ON THE 3RD AND 4TH ZOLOTAREV PROBLEMS IN THE COMPLEX-PLANE

In relation to the determination of optimal parameters for the alternating direction implicit (ADI) method, the authors consider the third Zolotarev problem in the complex plane, which has been the topic of several recent contributions. In the real case, Zolotarev also posed a fourth problem that involves best rational approximants in the Chebyshev sense, and Achieser proved later that the third and fourth Zolotarev problems are actually equivalent. By generalizing Achieser's argument, it is shown that the complex third Zolotarev problem can be stated as a best Chebyshev approximation by complex rational functions. This enables the authors to use known optimality and uniqueness conditions as well as an iterative algorithm for computing the exact solutions. Some numerical results are reported for rectangular domains of the complex plane that are of practical importance in the ADI method.