  In this paper, we propose an improved method for computing the
$\mathcal{H}_\infty$ norm of linear dynamical systems that results in a code
that is often several times faster than existing methods. By using standard
optimization tools to rebalance the work load of the standard algorithm due to
Boyd, Balakrishnan, Bruinsma, and Steinbuch, we aim to minimize the number of
expensive eigenvalue computations that must be performed. Unlike the standard
algorithm, our modified approach can also calculate the $\mathcal{H}_\infty$
norm to full precision with little extra work, and also offers more opportunity
to further accelerate its performance via parallelization. Finally, we
demonstrate that the local optimization we have employed to speed up the
standard globally-convergent algorithm can also be an effective strategy on its
own for approximating the $\mathcal{H}_\infty$ norm of large-scale systems.
