Aitken-Hermite Interpolation

To employ Aitkert interpolatioIx in the construction of Hermite interpolating polynomials, a fundamental theorem for Ai%ken intcrpolatio~ is given which permits the development of an Aitken interpolation algorithm for a wide class of constraints. A set of constraints, f(~)(x~)-a¢i = 0 defined for (i,j) C A, a set of ordered pairs of nonnegative integers, is said to be defined over A. If t.he set A has a finite number of elements n, and if a polynomial of degree n-1 or less uniquely exists such that the constraints over A are satisfied, the polynomi~l is called the Hermite interpolating polynomial [2] for the constraints over A and is designated by fA (x) and its qth derivative by f (~) (x). A [![ermite set A is defined as a nonempty set of pairs of nonnegative integer,s (i,j) such that forj ~ 0, (i,j) C A implies (i,j-1) ~ A. LEMMA 1. For a Hermite set A, j ¢ O, (i,j) ~ A implies (i,tc) ~ A for O~Ic<j. By construction, we show that if A is a Hermite set, the Hermite interpoluti~lg polynomial L(x) exists. Also, see 0strowski [41. An Aitken pair (A,B) is defined as a pair of Hermite sets A and B such that A-B and B-A each contain a single pail'. Let A-B = { (i',j')} andB-A = A-B consists of two pairs in violation of the definition of an Aitken pair. Hence (i,j-1) ~. B or (ij-1) ~. A rl B. The following is the fundamental theorem for Aitken interpolation. TI~lnom~M 1. Given (A,B) an Aitlcen pair, let C = A O B, a Hermite set, the~z re(x) and its derivatives are determined by the generalized Aitken algorithm f~f')(x) = [(x-x,,)f~q)(x) + qf(~a-')(x)-(x-x,,)flq)(x)-qf(8"-')(x)]/(x, '-x~,) for all nonnegative q.