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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…
On the basis of the generalized argument principle, here we develop a numerical scheme for locating zeros and poles of a meromorphic function. A subdivision-transformation-calculation scheme is proposed to ensure the algorithm stability. A…
Rouch\'e's Theorem is among the most useful results in complex analysis for counting zeros of analytic functions. Rouch\'e's Theorem also admits a harmonic analogue for counting zeros of complex harmonic functions. Previously, this analogue…
The AAA algorithm for rational approximation is employed to illustrate applications of rational functions all across numerical analysis.
In this note, we use Rouch\'e's theorem and the pleasant properties of the arithmetic of the logarithmic derivative to establish several new results regarding the geometry of the zeros, poles, and critical points of a rational function.…
In this work, we propose an extensive numerical study on approximating the absolute value function. The methods presented in this paper compute approximants in the form of rational functions and have been proposed relatively recently, e.g.,…
In this article, various results will be demonstrated that enable the delimitation of a zero-free region for holomorphic functions on a set $K$, studying the behavior of their imaginary or real part on the boundary of $K$. These findings…
A numerical algorithm (implemented in Matlab) for computing the zeros of the parabolic cylinder function $U(a,z)$ in domains of the complex plane is presented. The algorithm uses accurate approximations to the first zero plus a highly…
We consider the problem of finding a rational function in barycentric form to approximate a given function or data set in $\mathbb{R}$ or $\mathbb{C}$. The famous AAA algorithm, introduced in 2018, constructs such a rational function: the…
We investigate the zeros of two one-parameter families of harmonic functions and describe how the number of zeros depends on the parameter. Our functions have the property that all zeros lie on certain rays in the complex plane and thus we…
A two-step method for solving planar Laplace problems via rational approximation is introduced. First complex rational approximations to the boundary data are determined by AAA approximation, either globally or locally near each corner or…
Algorithms for the computation of the real zeros of hypergeometric functions which are solutions of second order ODEs are described. The algorithms are based on global fixed point iterations which apply to families of functions satisfying…
We present fully polynomial approximation schemes for a broad class of Holant problems with complex edge weights, which we call Holant polynomials. We transform these problems into partition functions of abstract combinatorial structures…
Unlike polynomials, rational functions can represent functions having poles or branch cuts with root-exponential convergence and no Runge phenomenon. Recent developments of the AAA and greedy Thiele algorithms have sparked renewed interest…
Let $E = L_p, \; 1<p\leq 2,$ and $A : E \to E^*$ be a bounded monotone map such that $0 \in R(A)$. In this paper, we introduce and study an algorithm for approximating zeros of $A$. Furthermore, we study the application of this algorithm to…
We present a method for solving nonlinear eigenvalue problems using rational approximation. The method uses the AAA method by Nakatsukasa, S\`{e}te, and Trefethen to approximate the nonlinear eigenvalue problem by a rational eigenvalue…
We present a new algorithm for finding isolated zeros of a system of real-valued functions in a bounded interval in $\mathbb{R}^n$. It uses the Chebyshev proxy method combined with a mixture of subdivision, reduction methods, and…
We introduce and develop the root locus method in mathematics. And we study the distribution of zeros of meromorphic functions by root locus method.
We obtain an estimate for uniform approximation rate of bounded analytic in the unit disk functions by logarithmic derivatives of $C$-polynomials, i.e., polynomials, all of whose zeros lie on the unit circle $C:|z|=1$.
As we have shown several years ago [Y2], zeros of $L(s, \Delta )$ and $L^(2)(s, \Delta )$ can be calculated quite efficiently by a certain experimental method. Here $\Delta$ denotes the cusp form of weight 12 with respect to SL$(2, Z)$ and…