Spectral Sets: Numerical Range and Beyond
For numerical linear algebra practitioners, this work provides theoretical convergence bounds for iterative methods on matrices with spectra in specific non-convex regions.
The paper extends the proof that the numerical range is a (1+√2)-spectral set to other regions, showing that annular regions are (1+√2)-spectral sets and convex regions with a circular hole are (3+2√3)-spectral sets. These results provide bounds on the convergence rate of GMRES and rational Krylov subspace methods.
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.