| So far, the theory for polynomial interpolation is nearly perfect. While we benefit from this algorithm, we still find that in certain circumstances, polynomial interpolation is not such a good choice. Instead, we can use a rational interpolation. For example, for approximation of a kind of function with one or more extremal points, rational interpolation can get better results than ordinary polynomial interpolation.In the first chapter, we give some basic concepts and theorems of the rational interpolation for better understanding of this paper. And then in the second chapter, we introduce a quasi- Neville algorithm for solving ordinary rational interpolation. We know that most articles which introduce algorithms for solving rational interpolation adopt a continued fraction approximation method, but less mention the quasi-Neville method. So in this paper, we will give a very specific deduction for this method, which is an iterated interpolation method. Comparing to the continued fraction approximation method, this quasi-Neville method is much more effective when the expression of the approximating function is unnecessary.In the third chapter of this paper, we introduce a convex combination method for solving osculatory rational interpolation problem. Here the definition for "osculatory rational interpolation" means that the approximating rational function should satisfy both values and derivatives at given points. The usual methods for solving this osculatory rational interpolation are also related to continued fraction approximation method, which needs both suitable points and a large amount of calculation. The advantages for the convex combination method are less calculation, no limits for points and convenience for realization by computer.The main work of this paper is giving a specific deduction for the quasi-Neville method and generalizing a convex combination method for solving two-variable osculatory rational interpolation problem.
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