Abstract
The interpolation of Thiele-type continued fractions is thought of as the traditional rational interpolation and plays a significant role in numerical analysis and image interpolation. Different to the classical method, a novel type of bivariate Thiele-like rational interpolation continued fractions with parameters is proposed to efficiently address the interpolation problem. Firstly, the multiplicity of the points is adjusted strategically. Secondly, bivariate Thiele-like rational interpolation continued fractions with parameters is developed. We also discuss the interpolant algorithm, theorem, and dual interpolation of the proposed interpolation method. Many interpolation functions can be gained through adjusting the parameter, which is flexible and convenient. We also demonstrate that the novel interpolation function can deal with the interpolation problems that inverse differences do not exist or that there are unattainable points appearing in classical Thiele-type continued fractions interpolation. Through the selection of proper parameters, the value of the interpolation function can be changed at any point in the interpolant region under unaltered interpolant data. Numerical examples are given to show that the developed methods achieve state-of-the-art performance.
Highlights
The interpolation method plays a critical role in approximation theory and is a classical topic in numerical analysis [1,2,3,4,5,6]
To avoid the problems mentioned above, this paper aims to develop bivariate Thiele-like rational interpolation continued fractions by introducing one or more parameters, which can adjust the shape of the curves or surfaces without altering the given interpolation points, so as to meet the practical requirements
Thiele-Like Rational Interpolation Continued Fractions with Parameter. It is well known from the literature that if Thiele-type continued fractions rational interpolation functions exist, they are unique compared with the popular method
Summary
The interpolation method plays a critical role in approximation theory and is a classical topic in numerical analysis [1,2,3,4,5,6]. To avoid the problems mentioned above, this paper aims to develop bivariate Thiele-like rational interpolation continued fractions by introducing one or more parameters, which can adjust the shape of the curves or surfaces without altering the given interpolation points, so as to meet the practical requirements. To solve the above problem, in the paper [33,34,35], the authors developed a univariate Thiele-like rational interpolation continued fractions with parameters. The organization of the paper is as follows: We give a brief review on univariate Thiele-like rational interpolation continued fractions with parameters and discuss a special rational interpolation problem where unattainable points and inverse differences do not exist, and solve it through univariate.
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