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In this paper, we extend the original criss-cross algorithms for computing the $varepsilon$-pseudospectral abscissa and radius to general spectral value sets. Furthermore, we greatly reduce the total number of expensive Hamiltonian eigenvalue decompositions needed by performing the horizontal/radial search subphases with a new root-finding-based approach that is both more numerically resilient and substantially faster, particularly for typical spectral value sets. Finally, we propose a new, more robust way of handling singular pencils that can arise when computing the $varepsilon$-spectral value set radius. Compared to would-be direct extensions of the original criss-cross algorithms, that is, without the changes we propose here, our modified methods are more robust and often several times faster, improvements that can also be relevant for the special case of pseudospectra.

Extended and Improved Criss-Cross Algorithms for Computing the Spectral Value Set Abscissa and Radius