Extended and improved criss-cross algorithms for computing the spectral value set abscissa and radius

Abstract

In this paper, we extend the original criss-cross algorithms for computing the $\varepsilon$-pseudospectral abscissa and radius to general spectral value sets. By proposing new root-finding-based strategies for the horizontal/radial search subphases, we significantly reduce the number of expensive Hamiltonian eigenvalue decompositions incurred, which typically translates to meaningful speedups in overall computation times. Furthermore, and partly necessitated by our root-finding approach, we develop a new way of handling the singular pencils or problematic interior searches that can arise when computing the $\varepsilon$-spectral value set radius. Compared to would-be direct extensions of the original algorithms, that is, without our additional modifications, our improved criss-cross algorithms are not only noticeably faster but also more robust and numerically accurate, for both spectral value set and pseudospectral problems.