The time taken for a call of nag_2d_spline_fit_scat (e02ddc) depends on the complexity of the shape of the data, the value of the smoothing factor , and the number of data points. If nag_2d_spline_fit_scat (e02ddc) is to be called for different values of , much time can be saved by setting  after the first call.

It should be noted that choosing  very small considerably increases computation time.

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.

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 bicubic 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.

To choose  very small is strongly discouraged. This considerably increases computation time and memory requirements. It may also cause rank-deficiency (as indicated by the argument rank) and endanger numerical stability.

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_2d_spline_fit_scat (e02ddc) 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_2d_spline_fit_scat (e02ddc) once more with the selected value for  but now using . Often, nag_2d_spline_fit_scat (e02ddc) then returns an approximation with the same quality of fit but with fewer knots, which is therefore better if data reduction is also important.

The number of knots may also depend on the upper bounds nxest and nyest. Indeed, if at a certain stage in nag_2d_spline_fit_scat (e02ddc) the number of knots in one direction (say ) has reached the value of its upper bound (nxest), then from that moment on all subsequent knots are added in the other  direction. This may indicate that the value of nxest is too small. On the other hand, it gives the user the option of limiting the number of knots the function locates in any direction. For example, by setting  (the lowest allowable value for nxest), the user can indicate that he wants an approximation which is a simple cubic polynomial in the variable .

The fit obtained is not defined outside the rectangle . The reason for taking the extreme data values of  and  for these four knots is that, as is usual in data fitting, the fit cannot be expected to give satisfactory values outside the data region. If, nevertheless, the user requires values over a larger rectangle, this can be achieved by augmenting the data with two artificial data points  and  with zero weight, where  denotes the enlarged rectangle.

First suitable knot sets are 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 bicubic spline is fitted to the data by least-squares and , the sum of squares of residuals, is computed. If , a new knot is added to one knot set or the other so as to reduce  at the next stage. The new knot is located in an interval where the fit is particularly poor. Sooner or later, we find that  and at that point the knot sets are accepted. The function then goes on to compute a spline which has these knot sets 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. The minimal least-squares solution is computed wherever the linear system is found to be rank-deficient.

An exception occurs when the function finds at the start that, even with no interior knots , the least-squares spline already has its sum of squares of residuals . In this case, since this spline (which is simply a bicubic 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 (1981b).

The values of the computed spline at the points , for , may be obtained in the array ff, of length at least n, by the following code:

e02dec(tx, ty, ff, spline,'m'=n)

where spline_data is a structure of type which is an output argument of nag_2d_spline_fit_scat (e02ddc).

To evaluate the computed spline on a kx by ky rectangular grid of points in the - plane, which is defined by the  co-ordinates stored in , for  kx, and the  co-ordinates stored in , for  ky, returning the results in the array fg which is of length at least , the following call may be used:

e02dfc(tx, ty, fg, spline,'mx'=kx,'my'=ky)

where spline_data is a structure of type which is an output argument of nag_2d_spline_fit_scat (e02ddc). The result of the spline evaluated at grid point  is returned in element  of the array fg.