nag_1d_cheb_interp_fit (e02afc) computes the coefficients of a polynomial, in its Chebyshev series form, which interpolates (passes exactly through) data at a special set of points. Least-squares polynomial approximations can also be obtained.

nag_1d_cheb_interp_fit (e02afc) computes the coefficients , for , in the Chebyshev series

which interpolates the data  at the points

Here  denotes the Chebyshev polynomial of the first kind of degree  with argument . The use of these points minimizes the risk of unwanted fluctuations in the polynomial and is recommended when the data abscissae can be chosen by the user, e.g. when the data is given as a graph. For further advantages of this choice of points, see Clenshaw (1962).

In terms of the user's original variables,  say, the values of  at which the data  are to be provided are

where  and  are respectively the upper and lower ends of the range of  over which the user wishes to interpolate.

Truncation of the resulting series after the term involving , say, yields a least-squares approximation to the data. This approximation, , say, is the polynomial of degree  which minimizes

where the residual , for .

The method employed is based on the application of the three-term recurrence relation due to Clenshaw (1955) for the evaluation of the defining expression for the Chebyshev coefficients (see, for example, Clenshaw (1962)). The modifications to this recurrence relation suggested by Reinsch and Gentleman (see Gentleman (1969)) are used to give greater numerical stability.

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

Subsequent evaluation of the computed polynomial, perhaps truncated after an appropriate number of terms, should be carried out using e02aec (nag_1d_cheb_eval).

"NE_INT_ARG_LT"

On entry, nplus1 must not be less than 2: .

The rounding errors committed are such that the computed coefficients are exact for a slightly perturbed set of ordinates . The ratio of the sum of the absolute values of the  to the sum of the absolute values of the  is less than a small multiple of , where  is the machine precision.

The time taken by nag_1d_cheb_interp_fit (e02afc) is approximately proportional to .

For choice of degree when using the function for least-squares approximation, see the the e02 Chapter Introduction.