spline_value
Computes the value of a spline or the value of one of its derivatives.
Synopsis
#include <imsl.h>
float imsl_f_spline_value (float x, Imsl_f_spline *sp, …, 0)
The type double function is imsl_d_spline_value.
Required Arguments
float x (Input)
Evaluation point for the spline.
Imsl_f_spline *sp (Input)
Pointer to the structure that represents the spline.
Return Value
The value of a spline or one of its derivatives at the point x. If no value can be computed, NaN is returned.
Synopsis with Optional Arguments
#include <imsl.h>
float imsl_f_spline_value (float x, Imsl_f_spline *sp,
IMSL_DERIV, int deriv,
IMSL_GRID, int n, float *xvec, float **value,
IMSL_GRID_USER, int n, float *xvec, float value_user[],
0)
Optional Arguments
IMSL_DERIV, int deriv (Input)
Let d = deriv and let s be the spline that is represented by the structure *sp. Then, this option produces the d-th derivative of s at x, s(d) (x).
Default: deriv = 0
IMSL_GRID, int n, float *xvec, float **value (Input/Output)
The argument xvec is the array of length n containing the points at which the spline is to be evaluated. The d-th derivative of the spline at the points in xvec is returned in value.
IMSL_GRID_USER int n, float *xvec, float value_user[] (Input/Output)
The argument xvec is the array of length n containing the points at which the spline is to be evaluated. The d-th derivative of the spline at the points in xvec is returned in value_user.
Description
The function imsl_f_spline_value computes the value of a spline or one of its derivatives. This function is based on the routine BVALUE by de Boor (1978, p. 144).
Examples
Example 1
In this example, a cubic spline interpolant to a function f is computed. The values of this spline are then compared with the exact function values. Since the default settings are used, the interpolant is determined by the “not-a-knot” condition (see de Boor 1978).
#include <imsl.h>
#include <stdio.h>
#include <math.h>
#define NDATA 11
/* Define function */
#define F(x) (float)(sin(15.0*x))
int main()
{
int i;
float fdata[NDATA], xdata[NDATA], x, y;
Imsl_f_spline *sp;
/* Set up a grid */
for (i = 0; i < NDATA; i++) {
xdata[i] = (float)i /((float)(NDATA-1));
fdata[i] = F(xdata[i]);
}
/* Compute cubic spline interpolant */
sp = imsl_f_spline_interp (NDATA, xdata,fdata, 0);
/* Print results */
printf(" x F(x) Interpolant Error\n");
for (i = NDATA/2; i < 3*NDATA/2; i++){
x = (float) i /(float)(2*NDATA-2);
y = imsl_f_spline_value(x, sp, 0);
printf(" %6.3f %10.3f %10.3f %10.4f\n", x, F(x), y,
fabs(F(x)-y));
}
}
Output
x F(x) Interpolant Error
0.250 -0.572 -0.549 0.0228
0.300 -0.978 -0.978 0.0000
0.350 -0.859 -0.843 0.0162
0.400 -0.279 -0.279 0.0000
0.450 0.450 0.441 0.0093
0.500 0.938 0.938 0.0000
0.550 0.923 0.903 0.0199
0.600 0.412 0.412 0.0000
0.650 -0.320 -0.315 0.0049
0.700 -0.880 -0.880 0.0000
0.750 -0.968 -0.938 0.0295
Example 2
Recall that in the first example, a cubic spline interpolant to a function f is computed. The values of this spline are then compared with the exact function values. This example compares the values of the first derivatives.
#include <imsl.h>
#include <stdio.h>
#include <math.h>
#define NDATA 11
/* Define function */
#define F(x) (float)(sin(15.0*x))
#define FP(x) (float)(15.*cos(15.0*x))
int main()
{
int i;
float fdata[NDATA], xdata[NDATA], x, y;
Imsl_f_spline *sp;
/* Set up a grid */
for (i = 0; i < NDATA; i++) {
xdata[i] = (float)i /((float)(NDATA-1));
fdata[i] = F(xdata[i]);
}
/* Compute cubic spline interpolant */
sp = imsl_f_spline_interp (NDATA, xdata, fdata, 0);
/* Print results */
printf(" x FP(x) Interpolant Deriv Error\n");
for (i = NDATA/2; i < 3*NDATA/2; i++) {
x = (float) i /(float)(2*NDATA-2);
y = imsl_f_spline_value(x, sp, IMSL_DERIV, 1, 0);
printf(" %6.3f %10.3f %10.3f %10.4f \n", x, FP(x), y,
fabs(FP(x)-y));
}
}
Output
x FP(x) Interpolant Deriv Error
0.250 -12.308 -12.559 0.2510
0.300 -3.162 -3.218 0.0560
0.350 7.681 7.796 0.1151
0.400 14.403 13.919 0.4833
0.450 13.395 13.530 0.1346
0.500 5.200 5.007 0.1926
0.550 -5.786 -5.840 0.0535
0.600 -13.667 -13.201 0.4660
0.650 -14.214 -14.393 0.1798
0.700 -7.133 -6.734 0.3990
0.750 3.775 3.911 0.1359
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. |