int nx, float *xvec, int ny, float *yvec, float value_user[] (Input/Output)
The argument xvec is the array of length nx containing the X coordinates at which the spline is to be evaluated. The argument yvec is the array of length ny containing the Y coordinates at which the spline is to be evaluated. The value of the spline on the nx by ny grid is returned in the user-supplied space value_user.
Description
The function imsl_f_spline_2d_value computes the value of a tensor-product spline or one of its derivatives. This function is based on the discussion in de Boor (1978, pp. 351−353).
Examples
Example 1
In this example, a spline interpolant
s to a function
f is constructed. Using the procedure
imsl_f_spline_2d_interp to compute the interpolant,
imsl_f_spline_2d_value is employed to compute
s(
x,
y). The values of this partial derivative and the error are computed on a 4
× 4 grid and then displayed.
#include <imsl.h>
#include <stdio.h>
#include <math.h>
#define NDATA 11
#define OUTDATA 2
/* Define function */
#define F(x,y) (float)(x*x*x+y*y)
int main()
{
int i, j, num_xdata, num_ydata;
float fdata[NDATA][NDATA], xdata[NDATA], ydata[NDATA];
float x, y, z;
Imsl_f_spline *sp;
/* Set up grid */
for (i = 0; i < NDATA; i++) {
xdata[i] = ydata[i] = (float) i / ((float) (NDATA - 1));
}
for (i = 0; i < NDATA; i++) {
for (j = 0; j < NDATA; j++) {
fdata[i][j] = F(xdata[i], ydata[j]);
}
}
num_xdata = num_ydata = NDATA;
/* Compute tensor-product interpolant */
sp = imsl_f_spline_2d_interp(num_xdata, xdata, num_ydata,
ydata, fdata, 0);
/* Print results */
printf(" x y F(x, y) Value Error\n");
for (i = 0; i < OUTDATA; i++) {
x = (float) (1+i) / (float) (OUTDATA+1);
for (j = 0; j < OUTDATA; j++) {
y = (float) (1+j) / (float) (OUTDATA+1);
z = imsl_f_spline_2d_value(x, y, sp, 0);
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) Value Error
0.333 0.333 0.148 0.148 0.0000
0.333 0.667 0.481 0.481 0.0000
0.667 0.333 0.407 0.407 0.0000
0.667 0.667 0.741 0.741 0.0000
Example 2
In this example, a spline interpolant s to a function f is constructed. Using function imsl_f_spline_2d_interp to compute the interpolant, then imsl_f_spline_2d_value is employed to compute s(2,1) (x, y). The values of this partial derivative and the error are computed on a 4 × 4 grid and then displayed.
#include <imsl.h>
#include <stdio.h>
#include <math.h>
#define NDATA 11
#define OUTDATA 2
/* Define function */
#define F(x, y) (float)(x*x*x*y*y)
#define F21(x,y) (float)(6.*x*2.*y)
int main()
{
int i, j, num_xdata, num_ydata;
float fdata[NDATA][NDATA], xdata[NDATA], ydata[NDATA];
float x, y, z;
Imsl_f_spline *sp;
/* Set up grid */
for (i = 0; i < NDATA; i++) {
xdata[i] = ydata[i] = (float)i / ((float)(NDATA-1));
}
for (i = 0; i < NDATA; i++) {
for (j = 0; j < NDATA; j++) {
fdata[i][j] = F(xdata[i], ydata[j]);
}
}
num_xdata = num_ydata = NDATA;
/* Compute tensor-product interpolant */
sp = imsl_f_spline_2d_interp(num_xdata, xdata, num_ydata,
ydata, fdata, 0);
/* Print results */
printf(" x y F21(x, y) 21InterpDeriv Error\n");
for (i = 0; i < OUTDATA; i++) {
x = (float) (1+i) / (float) (OUTDATA+1);
for (j = 0; j < OUTDATA; j++) {
y = (float) (1+j) / (float) (OUTDATA+1);
z = imsl_f_spline_2d_value(x, y, sp,
IMSL_DERIV, 2, 1,
0);
printf(" %6.3f %6.3f %10.3f %10.3f %10.4f\n",
x, y, F21(x, y), z, fabs(F21(x,y)-z));
}
}
}
Output
x y F21(x, y) 21InterpDeriv Error
0.333 0.333 1.333 1.333 0.0000
0.333 0.667 2.667 2.667 0.0000
0.667 0.333 2.667 2.667 0.0000
0.667 0.667 5.333 5.333 0.0001
Warning Errors
IMSL_X_NOT_WITHIN_KNOTS | The value of x does not lie within the knot sequence. |
IMSL_Y_NOT_WITHIN_KNOTS | The value of y does not lie within the knot sequence. |
Fatal Errors
IMSL_KNOT_MULTIPLICITY | Multiplicity of the knots cannot exceed the order of the spline. |
IMSL_KNOT_NOT_INCREASING | The knots must be nondecreasing. |