21.5: Computation of {mu}

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In general, there is no closed-form method for computing \(\mu\). Upper and lower bounds may be computed and refined, however. In these notes we will only be concerned with computing the upper bound. If \(\Delta_{0}=\operatorname{diag}\left(\Delta_{1}, \ldots, \Delta_{n}\right)\), then the upper bound on \(\mu\) is something that is easy to calculate. Furthermore, property 6 above suggests that by infimizing \(\sigma_{\max }\left(D^{-1} M D\right)\) over all possible diagonal scaling matrices, we obtain a better approximation of \(\mu\). This turns out to be a convex optimization problem at each frequency, so that by infimizing over \(\mathcal{D}\) at each frequency, the tightest upper bound over the set of \(\mathcal{D}\) may be found for \(\mu\).

We may then ask when (if ever) this bound is tight. In other words, when is it truly a least upper bound. The answer is that for three or fewer \(\Delta\)'s, the bound is tight. The proof of this is involved, and is beyond the scope of this class. Unfortunately, for four or more perturbations, the bound is not tight, and there is no known method for computing \(\mu\) exactly for more than three perturbations.