It is shown that quadrature formulas in many different applications can be derived from rational approximation of the Cauchy transform of a weight function. Since rational approximation is now a routine technology, this provides an easy new method to derive all kinds of quadrature formulas as well as fundamental insight into the mathematics of quadrature. Intervals or curves of quadrature nodes correspond to near-optimal branch cuts of the Cauchy transform.

This paper introduces a novel algorithm that, employing rational approximants of randomly scalarized boundary integral resolvents, efficiently evaluates acoustic and electromagnetic resonances in both open and closed cavities. The desired cavity resonances (also known as ``eigenvalues"" for interior problems and ``scattering poles"" or ``complex eigenvalues"" for exterior and open-cavity problems) are obtained as the poles of associated rational approximants; both the approximants and their poles are produced by means of the recently introduced AAA rational approximation algorithm. In fact, the proposed resonance search method applies to any nonlinear eigenvalue problem associated with a given function F : U ightarrow Cdxd, wherein, denoting F(k) = Fk, a complex value k is sought for which Fkw = 0 for some nonzero w \in Cd. For the scattering problems considered in this paper, which include interior, exterior, and open-cavity problems, Fk is taken to equal a spectrally discretized version of a Green function--based boundary integral operator at spatial frequency k. In all cases, the scalarized resolvent is given by an expression of the form u*F-1 k v, where u, v \in Cd are fixed random vectors. The proposed adaptive search strategy relies on use of a rectangular subdivision of the resonance search domain which is locally refined to ensure that all resonances in the domain are captured. The approach works equally well in the case in which the search domain is a one-dimensional set, such as, e.g., an interval of the real line, in which case the rectangles used degenerate into subintervals of the search domain. A variety of numerical results are presented, including comparisons with well-known methods based on complex contour integration, and a discussion of the asymptotics that result as open cavities approach closed cavities---in all, demonstrating the accuracy provided by the method, for low-and high-frequency states alike.

The AAA algorithm for rational approximation is employed to illustrate applications of rational functions all across numerical analysis. For example, rational functions enable strikingly effective methods for numerical differentiation and integration; interpolation of equispaced data; locating zeros, poles, and branch points; analytic continuation; computation of inverse functions; imputation of missing data; computation of nonlinear eigenvalues and resonances; model order reduction/reduced order modelling; solution of Wiener–Hopf and Hilbert transform problems; conformal mapping; and solution of the two-dimensional Laplace, biharmonic and Helmholtz equations. The paper also surveys various improvements and generalizations of the AAA algorithm that have developed since its original appearance in 2018.

An algorithm is presented to compute Zolotarev rational functions, that is, rational functions r\astn of a given degree that are as small as possible on one set E C \BbbC \cup {oo\} relative to their size on another set F C \BbbC\cup {oo\} (the third Zolotarev problem). Along the way we also approximate the sign function relative to E and F (the fourth Zolotarev problem).

An analytic function can be continued across an analytic arc Γ with the help of the Schwarz function S(z), the analytic function satisfying S(z) = z̅ for z∈Γ. We show how S(z) can be computed with the AAA algorithm of rational approximation, an operation that is the basis of the AAALS method for solution of Laplace and related PDE problems in the plane. We discuss the challenge of computing S(z) further away from from Γ, where it becomes multi-valued.

It is shown that a band-limited function bounded by 1 for negative x can grow arbitrarily fast for positive x.

Potential theory for rational approximation is reviewed by means of examples computed with the AAA algorithm.

Low Reynolds number fluid flows are governed by the Stokes equations. In two dimensions, Stokes flows can be described by two analytic functions, known as Goursat functions. Brubeck and Trefethen (2022) recently introduced a lightning Stokes solver that uses rational functions to approximate the Goursat functions in polygonal domains. In this paper, we present the "LARS" algorithm (Lightning-AAA Rational Stokes) for computing 2D Stokes flows in domains with smooth boundaries and multiply-connected domains using lightning and AAA rational approximation (Nakatsukasa et al., 2018). After validating our solver against known analytical solutions, we solve a variety of 2D Stokes flow problems with physical and engineering applications. Using these examples, we show rational approximation can now be used to compute 2D Stokes flows in general domains. The computations take less than a second and give solutions with at least 6-digit accuracy.

AAA rational approximation has normally been carried out on a discrete set, typically hundreds or thousands of points in a real interval or complex domain. Here we introduce a continuum AAA algorithm that discretizes a domain adaptively as it goes. This enables fast computation of high-accuracy rational approximations on domains such as the unit interval, the unit circle, and the imaginary axis, even in some cases where resolution of singularities requires exponentially clustered sample points, support points, and poles. Prototype MATLAB (or Octave) and Julia codes aaax, aaaz, and aaai are provided for these three special domains; the latter two are equivalent by a Moebius transformation. Execution is very fast since the matrices whose SVDs are computed have only three times as many rows as columns. The codes include a AAA-Lawson option for improvement of a AAA approximant to minimax, so long as the accuracy is well above machine precision. The result returned is pole-free in the approximation domain.

