Optimal Rational Functions for the Generalized Zolotarev Problem in the Complex Plane

Abstract

It has been long recognized that the determination of optimal parameters for the classical alternating direction implicit (ADI) method leads to the Zolotarev problem \[ \min_{r \in R_{nn}} \frac{\max \{|r(z)|, z \in E \}}{\min \{|r(z)|, z \in F\}} \] for disjoint compact sets E, F \subseteq \mathbb{C}$, where Rnn isthe collection of rational functions of order n. In the case where E and F are real intervals, it was more recently pointed out that if they have different lengths, it is of interest to generalize the foregoing problem to the set Rmn with unequal numerator degree m and denominator degree n. The object of the paper is to investigate the generalized Zolotarev problem in the complex plane. A method is proposed to construct the optimal rational function, which is then applied to the particular example of two line segments---E on the real axis, F parallel to the imaginary axis. Numerical experiments show that the improvement on the classical solution may be amazingly great. Explicit expressions are also provided for special values of data.