Fast Evaluation of Zolotarev Coefficients

We review the theory of elliptic functions leading to Zolotarev’s formula for the sign function over the range e≤|x|≤1. We show how Gauss’ arithmetico-geometric mean allows us to evaluate elliptic functions cheaply, and thus to compute Zolotarev coefficients “on the fly” as a function of e. This in turn allows us to calculate the matrix functions sgnH, \(\sqrt H \), and \(1/\sqrt H \) both quickly and accurately for any Hermitian matrix H whose spectrum lies in the specified range.