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Mathematics of Computation

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On the osculatory rational interpolation problem

Author: Luc Wuytack
Journal: Math. Comp. 29 (1975), 837-843
MSC: Primary 65D05
MathSciNet review: 0371008
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Abstract: The problem of the existence and construction of a table of osculating rational functions $ {r_{1,m}}$ for $ 1,m \geqslant 0$ is considered. First, a survey is given of some results from the theory of osculatory rational interpolation of order $ {s_i} - 1$ at points $ {x_i}$ for $ i \geqslant 0$. Using these results, we prove the existence of continued fractions of the form

$\displaystyle {c_0} + {c_1} \cdot (x - {y_0}) + \ldots + {c_k} \cdot (x - {y_0}... ... {y_{k + 1}})}}{1} + \frac{{{c_{k + 3}} \cdot (x - {y_{k + 2}})}}{1} + \ldots ,$

with the $ {y_k}$ suitably selected from among the $ {x_i}$, whose convergents form the elements $ {r_{k,0}},{r_{k + 1,0}},{r_{k + 1,1}},{r_{k + 2,1}}, \ldots $ of the table. The properties of these continued fractions make it possible to derive an algorithm for constructing their coefficients $ {c_i}$ for $ i \geqslant 0$. This algorithm is a generalization of the qd-algorithm.

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Article copyright: © Copyright 1975 American Mathematical Society