Near-circularity for the rational Zolotarev problem in the complex plane

Abstract

We consider the rational Zolotarev problem min rϵR H max zϵE|r(Z)| minzϵF|r(Z)| for compact sets E, F⊈ C, where R ll denotes the set of all rational functions of degree ⩽ l. This problem is of importance, e.g., for the determination of optimal parameters for the method of alternating directions (ADI method) which is used for the iterative solution of large linear systems. For E and F being real intervals, the solution of this problem was given explicitly in terms of elliptic functions by Zolotarev in the last century. For complex domains, however, little is known as yet about this problem. In this paper, after reviewing some results on the asymptotic behavior, we prove a result which is similar to the near-circularity criterion as it is well known in connection to classical approximation by polynomials or rational functions. If we assume that both sets E and F are bounded by Jordan curves, this gives us a lower bound for the minimal value in the rational Zolotarev problem. Moreover, we derive upper bounds for the modulus of the doubly connected region D: C ̂ \\(E∪:F) and show how the near-circularity criterion can be used for the construction of the rational minimal solutions for small degrees.