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We propose a new approach to computing global minimizers of singular valuefunctions in two real variables. Specifically, we present new algorithms tocompute the Kreiss constant of a matrix and the distance to uncontrollabilityof a linear control system, both to arbitrary accuracy. Previousstate-of-the-art methods for these two quantities rely on 2D level-set teststhat are based on solving large eigenvalue problems. Consequently, thesemethods are costly, i.e., $\mathcal{O}(n^6)$ work using dense eigensolvers, andoften multiple tests are needed before convergence. Divide-and-conquertechniques have been proposed that reduce the work complexity to$\mathcal{O}(n^4)$ on average and $\mathcal{O}(n^5)$ in the worst case, butthese variants are nevertheless still very expensive and can be numericallyunreliable. In contrast, our new interpolation-based globality certificatesperform level-set tests by building interpolant approximations to certainone-variable continuous functions that are both relatively cheap andnumerically robust to evaluate. Our new approach has a $\mathcal{O}(kn^3)$ workcomplexity and uses $\mathcal{O}(n^2)$ memory, where $k$ is the number offunction evaluations necessary to build the interpolants. Not only is thisinterpolation process mostly "embarrassingly parallel," but also low-fidelityapproximations typically suffice for all but the final interpolant, which mustbe built to high accuracy. Even without taking advantage of the aforementionedparallelism, $k$ is sufficiently small that our new approach is generallyorders of magnitude faster than the previous state-of-the-art.

Fast Interpolation-based Globality Certificates for Computing Kreiss Constants and the Distance to Uncontrollability