# Numerical Analysis and Scientific Computing Seminar

## Algorithms Based on Analytic Function Values at Roots of Unity

**Speaker:**
Nick Trefethen, Oxford University and CIMS

**Location:**
Warren Weaver Hall 1302

**Date:**
Sept. 20, 2013, 10 a.m.

**Synopsis:**

Let $f(z)$ be an analytic or meromorphic function in the closed unit disk sampled at the $n$th roots of unity. Based on these data, how can we approximately evaluate $f(z)$ or $f^{(m)}(z)$ at a point $z$ in the disk? How can we calculate the zeros or poles of $f$ in the disk? These questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations, partial differential equations, and large-scale linear algebra (e.g. the FEAST eigenvalue code). We survey what is already known and suggest new possibilities springing from a connection between Cauchy integrals and rational interpolation. This is joint work with Anthony Austin and Peter Kravanja.