Exponential node clustering at singularities for rational approximation, quadrature, and PDEs.

Rational approximations of functions with singularities can converge at a root-exponential rate if the poles are exponentially clustered. We begin by reviewing this effect in minimax, least-squares, and AAA approximations on intervals and complex domains, conformal mapping, and the numerical solution of Laplace, Helmholtz, and biharmonic equations by the "lightning" method. Extensive and wide-ranging numerical experiments are involved.

Stable polefinding and rational least-squares fitting via eigenvalues.

A common way of finding the poles of a meromorphic function in a domain, where an explicit expression of is unknown but can be evaluated at any given , is to interpolate by a rational function such that at prescribed sample points , and then find the roots of . This is a two-step process and the type of the rational interpolant needs to be specified by the user. Many other algorithms for polefinding and rational interpolation (or least-squares fitting) have been proposed, but their numerical stability has remained largely unexplored.