2020/07/23 by Lloyd N. Trefethen, Yuji Nakatsukasa, Trefethen, Lloyd N. +3 · 5 citations
Mathematics · #41A20 #65D32 #65N35 #FOS: Mathematics #Fractional Differential Equations Solutions #Iterative Methods for Nonlinear Equations #Numerical Analysis (math.NA) #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.2007.11828
openalex publication_date 2020/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Rational approximations of functions with singularities can converge at a\nroot-exponential rate if the poles are exponentially clustered. We begin by\nreviewing this effect in minimax, least-squares, and AAA approximations on\nintervals and complex domains, conformal mapping, and the numerical solution of\nLaplace, Helmholtz, and biharmonic equations by the "lightning" method.\nExtensive and wide-ranging numerical experiments are involved. We then present\nfurther experiments showing that in all of these applications, it is\nadvantageous to use exponential clustering whose density on a logarithmic scale\nis not uniform but tapers off linearly to zero near the singularity. We give a\ntheoretical explanation of the tapering effect based on the Hermite contour\nintegral and potential theory, showing that tapering doubles the rate of\nconvergence. Finally we show that related mathematics applies to the\nrelationship between exponential (not tapered) and doubly exponential (tapered)\nquadrature formulas. Here it is the Gauss--Takahasi--Mori contour integral that\ncomes into play.\n