Estimates for the asymptotic convergence factor of two intervals

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Let E be the union of two real intervals not containing zero. Then Lnr(E) denotes the supremum norm of that polynomial Pn of degree less than or equal to n, which is minimal with respect to the supremum norm provided that Pn(0)=1. It is well known that the limit κ(E):=lim n→∞Lnr(E)n exists, where 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) 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) in terms of elementary functions of the endpoints of E.

Original languageEnglish
Pages (from-to)28-38
Number of pages11
JournalJournal of Computational and Applied Mathematics
Issue number1
Publication statusPublished - 1 Aug 2011


  • Estimated asymptotic convergence factor
  • Inequality
  • Jacobian elliptic functions
  • Jacobian theta functions
  • Two intervals


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