The AAA algorithm for rational approximation

Journal article

We introduce a new algorithm for approximation by rational functions on a real interval or a set in the complex plane, implementable in 40 lines of Matlab. Even on a disk or interval the algorithm may outperform existing methods, and on more complicated domains it is especially competitive. The core ideas are (1) representation of the rational approximant in barycentric form with interpolation at certain support points, (2) greedy selection of the support points to avoid exponential instabilities, and (3) least-squares rather than interpolatory formulation of the overall problem. The name AAA stands for "aggressive Antoulas--Anderson" in honor of the authors who introduced a scheme based on (1). We present the core algorithm with a Matlab code and eight applications and describe variants targeted at problems of different kinds.

Authors: Nakatsukasa, Y; Sète, O; Trefethen, L

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Scientific Computing
• Volume: 40
• Issue: 3
• Pages: A1494–A1522
• Publication date: 2018-05-24
• Acceptance date: 2017-12-11
• DOI: 10.1137/16M1106122
• EISSN: 1095-7197
• ISSN: 1064-8275
• Keywords: barycentric formula; analytic continuation; Froissart doublet; AAA algorithm; rational approximation