nag_1d_cheb_fit (e02adc) computes weighted least-squares polynomial approximations to an arbitrary set of data points.

nag_1d_cheb_fit (e02adc) determines least-squares polynomial approximations of degrees  to the set of data points  with weights , for .

The approximation of degree  has the property that it minimizes  the sum of squares of the weighted residuals , where

and  is the value of the polynomial of degree  at the th data point.

Each polynomial is represented in Chebyshev series form with normalized argument . This argument lies in the range  to  and is related to the original variable  by the linear transformation

Here  and  are respectively the largest and smallest values of . The polynomial approximation of degree  is represented as

where  is the Chebyshev polynomial of the first kind of degree  with argument .

For , the function produces the values of , for , together with the value of the root mean square residual . In the case  the function sets the value of  to zero.

The method employed is due to Forsythe (1957) and is based upon the generation of a set of polynomials orthogonal with respect to summation over the normalized data set. The extensions due to Clenshaw (1960) to represent these polynomials as well as the approximating polynomials in their Chebyshev series forms are incorporated. The modifications suggested by Reinsch and Gentleman (Gentleman (1969)) to the method originally employed by Clenshaw for evaluating the orthogonal polynomials from their Chebyshev series representations 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 Chebyshev series representations of the polynomial approximations should be carried out using e02aec (nag_1d_cheb_eval).

"NE_2_INT_ARG_GT"

On entry,  while the number of distinct  values, . These arguments must satisfy .

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_INT_ARG_LT"

On entry, kplus1 must not be less than 1: .

"NE_NO_NORMALISATION"

On entry, all the  in the sequence ,  are the same.

"NE_NOT_NON_DECREASING"

On entry, the sequence ,  is not in non-decreasing order.

"NE_WEIGHTS_NOT_POSITIVE"

On entry, the weights are not strictly positive: .

No error analysis for the method has been published. Practical experience with the method, however, is generally extremely satisfactory.

The time taken by nag_1d_cheb_fit (e02adc) is approximately proportional to .

The approximating polynomials may exhibit undesirable oscillations (particularly near the ends of the range) if the maximum degree  exceeds a critical value which depends on the number of data points  and their relative positions. As a rough guide, for equally spaced data, this critical value is about . For further details see page 60 of Hayes (1970).