QUADRATURE = RATIONAL APPROXIMATION

Abstract :

Whenever you see a string of quadrature nodes, you can consider it as a branch cut defined by the poles of a rational approximation to the Cauchy transform of a weight function. The aim of this talk is to explain this strange statement and show how it opens the way to calculation of targeted quadrature formulas for all kinds of applications. Gauss quadrature is an example, but it is just the starting point, and many more examples will be shown. I hope this talk will change your understanding of quadrature formulas. This is joint work with Andrew Horning.

Nick Trefethen - Bio

Nick Trefethen

_{Nick Trefethen is Professor of Applied Mathematics in Residence at Harvard University; during 1997-2023 he was Professor of Numerical Analysis and head of the Numerical Analysis Group at the University of Oxford.}_{He was educated at Harvard and Stanford and held positions at NYU, MIT, and Cornell in earlier years.}_{He is a Fellow of the Royal Society, a member of the US National Academies of Sciences and Engineering, and a foreign member of the Chinese Academy of Sciences and served during 2011-2012 as President of SIAM.}

_{He has won many prizes including the Gold Medal of the Institute for Mathematics and its Applications, the Naylor Prize of the London Mathematical Society, the Polya and von Neumann Prizes from SIAM, and the Gibbs Lectureship from the American Mathematical Society. He holds honorary doctorates from the University of Fribourg and Stellenbosch University.}

_{As an author Trefethen is known for his books including Numerical Linear Algebra (1997), Spectral Methods in MATLAB (2000), Spectra and Pseudospectra (2005), Approximation Theory and Approximation Practice (2013/2019), Exploring ODEs (2018), and An Applied Mathematician's Apology (2022).}_{He organized the SIAM 100-Dollar, 100-Digit Challenge in 2002 and is the inventor of Chebfun and co-inventor of the AAA algorithm for rational approximation. Trefethen' s research interests since 2017 have been focused on algorithms for computing with rational functions.}