IMSL C Math Library
spline_2d_integral
Evaluates the integral of a tensor-product spline on a rectangular domain.
Synopsis
#include <imsl.h>
float imsl_f_spline_2d_integral (float a, float b, float c, float d, Imsl_f_spline *sp)
The type double function is imsl_d_spline_2d_integral.
Required Arguments
float a (Input)
blah
float b (Input)
The integration limits for the first variable of the tensor-product spline.
float c (Input)
blah
float d (Input)
The integration limits for the second variable of the tensor-product spline.
Imsl_f_spline *sp (Input)
Pointer to the structure that represents the spline.
Return Value
The value of the integral of the tensor-product spline over the rectangle [a, b] × [c, d]. If no value can be computed, NaN is returned.
Description
The function imsl_f_spline_2d_integral computes the integral of a tensor-product spline. If s is the spline, then this function returns
This function uses the (univariate integration) identity (22) in de Boor (1978, p. 151)
where t0 ≤ x ≤ tr.
It assumes (for all knot sequences) that the first and last k knots are stacked, that is, t0 = … = tk-1 and tn = … = tn+k-1 , where k is the order of the spline in the x or y direction.
Example
This example integrates a two-dimensional, tensor-product spline over the rectangle [0, x] × [0, y].
 
#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)
/* The integral of F from 0 to x */
/* and 0 to y */
#define FI(x,y) (float)(y*x*x*x*x/4. + x*y*y*y/3.)
 
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 FI(x, y) Integral 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_integral(0.0, x, 0.0, y, sp);
printf(" %6.3f %6.3f %10.3f %10.3f %10.4f\n",
x, y, FI(x, y), z, fabs(FI(x,y)-z));
}
}
}
 
Output
 
x y FI(x, y) Integral Error
0.333 0.333 0.005 0.005 0.0000
0.333 0.667 0.035 0.035 0.0000
0.667 0.333 0.025 0.025 0.0000
0.667 0.667 0.099 0.099 0.0000
Warning Errors
IMSL_SPLINE_LEFT_ENDPT
The left endpoint of X integration is not within the knot sequence. Integration occurs only from torder-1 to b.
IMSL_SPLINE_RIGHT_ENDPT
The right endpoint of X integration is not within the knot sequence. Integration occurs only from torder-1 to a.
IMSL_SPLINE_LEFT_ENDPT_1
The left endpoint of X integration is not within the knot sequence. Integration occurs only from b to tspline_space_dim‑1.
IMSL_SPLINE_RIGHT_ENDPT_1
The right endpoint of X integration is not within the knot sequence. Integration occurs only from a to tspline_space_dim‑1.
IMSL_SPLINE_LEFT_ENDPT_2
The left endpoint of Y integration is not within the knot sequence. Integration occurs only from torder-1 to d.
IMSL_SPLINE_RIGHT_ENDPT_2
The right endpoint of Y integration is not within the knot sequence. Integration occurs only from torder-1 to c.
IMSL_SPLINE_LEFT_ENDPT_3
The left endpoint of Y integration is not within the knot sequence. Integration occurs only from d to tspline_space_dim‑1.
IMSL_SPLINE_RIGHT_ENDPT_3
The right endpoint of Y integration is not within the knot sequence. Integration occurs only from c to tspline_space_dim‑1.
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.