Related papers: A refined nonlinear least-squares method for the r…
We introduce a new algorithm for approximation by rational functions on a real or complex set of points, implementable in 40 lines of Matlab and requiring no user input parameters. Even on a disk or interval the algorithm may outperform…
Many algorithms for approximating data with rational functions are built on interpolation or least-squares approximation. Inspired by the adaptive Antoulas-Anderson (AAA) algorithm for the univariate case, the parametric adaptive…
Rational approximation appears in many contexts throughout science and engineering, playing a central role in linear systems theory, special function approximation, and many others. There are many existing methods for solving the rational…
We consider the Adaptive Antoulas-Anderson (AAA) rational interpolation algorithm recently developed by Trefethen and co-authors, which can be viewed as a type of moment-matching technique for system realization and approximation. We…
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…
Approximations based on rational functions are widely used in various applications across computational science and engineering. For univariate functions, the adaptive Antoulas-Anderson algorithm (AAA), which uses the barycentric form of a…
We present two new methods for multivariate exponential analysis. In [7], we developed a new algorithm for reconstruction of univariate exponential sums by exploiting the rational structure of their Fourier coefficients and reconstructing…
Rational minimax approximation of real functions on real intervals is an established topic, but when it comes to complex functions or domains, there appear to be no algorithms currently in use. Such a method is introduced here, the {\em…
A selection of algorithms for the rational approximation of matrix-valued functions are discussed, including variants of the interpolatory AAA method, the RKFIT method based on approximate least squares fitting, vector fitting, and a method…
We provide rigorous theoretical bounds for Anderson acceleration (AA) that allow for approximate calculations when applied to solve linear problems. We show that, when the approximate calculations satisfy the provided error bounds, the…
We describe a strategy for solving nonlinear eigenproblems numerically. Our approach is based on the approximation of a vector-valued function, defined as solution of a non-homogeneous version of the eigenproblem. This approximation step is…
In recent years, the Adaptive Antoulas-Anderson AAA algorithm has established itself as the method of choice for solving rational approximation problems. Data-driven Model Order Reduction (MOR) of large-scale Linear Time-Invariant (LTI)…
Computing the discrete rational minimax approximation in the complex plane is challenging. Apart from Ruttan's sufficient condition, there are few other sufficient conditions for global optimality. The state-of-the-art rational…
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…
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…
In this paper, we introduce adaptive neuron enhancement (ANE) method for the best least-squares approximation using two-layer ReLU neural networks (NNs). For a given function f(x), the ANE method generates a two-layer ReLU NN and a…
We propose AAA rational approximation as a method for interpolating or approximating smooth functions from equispaced data samples. Although it is always better to approximate from large numbers of samples if they are available, whether…
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…
Unitary best approximation to the exponential function on an interval on the imaginary axis has been introduced recently. In the present work two algorithms are considered to compute this best approximant: an algorithm based on rational…
The AAA algorithm for rational approximation is employed to illustrate applications of rational functions all across numerical analysis.