Quadrature By Rational Approximation

Many numerical algorithms rely on quadrature formulas such as Gauss quadrature, the trapezoidal rule, and their conformal transplantations to specialized domains. Each quadrature formula can be interpreted as a rational approximation to an analytic function with a branch cut. Reversing the logic, new quadrature formulas can be quickly derived even for specialized domains by numerical rational approximation via the AAA algorithm, avoiding the need for conformal maps or other analysis. The poles of the rational approximations delineate branch cuts, and the poles and residues are the quadrature nodes and weights. The talk will present ten examples: five known problems, plus a variant for each one. I hope it will change your understanding of quadrature formulas.

• Speaker: Nick Trefethen (Oxford and Harvard)
• Thursday 19 June 2025, 15:00–16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Matthew Colbrook.