Convergence acceleration as a dynamical system

We survey recent results on local and global aspects of generalized Steffensen iteration. The main idea behind this algorithm is to replace the dynamical system z ↦ f( z) with z ↦ F n ( z), where F n is an appropri ately constructed quotient of two Hankel determinants. We show that F n retains all f...

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Bibliographic Details
Published inApplied numerical mathematics Vol. 15; no. 2; pp. 101 - 121
Main Author Iserles, A.
Format Journal Article Conference Proceeding
LanguageEnglish
Published Amsterdam Elsevier B.V 01.09.1994
Elsevier
Subjects
Acceleration of convergence
Exact sciences and technology
Global analysis, analysis on manifolds
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Sciences and techniques of general use
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Complex function
Fix point
Dynamical system
Convergence acceleration
Attractor
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Summary:We survey recent results on local and global aspects of generalized Steffensen iteration. The main idea behind this algorithm is to replace the dynamical system z ↦ f( z) with z ↦ F n ( z), where F n is an appropri ately constructed quotient of two Hankel determinants. We show that F n retains all finite fixed points of f and determine the local speedup in convergence. Moreover, we investigate how the basin of attraction varies with n, proving that for polynomial f most poles and zeros of F n accumulate on and inside the Julia set of f as n → ∞. This is in close agreement with computational results.
ISSN:0168-9274
1873-5460
DOI:10.1016/0168-9274(94)00020-4