Optimal rational functions for the generalized Zolotarev problem in the complex plane

It has been long recognized that the determination of optimal parameters fo r the classical alternating direction implicit (ADI) method leads to the Zo lotarev problem [GRAPHICS] for disjoint compact sets E, F subset of or equal to C, where R-nn is the c ollection of rational functions of order n. In the case where E and F are r eal 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 R-mn with unequal numerator degree m and denominator degree n. The object o f the paper is to investigate the generalized Zolotarev problem in the comp lex 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. Exp licit expressions are also provided for special values of data.