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