Optimum parameters for the generalized ADI method

Summary.

In the context of the generalized ADI method, we are concerned with the problem of finding in the set of rational functions r with numerator degree m and denominator degree n an element $r^*$ that minimizes \[ \max_{z\in E} | r(z)| / \min_{z\in F} |r(z)| \; , \] where E,F are disjoint real intervals. By extending a recent analysis by Levin and Saff, we present an explicit formula for choosing the pair (m,n) for given m +n. Furthermore, we provide a characterization of $r^*$ and a Remes type algorithm for its determination. Extensive numerical computations furnish some comparison of $r^*$ with asymptotically optimal solutions based on Fejér-Walsh and Leja-Bagby points.