自适应 Newton-Thiele 有理插值及应用

二元连分式插值是二元有理插值的重要组成部分；文章在前人研究的基础上，对Newton-Thiele有理插值构造过程进行改进。针对Newton-Thiele有理插值在插值过程出现逆差商不存在的情况，传统的解决方法是将相应的Thiele型插值连分式转换为Newton插值多项式，然而该处理方法会导致计算复杂度的增加。借鉴相关文献在一元有理插值上的选点方法，文章给出一种带终止条件的自适应贪婪选点算法，即在给定插值点中根据自适应条件筛选出局部点对函数进行构造，以提高Newton-Thiele有理插值函数构造过程的稳定性，提升运算效率。对非线性函数的插值结果表明：该算法的插值效果较好、误差较小；同时将该算法应用到图像修复中，并与其他相关算法的修复效果进行对比，进一步验证了该算法的有效性。

Binary continued fraction interpolation plays an important role in the field of binary rational interpolation functions. Based on previous studies, this paper improves Newton-Thiele's rational interpolation in practical application. In the construction of Newton-Thiele's rational interpolation, there are situations where the inverse differences do not exist. In traditional methods, when the inverse differences do not exist, the corresponding Thiele-type continued fraction is replaced by a Newton-type polynomial to solve this problem. However, this processing method leads to an increase in computational complexity. Drawing inspiration from the point selection methods in univariate rational interpolation in relevant literature, the paper explores a Newton-Thiele's rational interpolation algorithm based on adaptive greedy point selection strategy with termination conditions. By selecting partial points among the given points based on adaptive conditions, this algorithm can improve the stability of the construction process of Newton-Thiele's rational interpolation function and improve the efficiency of computation. Through nonlinear function interpolation, it is proved that the algorithm can produce favorable interpolation effect and maintain the error at a low level. Meanwhile, in the application of image inpainting, the algorithm is compared with other related algorithms to further verify the effectiveness of the algorithm.