Estimates for the asymptotic convergence factor of two intervals

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 κ ( E ) ≔ lim n → ∞ L n...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2011-08, Vol.236 (1), p.28-38
Main Author: Schiefermayr, Klaus
Format: Article
Language:English
Subjects:
Algebra
Algebraic geometry
Asymptotic properties
Convergence
Estimated asymptotic convergence factor
Exact sciences and technology
Inequality
Intervals
Jacobian elliptic functions
Jacobian theta functions
Mathematical analysis
Mathematical models
Mathematics
Norms
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Representations
Sciences and techniques of general use
Sequences, series, summability
Two intervals
Unions
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Summary: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 κ ( E ) ≔ lim n → ∞ L n r ( 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 .
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2010.06.008