Estimates for the asymptotic convergence factor of two intervals

TL;DR

This paper provides simple bounds for the asymptotic convergence factor of polynomial approximation on two intervals, which is important for iterative methods in large-scale matrix computations.

Contribution

It offers explicit elementary bounds for the asymptotic convergence factor expressed through the endpoints of the intervals, simplifying previous complex representations.

Findings

Derived precise upper bounds for (E)

Derived precise lower bounds for (E)

Bounds are expressed using elementary functions of interval endpoints

Abstract

Let $E$ be the union of two real intervals not containing zero. Then $L_n^r(E)$ denotes the supremum norm of that polynomial $P_n$ of degree less than or equal to $n$, which is minimal with respect to the supremum norm provided that $P_n(0)=1$. It is well known that the limit $\kappa(E):=\lim_{n\to\infty}\sqrt[n]{L_n^r(E)}$ exists, where $\kappa(E)$ is called the asymptotic convergence factor, since it plays a crucial role for certain iterative methods solving large-scale matrix problems. The factor $\kappa(E)$ 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 $\kappa(E)$ in terms of elementary functions of the endpoints of $E$.