  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.
