The time taken for a call of nag_2d_spline_fit_grid (e02dcc) 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_grid (e02dcc) is to be called for different values of , much time can be saved by setting  after the first call.

nag_2d_spline_fit_grid (e02dcc) does not allow individual weighting of the data values. If these were determined to widely differing accuracies, it may be better to use e02ddc (nag_2d_spline_fit_scat). The computation time would be very much longer, however.

If the standard deviation of  is the same for all  and  (the case for which this function is designed – see Section [Weighting of Data Points ]) and known to be equal, at least approximately, to , say, then following Reinsch (1967) and choosing the smoothing factor  in the range , where , is likely to give a good start in the search for a satisfactory value. If the standard deviations vary, the sum of their squares over all the data points could be used. 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 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.

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_grid (e02dcc) 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_grid (e02dcc) once more with the selected value for  but now using . Often, nag_2d_spline_fit_grid (e02dcc) 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_grid (e02dcc) 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. Therefore the user has 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 .

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

If , 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 , new knots are added to one knot set or the other so as 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 sets are accepted. The function then goes on to compute the (unique) 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, done in a similarly stable and efficient manner as in e02bac (nag_1d_spline_fit_knots) for least-squares curve fitting.

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 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 (1982).

The values of the computed spline at the points , for  may be obtained in the array ff, of length at least , 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_grid (e02dcc).

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 , and the  co-ordinates stored in , for , 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_grid (e02dcc). The result of the spline evaluated at grid point  is returned in element  of the array fg.