In this short, conceptual paper we observe that closely related mathematics applies in four contexts with disparate literatures: (1) sigmoidal and RBF approximation of smooth functions, (2) rational approximation of analytic functions with singularities, (3) $hp\kern .7pt$-mesh refinement for solution of \pdes, and (4) double exponential (DE) and generalized Gauss quadrature. The relationships start from the change of variables $s = \log(x)$, and they suggest possibilities for new analyses and new methods in several areas. Concerning (2) and (3), we show that both problems feature the same effect of “linear tapering” near the singularity---of clustered poles in rational approximation and of polynomial orders in $hp\kern .7pt$-mesh refinement. Concerning (4), we note that the tapering effect appears here too, and that the change of variables interpretation sheds new light on why the DE and generalized Gauss methods are effective at integrating arbitrary singularities.

Laplace problems on planar domains can be solved by means of least-squares expansions associated with polynomial or rational approximations. Here it is shown that, even in the context of an analytic domain with analytic boundary data, the difference in convergence rates may be huge when the domain is nonconvex. Our proofs combine the theory of the Schwarz function for analytic continuation, potential theory for polynomial and rational approximation rates, and the theory of crowding of conformal maps.

We propose AAA rational approximation as a method for interpolating or approximating smooth functions from equispaced samples. Although it is always better to approximate from large numbers of samples if they are available, whether equispaced or not, this method often performs impressively even when the sampling grid is coarse. In most cases it gives more accurate approximations than other methods. We support this claim with a review and discussion of nine classes of existing methods in the light of general properties of approximation theory as well as the "impossibility theorem" for equispaced approximation. We make careful use of numerical experiments, which are summarized in a sequence of nine figures. Among our new contributions is the observation, summarized in Fig. 7, that methods such as polynomial least-squares and Fourier extension may be either exponentially accurate and exponentially unstable, or less accurate and stable, depending on implementation.

In this dissertation we have applied Trefethen and Gopal's Lightning Method to solve the Helmholtz equation in the exterior of two dimensional piecewise smooth domains. The background theory motivating the method is presented, and we explore the optimal method implementation for the unit square, which is subsequently used to give a guide on parameter selection for a general region. The behaviour of the computed solutions is verified to act in accordance with our intuition and current understanding of wave propagation, and we show that the wave decays to approximately 0 in the shadow region.

Let f be an analytic function on a simply-connected compact continuum E of the complex z -plane. This might be an interval of the real line, where f might be real analytic. How can we calculate good estimates of the analytic continuation of f to other points z∈ℂ ? How can we estimate the locations of real or complex singularities of f ? We review both the theory and the practice of some existing methods for these problems and propose that excellent results can be obtained from the computation of rational approximations of f by the AAA algorithm. In the case of analytic functions of two or more variables, the rational approximations are applied along line segments or other analytic arcs.

In this short, conceptual paper we observe that essentially the same mathematics applies in three contexts with disparate literatures: (1) sigmoidal and RBF approximation of smooth functions, (2) rational approximation of analytic functions near singularities, and (3) $hp$ mesh refinement for solution of PDEs. The relationship of (1) and (2) is as simple as the change of variables $s = \log(x)$, and our informal mnemonic for this relationship is ``sigmoid = log(ratapprox).''

The AAA algorithm, introduced in 2018, computes best or near-best rational approximations to functions or data on subsets of the real line or the complex plane. It is much faster and more robust than previous algorithms for such problems and has been used in many applications since its appearance, including the numerical solution of Laplace, Poisson, and biharmonic PDE problems in irregular domains. AAA has also been extended in new directions and seems likely to be a tool of lasting importance in the future.

Müntz’s theorem asserts, for example, that the linear span of the even powers 1, x^2, x^4,… is dense in C([0,1]) . We show that the associated expansions are so inefficient as to have no conceivable relevance to any actual computation. For example, approximating f(x)=x to accuracy ε = 10^-6 in this basis requires powers larger than x^280,000 and coefficients larger than 10^107,000 . We present a theorem establishing exponential growth of coefficients with respect to 1/ε .

Results on the rational approximation of functions containing singularities are presented. We build further on the ''lightning method'', recently proposed by Trefethen and collaborators, based on exponentially clustering poles close to the singularities. Our results are obtained by augmenting the lightning approximation set with either a low-degree polynomial basis or poles clustering towards infinity, in order to obtain a robust approximation of the smooth behaviour of the function. This leads to a significant increase in the achievable accuracy as well as the convergence rate of the numerical scheme. For the approximation of $x^\alpha$ on $[0,1]$, the optimal convergence rate as shown by Stahl in 1993 is now achieved simply by least-squares fitting.

Often the easiest way to discretize an ordinary or partial differential equation is by a rectangular numerical method, in which n basis functions are sampled at m ≫ n collocation points. We show how eigenvalue problems can be solved in this setting by QR reduction to square matrix generalized eigenvalue problems. The method applies equally in the limit “ m=∞ ” of eigenvalue problems for quasimatrices. Numerical examples are presented as well as pointers to related literature.

Gopal and Trefethen recently introduced"lightning solvers"for the 2D Laplace and Helmholtz equations, based on rational functions with poles exponentially clustered near singular corners. Making use of the Goursat representation in terms of analytic functions, we extend these methods to the biharmonic equation, specifically to 2D Stokes flow. Solutions to model problems are computed to 10-digit accuracy in less than a second of laptop time. As an illustration of the high accuracy, we resolve two or more counter-rotating Moffatt eddies near a singular corner.