nag_1d_spline_interpolant (e01bac) determines a cubic spline interpolant to a given set of data.

nag_1d_spline_interpolant (e01bac) determines a cubic spline , defined in the range , which interpolates (passes exactly through) the set of data points , for , where  and . Unlike some other spline interpolation algorithms, derivative end conditions are not imposed. The spline interpolant chosen has  interior knots , which are set to the values of  respectively. This spline is represented in its B-spline form (see Cox (1975a)):

where  denotes the normalized B-spline of degree 3, defined upon the knots , and  denotes its coefficient, whose value is to be determined by the function.

The use of B-splines requires eight additional knots , , , , , ,  and  to be specified; the function sets the first four of these to  and the last four to .

The algorithm for determining the coefficients is as described in Cox (1975a) except that  factorization is used instead of  decomposition. The implementation of the algorithm involves setting up appropriate information for the related function e02bac (nag_1d_spline_fit_knots) followed by a call of that function. (For further details of e02bac (nag_1d_spline_fit_knots), see the function document.)

Values of the spline interpolant, or of its derivatives or definite integral, can subsequently be computed as detailed in Section [Further Comments].

"NE_ALLOC_FAIL"

 Dynamic memory allocation failed.

"NE_INT_ARG_LT"

On entry, m must not be less than 4: .

"NE_NOT_STRICTLY_INCREASING"

 The sequence x is not strictly increasing: , .

The rounding errors incurred are such that the computed spline is an exact interpolant for a slightly perturbed set of ordinates . The ratio of the root-mean-square value of the  to that of the  is no greater than a small multiple of the relative machine precision.

The time taken by nag_1d_spline_interpolant (e01bac) is approximately proportional to .

All the  are used as knot positions except  and . This choice of knots (see Cox (1977)) means that  is composed of  cubic arcs as follows. If , there is just a single arc space spanning the whole interval  to . If , the first and last arcs span the intervals  to  and  to  respectively. Additionally if , the th arc, for  spans the interval  to .

After the call

e01bac(x, y, spline)

the following operations may be carried out on the interpolant .

The value of  at  can be provided in the variable sval by calling the function

e02bbc(xval, sval, spline)

The values of  and its first three derivatives at  can be provided in the array sdif of dimension 4, by the call

e02bcc(derivs, xval, sdif, spline)

Here derivs must specify whether the left- or right-hand value of the third derivative is required (see e02bcc (nag_1d_spline_deriv) for details). The value of the integral of  over the range  to  can be provided in the variable sint by

e02bdc(spline, sint)