OPTIMAL DOMAIN SPLITTING FOR INTERPOLATION BY CHEBYSHEV POLYNOMIALS

• Published in: SIAM journal on numerical analysis Vol. 52; no. 4; pp. 1913 - 1927
• Main Authors: DRISCOLL, TOBIN A., WEIDEMAN, J. A. C.
• Format: Journal Article
• Language: English
• Published: Society for Industrial and Applied Mathematics 01-01-2014
• Subjects: Approximation; Assembly lines; Chebyshev approximation; Ellipses; Geometric planes; Interpolation; Intervals; Intervals of convergence; Mathematical analysis; Mathematical functions; Mathematical intervals; Mathematical models; Mathematical theorems; Optimization; Polynomials; Singularities; Splitting
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/130919428

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 intermediate function evaluations that are not needed in the final approximant. A Chebyshev–Padé technique generates approximate singularity locations that are good enough to get close to the optimal splitting more directly in test cases. The technique is applied to a singularly perturbed boundary-value problem.