# Spectral Sets: Numerical Range and Beyond

## TL;DR

This paper extends the concept of spectral sets to various regions in the complex plane, providing bounds useful for analyzing convergence rates of iterative methods like GMRES and rational Krylov subspace methods.

## Contribution

It generalizes the spectral set results to annular and convex regions with holes, offering new bounds for convergence analysis of iterative algorithms.

## Key findings

- Annular regions are $(1 + \sqrt{2})$-spectral sets.
- Convex regions with holes are $(3 + 2 \sqrt{3})$-spectral sets.
- Results improve bounds on GMRES and rational Krylov methods.

## Abstract

We extend the proof in [M.~Crouzeix and C.~Palencia, {\em The numerical range is a $(1 + \sqrt{2})$-spectral set}, SIAM Jour.~Matrix Anal.~Appl., 38 (2017), pp.~649-655] to show that other regions in the complex plane are $K$-spectral sets. In particular, we show that various annular regions are $(1 + \sqrt{2} )$-spectral sets and that a more general convex region with a circular hole or cutout is a $(3 + 2 \sqrt{3} )$-spectral set. We demonstrate how these results can be used to give bounds on the convergence rate of the GMRES algorithm for solving linear systems and on that of rational Krylov subspace methods for approximating $f(A)b$, where $A$ is a square matrix, $b$ is a given vector, and $f$ is a function that can be uniformly approximated on such a region by rational functions with poles outside the region.