We present a novel algorithm for computing best uniform rational approximations to real scalar functions in the setting of zero defect. The method, dubbed BRASIL (best rational approximation by successive interval length adjustment), is based on the observation that the best rational approximation r to a function f must interpolate f at a certain number of interpolation nodes (x_j). Furthermore, the sequence of local maximum errors per interval (x_j− 1,x_j) must equioscillate. The proposed algorithm iteratively rescales the lengths of the intervals with the goal of equilibrating the local errors. The required rational interpolants are computed in a stable way using the barycentric rational formula. The BRASIL algorithm may be viewed as a fixed-point iteration for the interpolation nodes and converges linearly. We demonstrate that a suitably designed rescaled and restarted Anderson acceleration (RAA) method significantly improves its convergence rate. The new algorithm exhibits excellent numerical stability and computes best rational approximations of high degree to many functions in a few seconds, using only standard IEEE double-precision arithmetic. A free and open-source implementation in Python is provided. We validate the algorithm by comparing to results from the literature. We also demonstrate that it converges quickly in some situations where the current state-of-the-art method, the minimax function from the Chebfun package which implements a barycentric variant of the Remez algorithm, fails.

我们提出了一种新颖的算法，用于在零缺陷的情况下计算与实标量函数的最佳均匀有理逼近。该方法被称为BRASIL（通过连续间隔长度调整的最佳有理逼近），是基于以下观察结果：函数f的最佳有理逼近r必须在一定数量的插值节点（x _j）处插值f。此外，每个间隔的局部最大误差序列（x _{j − 1}，x _j）必须振荡。所提出的算法以平衡局部误差为目标，迭代地重新调整间隔的长度。使用重心有理公式以稳定的方式计算所需的有理插值。BRASIL算法可以看作是插值节点的定点迭代，并且可以线性收敛。我们证明，适当设计的重新缩放和重新启动的安德森加速（RAA）方法可以大大提高其收敛速度。该新算法具有出色的数值稳定性，并且仅使用标准IEEE双精度算法即可在几秒钟内计算出对许多函数的高阶最佳有理逼近。提供了Python的免费开源实现。我们通过与文献结果进行比较来验证算法。极小极大从函数Chebfun包它实现了雷米兹算法的重心变型中，将失败。