In this paper, applied strictly monotonic increasing scaled maps, a kind of well-conditioned linear barycentric rational interpolations are proposed to approximate functions of singularities at the origin, such as $x^\alpha$ for $\alpha \in (0,1)$ and $\log(x)$. It just takes $O(N)$ flops and can achieve fast convergence rates with the choice the scaled parameter, where $N$ is the maximum degree of the denominator and numerator. The construction of the rational interpolant couples rational polynomials in the barycentric form of second kind with the transformed Jacobi-Gauss-Lobatto points. Numerical experiments are considered which illustrate the accuracy and efficiency of the algorithms. The convergence of the rational interpolation is also considered.

中文翻译：

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通过缩放变换对奇异函数进行快速线性重心有理插值

在本文中，应用严格单调递增的比例映射，提出了一种条件良好的线性重心有理插值来逼近原点的奇异函数，例如$ x ^ \ alpha $ for $ \ alpha \ in（0,1 ）$和$ \ log（x）$。它只需要$ O（N）$触发器，并且可以通过选择scaled参数来获得快速收敛速度，其中$ N $是分母和分子的最大程度。有理插值的构造将第二种重心形式的有理多项式与转换的Jacobi-Gauss-Lobatto点耦合。考虑了数值实验，这些实验说明了算法的准确性和效率。还考虑了有理插值的收敛性。