Results on the rational approximation of functions containing singularities are presented. We build further on the ''lightning method'', recently proposed by Trefethen and collaborators, based on exponentially clustering poles close to the singularities. Our results are obtained by augmenting the lightning approximation set with either a low-degree polynomial basis or poles clustering towards infinity, in order to obtain a robust approximation of the smooth behaviour of the function. This leads to a significant increase in the achievable accuracy as well as the convergence rate of the numerical scheme. For the approximation of $x^\alpha$ on $[0,1]$, the optimal convergence rate as shown by Stahl in 1993 is now achieved simply by least-squares fitting.
翻译:报告介绍了包含单数的函数合理近似结果。 我们进一步借鉴了特雷菲森和协作者最近提出的“ 亮光方法 ”, 其基础是接近奇数的指数聚集极。 我们的成果是通过提高闪电近近似值获得的, 要么是低度多元基数, 要么是向无限度集中的极点, 以获得该函数顺利运行的稳健近似值。 这导致可实现的精确度和数字方法的趋同率大幅提高。 对于以$$[0,1]为基数的“亮光方法 ” 近似值, Stahl 于1993年显示的最佳趋同率现在仅仅通过最差的安装来实现。