Optimum parameters for the generalized ADI method (Q1272812)

The generalized alternating direction implicit (ADI) method leads to the following rational approximation problem. Let \(E\), \(F\) be two disjoint real intervals and \((m,n)\) be a pair of nonnegative integers. We look for the minimum \[ \sigma_{(m,n)}(E, F)= \min_{r\in R_{m,n}} {\max\{| r(z)|: z\in E\}\over \min\{| r(z)|: z\in F\}}, \] where \(R_{m,n}\) is the set of rational functions with numerator degree \(m\) and denominator degree \(n\). In the case \(m= n\) this problem was posed and solved by Zolotarev in the nineteenth century. However, recent developments led the authors to consider the case \(m\neq n\) as well. They present an explicit formula for choosing the pair \((m,n)\) in an optimal way if \(m+n\) is given. Also, they provide a characterization for the optimal rational function and an algorithm to compute it. The method is supported by extensive numerical computations.