The general Neville-Aitken-algorithm and some applications

Abstract

Summary. In this note we will present the most general linear form of a NevilleAitken-algorithm for interpolation of functions by linear combinations of functions forming a t~eby~ev-system. Some applications are given. Expecially we will give simple new proofs of the recurrence formula for generalized divided differences [5] and of the author's generalization of the classical Neville-Aitkenalgorithm [-8] applying to complete t~eby~ev-systems. Another application of the general Neville-Aitken-algorithm deals with systems of linear equations. Also a numerical example is given. Subject Classifications. AMS (MOS): 65D05, 65B05, 65F05. IntroductionIn [8] the author has given an extension of the classical Neville-Aitken-algorithm [1,10] for polynomial interpolation to the more general case of interpolation by functions of a complete t~eby~ev-system which in [3] is called an outstanding problem even for simple ~eby~ev-systems. The aim of the present note is twofold: first to present a simple new proof of this extension and second to give the most general linear form of a Neville-Aitken-like interpolation algorithm computing the interpolant of a function by recurrence. This algorithm is applicable whenever a function is to be interpolated by means of linear combinations of functions forming a t~eby~ev-system such that one of its subsystems is again a (~eby~ev-system. Especially, it may be applied to interpolation by functions forming a complete or quasicomplete t~eby~ev-system where for complete (~eby~ev-systems also very general error estimates are known [9]. In case of interpolation by trigonometric polynomials the algorithm reduces to a Neville-Aitken-algorithm due to Baron [2]. Moreover, for complete t~eby~ev-systems the equivalence of the generalized Neville-Aitken-algorithm and the generalized Newton interpolation formula leads to a new simple proof of the recurrence relation for generalized divided differences [5]. Engels [4] has given another generalization of the classical Neville-Aitken-algorithm which in contrast to

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“…tp being analytic inside the simply closed contour W. From (5.1) it can be'seen that most of the present" paper's assertions apply also to the divided difference case, thus recovering some of the results in [Mühlbach 1973[Mühlbach , 1978 on classieal and generalized divided differences.…”

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“…He calls the set (fi) a Cebysev-system if given any function f, for any pair of subsets of I, L and M, having the same (finite) number of elements, there exist real numbers ai such that : for denoting the coefficients of fk in the development (6). Then, in [12], he shows that if one has a Cebysev system (theorem 4.1 pp. 106) :…”

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