Continued fractions associated with the Newton-Pad� table

Abstract

Summary. We discuss first the block structure of the Newton- Pad6 table (or, rational interpolation table)

In what sense is the rational interpolation problem well posed

“…Then the Newton coefficients Pk of the numerator are explicitly given by (3.3a). The principal result concerning the system (3.3) is [8], [3], [61 [5]: The following characterization follows easily from this result (but can be proved with little effort directly): We say that z' o ..... z'+, are well ordered for the interpolant r if the zeros of the polynomial s in (3.4) (i.e., the points where r does not interpolate) come last, i.e., s(z) = (z-z'+,)(z-z'~+,_l)...(z-z~,+n-0,). If r is not a true interpolant (i.e., 0s > 0), the Newton series off and r then coincide exactly up to the term tm+,-0s, and we write In general, if the two Newton series agree at least up to the term tt, we use the notation Proof.…”

Symbolic and Interval Rational Interpolation: the Problem of Unattainable Data

“…Given finite real data {(xo, /o), • • •, (x", /")}, the classical problem of univariate rational interpolation [2,3,4] consists in finding an irreducible rational function p{x)/q{x) that satisfies the interpolation conditions ^=f, / = 0,...,«…”

A new formal approach to the rational interpolation problem

“…3). In fact, our algorithm can be seen as a BerlekampMassey version of the Euclidean-like algorithm of Gutknecht [15]. In Sect.…”

“…2). Compared to E27], we pose the rational interpolation problem in a much more general way, inspired by [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]16], allowing confluent interpolation points, poles and ~ as interpolation point (Sect. 3).…”

“…Each step of this algorithm handles another data point. In [15], Gutknecht gives an algorithm to construct continued fractions along a diagonal or a staircase in the NPT without reordering the data points. He also shows that the algorithms of Werner and of GravesMorris and Hopkins are special cases starting with a reordered set of data points.…”