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We present a complete algorithm for finding an exact minimal polynomial from its approximate value by using an improved parameterized integer relation construction method. Our result is superior to the existence of error controlling on…
This paper proposes a novel localized Fourier extension method for approximating non-periodic functions via domain segmentation. By partitioning the computational domain into subregions with uniform discretization scales, the method…
Automatic algorithms attempt to provide approximate solutions that differ from exact solutions by no more than a user-specified error tolerance. This paper describes an automatic, adaptive algorithm for approximating the solution to a…
An analytic function can be continued across an analytic arc $\Gamma$ with the help of the Schwarz function $S(z)$, the analytic function satisfying $S(z) = \bar z$ for $z\in \Gamma$. We show how $S(z)$ can be computed with the AAA…
Rational approximation is a powerful tool to obtain accurate surrogates for nonlinear functions that are easy to evaluate and linearize. The interpolatory adaptive Antoulas--Anderson (AAA) method is one approach to construct such…
This paper presents a method for computing eigenvalues and eigenvectors for some types of nonlinear eigenvalue problems. The main idea is to approximate the functions involved in the eigenvalue problem by rational functions and then apply a…
Potential theory for rational approximation is reviewed by means of examples computed with the AAA algorithm.
We study the distribution of zeros of general solutions of the Airy and Bessel equations in the complex plane. Our results characterize the patterns followed by the zeros for any solution, in such a way that if one zero is known it is…
The need to compute small con-eigenvalues and the associated con-eigenvectors of positive-definite Cauchy matrices naturally arises when constructing rational approximations with a (near) optimally small $L^{\infty}$ error. Specifically,…
We present methods that provide all zeroes and extrema of a function that do not require differentiation. Using point process theory, we are able to describe the locations of zeroes or maxima, their number, as well as their distribution…
The dominant theme of this thesis is that random matrix valued analytic functions, generalizing both random matrices and random analytic functions, for many purposes can (and perhaps should) be effectively studied in that level of…
We present a continuation method to compute all zeros of a harmonic mapping $f$ in the complex plane. Our method works without any prior knowledge of the number of zeros or their approximate location. We start by computing all solution of…
We introduce a theoretical framework for the rational approximation of optical response functions in resonant photonic systems. The framework is based on the AAA algorithm and further allows to solve the underlying nonlinear eigenproblems…
The independence polynomial originates in statistical physics as the partition function of the hard-core model. The location of the complex zeros of the polynomial is related to phase transitions, and plays an important role in the design…
We study AAK-type meromorphic approximants to functions $F$, where $F$ is a sum of a rational function $R$ and a Cauchy transform of a complex measure $\lambda$ with compact regular support included in $(-1,1)$, whose argument has bounded…
The study of inner and cyclic functions in $\ell^p_A$ spaces requires a better understanding of the zeros of the so-called optimal polynomial approximants. We determine that a point of the complex plane is the zero of an optimal polynomial…
We consider the number of zeros of holomorphic functions in a bounded domain that depend on a small parameter and satisfy an exponential upper bound near the boundary of the domain and similar lower bounds at finitely many points along the…
Functional linear discriminant analysis offers a simple yet efficient method for classification, with the possibility of achieving a perfect classification. Several methods are proposed in the literature that mostly address the…
We utilize Cauchy's argument principle in combination with the Jacobian of a holomorphic function in several complex variables and the first moment of a ratio of two correlated complex normal random variables to prove explicit formulas for…
One of the most natural and challenging issues in discrete complex analysis is to prove the convergence of discrete holomorphic functions to their continuous counterparts. This article is to solve the open problem in the general setting. To…