Asymptotic analysis of a generalized Richardson extrapolation process on linear sequences

Sidi, Avram

doi:10.1090/s0025-5718-09-02318-7

Cited by 5 publications

(3 citation statements)

References 23 publications

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“…As will become clear, the analysis turns out to be quite involved and nontrivial, and the main results are interesting and nonintuitive. Our results here are analogous to, yet different from, those of Sidi [36] that pertain to the application of the author's generalization of the Richardson extrapolation process GREP (m) to general linear sequences with m > 1. For GREP (m) , see [34,Chapter 4], for example.…”

Section: Introductioncontrasting

confidence: 50%

“…Comparing Theorems 3.1 and 3.2 of this paper with Theorems 4.1 and 4.2 in [36], we see that they are very similar. Consequently, the implications of both sets of results are the same.…”

Section: Implications Of Theorems 31 and 32mentioning

confidence: 64%

“…Consequently, the implications of both sets of results are the same. These implications are already discussed in [36]. For completeness, we discuss briefly these implications as they are related to the Shanks transformation.…”

Section: Implications Of Theorems 31 and 32mentioning

confidence: 73%

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Acceleration of convergence of general linear sequences by the Shanks transformation

Sidi

2011

Numer. Math.

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The Shanks transformation is a powerful nonlinear extrapolation method that is used to accelerate the convergence of slowly converging, and even diverging, sequences {A n }. It generates a two-dimensional array of approximations A ( j) n to the limit or anti-limit of {A n } defined as solutions of the linear systemswhereβ k are additional unknowns. In this work, we study the convergence and stability properties of A ( j) n , as j → ∞ with n fixed, derived from general linear sequencesi=0 β ki n γ k −i as n → ∞, where ζ k = 1 are distinct and |ζ 1 | = · · · = |ζ m | = θ, and γ k = 0, 1, 2, . . .. Here A is the limit or the anti-limit of {A n }. Such sequences arise, for example, as partial sums of Fourier series of functions that have finite jump discontinuities and/or algebraic branch singularities. We show that definitive results are obtained with those values of n for which the integer programming problems max s 1 ,...,s m m k=1 ( γ k )s k − s k (s k − 1) , subject to s 1 ≥ 0, . . . , s m ≥ 0 and m k=1 s k = n, 123 726 A. Sidi have unique (integer) solutions for s 1 , . . . , s m . A special case of our convergence result concerns the situation in which γ 1 = · · · = γ m = α and n = mν with ν = 1, 2, . . . , for which the integer programming problems above have unique solutions, and it reads A

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Section: Introductioncontrasting

confidence: 50%

“…Comparing Theorems 3.1 and 3.2 of this paper with Theorems 4.1 and 4.2 in [36], we see that they are very similar. Consequently, the implications of both sets of results are the same.…”

Section: Implications Of Theorems 31 and 32mentioning

confidence: 64%