Estimates for the asymptotic convergence factor of two intervals

Let EE be the union of two real intervals not containing zero. Then Lnr(E) denotes the supremum norm of that polynomial PnPn of degree less than or equal to nn, which is minimal with respect to the supremum norm provided that Pn(0)=1Pn(0)=1. It is well known that the limit κ(E)≔limn→∞Lnr(E)n exists, where κ(E)κ(E) is called the asymptotic convergence factor, since it plays a crucial role for certain iterative methods solving large-scale matrix problems. The factor κ(E)κ(E) can be expressed with the help of Jacobi’s elliptic and theta functions, where this representation is very involved. In this paper, we give precise upper and lower bounds for κ(E)κ(E) in terms of elementary functions of the endpoints of EE.