Applications of QR-based Vector-Valued Rational Approximation

Several applications of the QR-AAA algorithm, a greedy scheme for vector-valued rational approximation, are presented. The focus is on demonstrating the flexibility and practical effectiveness of QR-AAA in a variety of computational settings, including Stokes flow computation, multivariate rational approximation, function extension, the development of novel quadrature methods and near-field approximation in the boundary element method.

💡 Research Summary

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The paper presents QR‑AAA, a fast greedy scheme for vector‑valued rational approximation, and demonstrates its versatility through five distinct computational applications. Traditional vector‑valued AAA (SV‑AAA) constructs a rational interpolant by repeatedly forming and compressing a Loewner matrix of size ((N-m)\times m) for each of the (n) component functions. When (n) is large, the cost scales as (O(m^{2}n(N-m))) per iteration, quickly becoming prohibitive. QR‑AAA circumvents this bottleneck by first applying a column‑pivoted QR decomposition to the data matrix (F\in\mathbb{F}^{N\times n}), obtaining a low‑rank factorisation (F\approx Q C) with (Q\in\mathbb{F}^{N\times k}) and (k\ll n). The SV‑AAA algorithm is then applied only to the basis matrix (Q). The resulting support points ({\zeta_\nu}) and barycentric weights ({w_\nu}) are shared across all components, yielding a vector‑valued rational approximant \