The time taken for a call of nag_1d_spline_fit (e02bec) depends on the complexity of the shape of the data, the value of the smoothing factor , and the number of data points. If nag_1d_spline_fit (e02bec) is to be called for different values of , much time can be saved by setting  after the first call.

If the weights have been correctly chosen (see the the e02 Chapter Introduction), the standard deviation of  would be the same for all , equal to , say. In this case, choosing the smoothing factor  in the range , as suggested by Reinsch (1967), is likely to give a good start in the search for a satisfactory value. Otherwise, experimenting with different values of  will be required from the start, taking account of the remarks in Section [Description].

In that case, in view of computation time and memory requirements, it is recommended to start with a very large value for  and so determine the least-squares cubic polynomial; the value returned for fp, call it , gives an upper bound for . Then progressively decrease the value of  to obtain closer fits – say by a factor of 10 in the beginning, i.e., , , and so on, and more carefully as the approximation shows more details.

The number of knots of the spline returned, and their location, generally depend on the value of  and on the behaviour of the function underlying the data. However, if nag_1d_spline_fit (e02bec) is called with , the knots returned may also depend on the smoothing factors of the previous calls. Therefore if, after a number of trials with different values of  and , a fit can finally be accepted as satisfactory, it may be worthwhile to call nag_1d_spline_fit (e02bec) once more with the selected value for  but now using . Often, nag_1d_spline_fit (e02bec) then returns an approximation with the same quality of fit but with fewer knots, which is therefore better if data reduction is also important.

If , the requisite number of knots is known in advance, i.e., ; the interior knots are located immediately as , for . The corresponding least-squares spline (see e02bac (nag_1d_spline_fit_knots)) is then an interpolating spline and therefore a solution of the problem.

If , a suitable knot set is built up in stages (starting with no interior knots in the case of a cold start but with the knot set found in a previous call if a warm start is chosen). At each stage, a spline is fitted to the data by least-squares (see e02bac (nag_1d_spline_fit_knots)) and , the weighted sum of squares of residuals, is computed. If , new knots are added to the knot set to reduce  at the next stage. The new knots are located in intervals where the fit is particularly poor, their number depending on the value of  and on the progress made so far in reducing . Sooner or later, we find that  and at that point the knot set is accepted. The function then goes on to compute the (unique) spline which has this knot set and which satisfies the full fitting criterion specified by 2 and 3. The theoretical solution has . The function computes the spline by an iterative scheme which is ended when  within a relative tolerance of 0.001. The main part of each iteration consists of a linear least-squares computation of special form, done in a similarly stable and efficient manner as in e02bac (nag_1d_spline_fit_knots).

An exception occurs when the function finds at the start that, even with no interior knots , the least-squares spline already has its weighted sum of squares of residuals . In this case, since this spline (which is simply a cubic polynomial) also has an optimal value for the smoothness measure , namely zero, it is returned at once as the (trivial) solution. It will usually mean that  has been chosen too large.

For further details of the algorithm and its use, see Dierckx (1981a).

The value of the computed spline at a given value x may be obtained in the variable sval by the call:

e02bbc(x, sval, spline)

where spline_data is a structure of type which is the output argument of nag_1d_spline_fit (e02bec).

The values of the spline and its first three derivatives at a given value x may be obtained in the array sdif of dimension at least 4 by the call:

e02bcc(derivs, x, sdif, spline)

where, if , left-hand derivatives are computed and, if , right-hand derivatives are calculated. The value of derivs is only relevant if x is an interior knot.

The value of the definite integral of the spline over the interval  to  can be obtained in the variable sint by the call:

e02bdc(spline, sint)