Note on osculatory rational interpolation

Author:
Herbert E. Salzer

Journal:
Math. Comp. **16** (1962), 486-491

MSC:
Primary 65.20

DOI:
https://doi.org/10.1090/S0025-5718-1962-0149648-7

MathSciNet review:
0149648

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Abstract: In *n*-point osculatory interpolation of order at points , , by a rational expression , where and are polynomials and , We use the lemma that the system (1) , is equivalent to (2) . This equivalence does not require or to be a polynomial or even a linear combination of given functions. The lemma implies that (1), superficially non-linear in and , being the same as (2), is actually linear. For the *n*-point interpolation problem, the linear system, of order , which might be large, is replaceable by separate linear Systems of orders (or even when conveniently small) by applying the lemma to the continued fraction (3) . In (3), which has the property (proven in two ways) that the determination of is independent of all a's that follow, we find stepwise, but several at a time (instead of singly which is more tedious), retrieving them readily from the solutions of those lower-order linear systems.

**[1]**L. M. Milne-Thomson,*The Calculus of Finite Differences*, Macmillan, London, 1933, Chapter V, p. 104-123. MR**0043339 (13:245c)****[2]**P. I. Richards,*Manual of Mathematical Physics*, Pergamon, London and New York, 1959, p. 257. MR**0108988 (21:7700)****[3]**O. Perron,*Die Lehre von den Kettenbrüchen*, B. G. Teubner, Stuttgart, 1954, Vol. I, p. 5-7. MR**0064172 (16:239e)**

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DOI:
https://doi.org/10.1090/S0025-5718-1962-0149648-7

Article copyright:
© Copyright 1962
American Mathematical Society