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The main two algorithms for computing the numerical radius are the level-setmethod of Mengi and Overton and the cutting-plane method of Uhlig. Via newanalyses, we explain why the cutting-plane approach is sometimes much faster ormuch slower than the level-set one and then propose a new hybrid algorithm thatremains efficient in all cases. For matrices whose fields of values are acircular disk centered at the origin, we show that the cost of Uhlig's methodblows up with respect to the desired relative accuracy. More generally, we alsoanalyze the local behavior of Uhlig's cutting procedure at outermost points inthe field of values, showing that it often has a fast Q-linear rate ofconvergence and is Q-superlinear at corners. Finally, we identify and addressinefficiencies in both the level-set and cutting-plane approaches and proposerefined versions of these techniques.

Convergence rate analysis and improved iterations for numerical radius computation