nag_1d_spline_fit_knots (e02bac) computes a weighted least-squares approximation to an arbitrary set of data points by a cubic spline with knots prescribed by the user. Cubic spline interpolation can also be carried out.

nag_1d_spline_fit_knots (e02bac) determines a least-squares cubic spline approximation  to the set of data points  with weights , for . The value of , where  is the number of intervals of the spline (one greater than the number of interior knots), and the values of the knots , interior to the data interval, are prescribed by the user.

 has the property that it minimizes , the sum of squares of the weighted residuals , for , where

The function produces this minimizing value of  and the coefficients , where , in the B-spline representation

Here  denotes the normalized B-spline of degree 3 defined upon the knots .

In order to define the full set of B-splines required, eight additional knots  and ,  are inserted automatically by the function. The first four of these are set equal to the smallest  and the last four to the largest .

The representation of  in terms of B-splines is the most compact form possible in that only  coefficients, in addition to the  knots, fully define .

The method employed involves forming and then computing the least-squares solution of a set of  linear equations in the coefficients . The equations are formed using a recurrence relation for B-splines that is unconditionally stable (Cox (1972a), De Boor (1972)), even for multiple (coincident) knots. The least-squares solution is also obtained in a stable manner by using orthogonal transformations, viz. a variant of Givens rotations (Gentleman (1974) and Gentleman (1973)). This requires only one equation to be stored at a time. Full advantage is taken of the structure of the equations, there being at most four non-zero values of  for any value of  and hence at most four coefficients in each equation.

For further details of the algorithm and its use see Cox (1974), Cox (1975b) and Cox and Hayes (1973).

Subsequent evaluation of  from its B-spline representation may be carried out using e02bbc (nag_1d_spline_evaluate). If derivatives of  are also required, e02bcc (nag_1d_spline_deriv) may be used. e02bdc (nag_1d_spline_intg) can be used to compute the definite integral of .

"NE_ALLOC_FAIL"

 Dynamic memory allocation failed.

"NE_INT_ARG_LT"

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

"NE_KNOTS_DISTINCT_ABSCI_CONS"

 Too many knots for the number of distinct abscissae, : , . These must satisfy the constraint .

"NE_KNOTS_OUTSIDE_DATA_INTVL"

On entry, user-specified knots must be interior to the data interval,  must be greater than  and  must be less than : , , , .

"NE_NOT_INCREASING"

 The sequence lamda is not increasing: , . This condition on lamda applies to user-specified knots in the interval , . The sequence x is not increasing: , .

"NE_SW_COND_FAIL"

 The conditions specified by Schoenberg and Whitney fail. The conditions specified by Schoenberg and Whitney (1953) fail to hold for at least one subset of the distinct data abscissae. That is, there is no subset of  strictly increasing values, , , among the abscissae such that

This means that there is no unique solution: there are regions containing too many knots compared with the number of data points.

"NE_WEIGHTS_NOT_POSITIVE"

On entry, the weights are not strictly positive: .

The rounding errors committed are such that the computed coefficients are exact for a slightly perturbed set of ordinates . The ratio of the root-mean-square value for the  to the root-mean-square value of the  can be expected to be less than a small multiple of machine precision, where  is a condition number for the problem. Values of  for 20-30 practical data sets all proved to lie between 4.5 and 7.8 (see Cox (1975b)). (Note that for these data sets, replacing the coincident end knots at the end-points  and  used in the function by various choices of non-coincident exterior knots gave values of  between 16 and 180. Again see Cox (1975b) for further details.) In general we would not expect  to be large unless the choice of knots results in near-violation of the Schoenberg–Whitney conditions.

A cubic spline which adequately fits the data and is free from spurious oscillations is more likely to be obtained if the knots are chosen to be grouped more closely in regions where the function (underlying the data) or its derivatives change more rapidly than elsewhere.

The time taken by nag_1d_spline_fit_knots (e02bac) is approximately  seconds, where  is a machine-dependent constant.

Multiple knots are permitted as long as their multiplicity does not exceed 4, i.e., the complete set of knots must satisfy , for , (see Section [Error Indicators and Warnings]). At a knot of multiplicity one (the usual case),  and its first two derivatives are continuous. At a knot of multiplicity two,  and its first derivative are continuous. At a knot of multiplicity three,  is continuous, and at a knot of multiplicity four,  is generally discontinuous.

The function can be used efficiently for cubic spline interpolation, i.e., if . The abscissae must then of course satisfy . Recommended values for the knots in this case are , for .