nag_1d_spline_deriv (e02bcc) evaluates a cubic spline and its first three derivatives from its B-spline representation.

nag_1d_spline_deriv (e02bcc) evaluates the cubic spline  and its first three derivatives at a prescribed argument . It is assumed that  is represented in terms of its B-spline coefficients , for  and (augmented) ordered knot set , for , (see e02bac (nag_1d_spline_fit_knots)), i.e.,

Here ,  is the number of intervals of the spline and  denotes the normalized B-spline of degree 3 (order 4) defined upon the knots . The prescribed argument  must satisfy .

At a simple knot  (i.e., one satisfying ), the third derivative of the spline is in general discontinuous. At a multiple knot (i.e., two or more knots with the same value), lower derivatives, and even the spline itself, may be discontinuous. Specifically, at a point  where (exactly)  knots coincide (such a point is termed a knot of multiplicity ), the values of the derivatives of order , for , are in general discontinuous. (Here  is not meaningful.) The user must specify whether the value at such a point is required to be the left- or right-hand derivative.

The method employed is based upon:

nag_1d_spline_deriv (e02bcc) can be used to compute the values and derivatives of cubic spline fits and interpolants produced by e02bac (nag_1d_spline_fit_knots), e02bec (nag_1d_spline_fit) or e01bac (nag_1d_spline_interpolant).

If only values and not derivatives are required, e02bbc (nag_1d_spline_evaluate) may be used instead of nag_1d_spline_deriv (e02bcc), which takes about 50% longer than e02bbc (nag_1d_spline_evaluate).

"NE_ABSCI_OUTSIDE_KNOT_INTVL"

On entry, x must satisfy : , , .

"NE_BAD_PARAM"

On entry, argument derivs had an illegal value.

"NE_INT_ARG_LT"

On entry, n must not be less than 8: .

"NE_SPLINE_RANGE_INVALID"

On entry, the cubic spline range is invalid:  while . These must satisfy .

The computed value of  has negligible error in most practical situations. Specifically, this value has an absolute error bounded in modulus by  machine precision, where  is the largest in modulus of  and , and  is an integer such that . If  and  are all of the same sign, then the computed value of  has relative error bounded by  machine precision. For full details see Cox (1978).

No complete error analysis is available for the computation of the derivatives of . However, for most practical purposes the absolute errors in the computed derivatives should be small.

The time taken by this function is approximately linear in .

Note: the function does not test all the conditions on the knots given in the description of lamda in Section [Parameters], since to do this would result in a computation time approximately linear in  instead of . All the conditions are tested in e02bac (nag_1d_spline_fit_knots), however, and the knots returned by e01bac (nag_1d_spline_interpolant) or e02bec (nag_1d_spline_fit) will satisfy the conditions.