Computation of Zolotarev rational functions

An algorithm is presented to compute Zolotarev rational functions, that is, rational functions $r_n^{\ast }$ of a given degree that are as small as possible on one set $E\subseteq \text{complex}\cup \{\mathrm{\infty }\}$ relative to their size on another set $F\subseteq \text{complex}\cup \{\mathrm{\infty }\}$ (the third Zolotarev problem). Along the way we also approximate the sign function relative to $E$ and $F$ (the fourth Zolotarev problem).