  We present quadratically convergent algorithms to compute the extremal value
of a real parameter for which a given rational transfer function of a linear
time-invariant system is passive. This problem is formulated for both
continuous-time and discrete-time systems and is linked to the problem of
finding a realization of a rational transfer function such that its passivity
radius is maximized. Our new methods make use of the Hybrid
Expansion-Contraction algorithm, which we extend and generalize to the setting
of what we call root-max problems.
