New Laplace and Helmholtz solvers

New numerical algorithms based on rational functions are introduced that can solve certain Laplace and Helmholtz problems on two-dimensional domains with corners faster and more accurately than the standard methods of finite elements and integral equations. The new algorithms point to a reconsideration of the assumptions underlying existing numerical analysis for partial differential equations

Adaptive Thiele interpolation

The current implementation of Thiele rational interpolation in Maple (the ThieleInterpolation routine) breaks down when the points are not well-ordered. In this article, it is shown how this breakdown can be avoided by ordering the interpolation points in an adaptive way

Rational minimax approximation via adaptive barycentric representations

Computing rational minimax approximations can be very challenging when there are singularities on or near the interval of approximation - precisely the case where rational functions outperform polynomials by a landslide. We show that far more robust algorithms than previously available can be developed by making use of rational barycentric representations whose support points are chosen in an adaptive fashion as the approximant is computed. Three variants of this barycentric strategy are all shown to be powerful: (1) a classical Remez algorithm, (2) a "AAA-Lawson" method of iteratively reweighted least-squares, and (3) a differential correction algorithm. Our preferred combination, implemented in the Chebfun MINIMAX code, is to use (2) in an initial phase and then switch to (1) for generically quadratic convergence. By such methods we can calculate approximations up to type (80, 80) of $|x|$ on $[-1, 1]$ in standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan, and Carpenter required 200-digit extended precision.Comment: 29 pages, 11 figure

Case study: Approximations of the Bessel Function

The purpose of this note is to compare various approximation methods as applied to the inverse of the Bessel function of the first kind, in a given domain of the complex plane.Comment: 18 pages, 44 figure