Optimal Domain Splitting for Interpolation by Chebyshev Polynomials
Optimal Domain Splitting for Interpolation by Chebyshev Polynomials
复制标题
切比雪夫多项式插值的最优域分割
DOI：
10.1137/130919428
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
J. Weideman
中科院分区：
文献类型：
--
作者：
T. Driscoll;J. Weideman
用切比雪夫极值点作为节点定义的多项式插值在对区间上解析的函数采样时，以几何速率均匀收敛。然而，当函数扩展到复平面时，如果在区间附近有一个奇异点，则收敛速度可以任意接近于1。在这种情况下，分割间隔并进行分段插值可能比全局插值在节点总数上更有效。由于收敛速度是由Joukowski保角映射得到的Bernstein椭圆决定的，因此可以计算区间内任意点的相对分裂效率，然后在区间内进行优化。最优分割可以递归地应用。Chebfun软件项目使用一个简单的经验法则来创建一个二进制搜索，它可以在大多数情况下出色地找到最优的分割。然而，该过程可以使用大量的中间体…
Polynomial interpolants defined using Chebyshev extreme points as nodes converge uniformly at a geometric rate when sampling a function that is analytic on an interval. However, the convergence rate can be arbitrarily close to unity if the function has a singularity close to the interval when extended to the complex plane. In such cases, splitting the interval and doing piecewise interpolation may be more efficient in the total number of nodes than the global interpolant. Because the convergence rate is determined by Bernstein ellipses obtained through a Joukowski conformal map, relative efficiency of splitting at any point in the interval can be calculated and then optimized over the interval. The optimal splitting may be applied recursively. The Chebfun software project uses a simple rule of thumb without prior singularity information to create a binary search that can be shown to do an excellent job of finding the optimal splitting in most cases. However, the process can use a large number of intermedi...