On the osculatory rational interpolation problem

Author:

Wuytack Luc

Abstract

The problem of the existence and construction of a table of osculating rational functions r 1 , m {r_{1,m}} for 1 , m 0 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 {s_i} - 1 at points x i {x_i} for i 0 i \geqslant 0 . Using these results, we prove the existence of continued fractions of the form \[ c 0 + c 1 ( x y 0 ) + + c k ( x y 0 ) ( x y k 1 ) + c k + 1 ( x y 0 ) ( x y k ) 1 + c k + 2 ( x y k + 1 ) 1 + c k + 3 ( x y k + 2 ) 1 + , {c_0} + {c_1} \cdot (x - {y_0}) + \ldots + {c_k} \cdot (x - {y_0}) \ldots (x - {y_{k - 1}}) + \frac {{{c_{k + 1}} \cdot (x - {y_0}) \ldots (x - {y_k})}}{1} + \frac {{{c_{k + 2}} \cdot (x - {y_{k + 1}})}}{1} + \frac {{{c_{k + 3}} \cdot (x - {y_{k + 2}})}}{1} + \ldots , \] with the y k {y_k} suitably selected from among the x i {x_i} , whose convergents form the elements r k , 0 , r k + 1 , 0 , r k + 1 , 1 , r k + 2 , 1 , {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 {c_i} for i 0 i \geqslant 0 . This algorithm is a generalization of the qd-algorithm.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,Computational Mathematics,Algebra and Number Theory

Reference7 articles.

1. Osculatory interpolation;Kahng, S. W.;Math. Comp.,1969

2. Note on osculatory rational interpolation;Salzer, Herbert E.;Math. Comp.,1962

3. Die Grundlehren der mathematischen Wissenschaften, Band 141;Angelitch, T. P.,1968

4. H. C. THACHER, JR., "A recursive procedure for osculatory interpolation by rational functions." (Unpublished manuscript.)

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