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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Note on osculatory rational interpolation

Author: Herbert E. Salzer
Journal: Math. Comp. 16 (1962), 486-491
MSC: Primary 65.20
MathSciNet review: 0149648
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Abstract: In n-point osculatory interpolation of order $ {r_i}\, - \,1$ at points $ {x_i}$, $ i\, = \,1,\,2,\, \cdots \,,\,n,$, by a rational expression $ {N(x)}/{D(x)}$, where $ N(x)$ and $ D(x)$ are polynomials $ \sum {{a_j}{x^j}} $ and $ \sum {{b_j}{x^j}} $, We use the lemma that the system (1) $ {{\{N({{x}_{i}})/D({{x}_{i}})\}}^{(m)}}\,=\,{{f}^{(m)}}({{x}_{i}}),\,m=0,1,\cdots ,{{r}_{i}}-1$, is equivalent to (2) $ {N^{(m)}}({{x_i}})\, = \,{\{{f({{x_i}})D({{x_i}})}\}^{(m)}},\,m\, = \,0,\,1,\, \cdots \,,\,{r_i}\, - \,1,\,D({{x_i}})\, \ne \,0$. This equivalence does not require $ N(x)$ or $ D(x)$ to be a polynomial or even a linear combination of given functions. The lemma implies that (1), superficially non-linear in $ {a_j}$ and $ {b_j}$, being the same as (2), is actually linear. For the n-point interpolation problem, the linear system, of order $ \sum\limits_{i = 1}^n {{r_i}} $, which might be large, is replaceable by separate linear Systems of orders $ {r_i}$ (or even $ {r_i}\, + \,{r_{i + 1}}\, + \, \cdots \, + \,{r_{i + j}}$ when conveniently small) by applying the lemma to the continued fraction (3) $ {N(x)}/{D(x)\,=\,{{a}_{1,0}}\,+\,\tfrac{x- {{x}_{1}}\vert}{\vert {{a}_{1,1}} }... ...1}} }\,+\,\cdots \,+\tfrac{x- {{x}_{n}} \vert}{\vert {{a}_{n,{{r}_{n}}-1}} }}\;$. In (3), which has the property (proven in two ways) that the determination of $ {a_{i,m}}$ is independent of all a's that follow, we find $ {a_{i,m}}$ 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.

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Article copyright: © Copyright 1962 American Mathematical Society