Computes the value of a Hermite quintic spline or the value of one of its derivatives. In particular, computes solutions to the Feynman-Kac PDE handled by function imsl_f_feynman_kac.
The type double function is imsl_d_feynman_kac_evaluate.
Required Arguments
intnw (Input) Length of the array containing the evaluation points of the spline.
intm (Input) Number of breakpoints for the Hermite quintic spline interpolation. It is required that m 2. When applied to imsl_f_feynman_kac, m is identical to argument nxgrid.
floatbreakpoints[] (Input) Array of length m containing the breakpoints for the Hermite quintic spline interpolation. The breakpoints must be in strictly increasing order. When applied to imsl_f_feynman_kac, breakpoints[] is identical to array xgrid[].
floatw[] (Input) Vector of length nw containing the evaluation points for the spline. It is required that breakpoints[0]w[i]breakpoints[m-1] for i=0,…,nw-1.
floatcoef[] (Input) Vector of length 3*m containing the coefficients of the Hermite quintic spline. When applied to imsl_f_feynman_kac, this vector is one of the rows of output arrays y or y_prime related to the spline coefficients at time points t=tgrid[j], j=1, …,ntgrid.
Return Value
A pointer to an array of length nw containing the values of the Hermite quintic spline or one of its derivatives at the evaluation points in array w[]. If no values can be computed, then NULL is returned.
IMSL_DERIV, intderiv (Input) Let d = deriv and let H(w) be the given Hermite quintic spline. Then, this option produces the d-th derivative of H(w) at w, Hd(w). It is required that deriv= 0,1,2 or 3. Default: deriv= 0.
IMSL_RETURN_USER, float value[] (Output) A user defined array of length nw to receive the d-th derivative of at the evaluation points in w[]. When using this option, the return value of the function is NULL.
Description
The Hermite quintic spline interpolation is done over the composite interval , where -breakpoints[i-1] = are given by .
The Hermite quintic spline function is constructed using three primary functions, defined by
For each
the spline is locally defined by
where
are the values of a given twice continuously differentiable function and its first two derivatives at the breakpoints. The approximating function is twice continuously differentiable on , whereas the third derivative is in general only continuous within the interior of the intervals . From the local representation of it follows that
The spline coefficients are stored as successive triplets in array coef[]. For a given , function imsl_f_feynman_kac_evaluate uses the information in coef[] together with the values of and its derivatives at to compute using the local representation on the particular subinterval containing .
Example
Consider function , a polynomial of degree 5, on the interval with breakpoints . Then, the end derivative values are
and
Since the Hermite quintic interpolates all polynomials up to degree 5 exactly, the spline interpolation on must agree with the exact function value up to rounding errors.
#include <imsl.h>
#include <stdio.h>
#include <math.h>
/* Define function */
#define F(x) pow(x,5.0)
int main()
{
int i;
int nw = 7;
int m = 2;
float breakpoints[] = { -1.0, 1.0 };
float w[] = { -0.75, -0.5, -0.25, 0.0,
0.25, 0.5, 0.75 };
float coef[] = { -1.0, 5.0, -20.0,
1.0, 5.0, 20.0 };
float *result = NULL;
result = imsl_f_feynman_kac_evaluate(nw, m, breakpoints, w, coef, 0);