Faster and more accurate computation of the $\mathcal{H}_\infty$ norm via optimization

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.