On computing best Chebyshev complex rational approximants

An Algorithm for Real and Complex Rational Minimax Approximation

“…Nevertheless, the approximations in such cases often converge quickly, and we have observed that they converge much faster still if (3. As has been mentioned, Ellacott--Williams (EW) and Istace--Thiran (IT) algorithms for complex rational minimax approximation have been published previously by Ellacott and Williams [22] and Istace and Thiran [40], building on earlier work by Osborne and Watson [60]. In this section, we comment on the relationship of these to the AAA-Lawson algorithm.…”

“…For complex functions or domains, however, the situation is very different. The theory developed by Walsh in the 1930s shows that existence and especially uniqueness may fail [86,87], and as for algorithms, there is not much available apart from a pair of methods introduced by Ellacott and Williams (EW) (1976) and Istace and Thiran (IT) (1993) based on earlier work by Osborne and Watson [60], which, as far as we are aware, are not in use today [22,40] (see section 7). This is a striking gap, since rational approximations are of growing importance in computational complex analysis (Figure 4.3) [29,83], systems theory and model order reduction (Figure 4.6) [2,3,7,12], low-rank data compression (Figures 4.6 and 4.7) [6,46], electronic structure calculation (Figure 5.2) [48,53], and solution of partial differential equations (Figure 5.4) [15,20,30,31].…”

“…When it comes to characterization of rational best approximants, the Kolmogorov condition diminishes to a necessary condition for local optimality: for a candidate approximation to be locally optimal, it must be a minimal point with respect to certain local linear perturbations. Discussions can be found in a number of sources, including [34,65,71,92], and we recommend in particular the papers [40,76] by Istace and Thiran. Sufficient conditions, and conditions for global optimality, are mostly not available, though there are some results in [51,65].…”

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

“…For that purpose, we make use of the algorithm presented in [10] (see [20] for a more complete description) in the case of E = −F . As this symmetry property no longer holds for the sets defined in Table 1, the number of parameters is more than doubled in the problem under investigation.…”

n-Widths, Faber Expansion, and Computation of Analytic Functions

“…It should be noted also that values of some analytic functions can be effectively computed by Faber rational approximants of Padé and Carathé odory-Fejé r type (see, e.g., Ellacott and Gutknecht, 1983;Istace and Thiran, 1993). Besides the Faber polynomials have some interesting applications in numerical linear algebra (e.g., Eiermann and Starke, 1990;Farkova, 1992;Starke and Varga, 1993).…”