Most algorithms for computing a matrix function $f(A)$ are based on finding a rational (or polynomial) approximant $r(A)≈f(A)$ to the scalar function on the spectrum of $A$. These functions are often in a composite form, that is, $f(z)≈r(z)=r_k(⋯r_2(r_1(z)))$ (where $k$ is the number of compositions, which is often the iteration count, and proportional to the computational cost); this way $r$ is a rational function whose degree grows exponentially in $k$. I will review algorithms that fall into this category and highlight the remarkable power of composite (rational) functions.

Rational functions are able to approximate functions containing branch point singularities with a root-exponential convergence rate. These appear for example in the solution of boundary value problems on domains containing corners or edges. Results from Newman in 1964 indicate that the poles of the optimal rational approximant are exponentially clustered near the branch point singularities. Trefethen and collaborators use this knowledge to linearize the approximation problem by fixing the poles in advance, giving rise to the Lightning approximation. The resulting approximation set is however highly ill-conditioned, which raises the question of stability. We show that augmenting the approximation set with polynomial terms greatly improves stability. This observation leads to a  decoupling of the approximation problem into two regimes, related to the singular and the smooth behaviour of the function. In addition, adding polynomial terms to the approximation set can result in a significant increase in convergence speed. The convergence rate is however very sensitive to the speed at which the clustered poles approach the singularity.

An occupational hazard of mathematicians is the investigation of objects that are "optimal" in a mathematically precise sense, yet may be far from optimal in practice. This talk will discuss an extreme example of this effect: Gauss-Hermite quadrature on the real line. For large numbers of quadrature points, Gauss-Hermite quadrature is a very poor method of integration, much less efficient than simply truncating the interval and applying Gauss-Legendre quadrature or the periodic trapezoidal rule. We will present a theorem quantifying this difference and explain where the standard notion of optimality has failed.

TBA

For the first time, a method has become available for fast computation of near-best rational approximations on arbitrary sets in the real line or complex plane: the AAA algorithm (Nakatsukasa-Sète-T. 2018).  After a brief presentation of the algorithm this talk will focus on twenty demonstrations of the kinds of things we can do, all across applied mathematics, with a black-box rational approximation tool.