Quadrature formulas from rational approximations
Andrew Horning, Lloyd N. Trefethen
TL;DR
This paper introduces a novel method for deriving quadrature formulas using rational approximation of the Cauchy transform, offering both practical tools and deeper mathematical understanding.
Contribution
It presents a new approach to generate quadrature formulas from rational approximations, linking node placement to branch cuts of the Cauchy transform.
Findings
Quadrature nodes correspond to near-optimal branch cuts.
The method simplifies deriving various quadrature formulas.
Provides insights into the mathematical structure of quadrature.
Abstract
It is shown that quadrature formulas in many different applications can be derived from rational approximation of the Cauchy transform of a weight function. Since rational approximation is now a routine technology, this provides an easy new method to derive all kinds of quadrature formulas as well as fundamental insight into the mathematics of quadrature. Intervals or curves of quadrature nodes correspond to near-optimal branch cuts of the Cauchy transform.
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