Abstract
LetG be a compact set in ℝd d≥1,M=G×G andϕ:M →G a map inC 3(M). Suppose thatϕ has a fixed pointξ, i.e.ϕ(ξ, ξ)=ξ. We investigate the rate of convergence of the iterationx n+2=φ(x n+1,x n) withx 0,x 1∈G andx n→ξ. Iff n=Q‖x n−ξ‖ with a suitable norm and a constantQ depending onξ, an exact representation forf n is derived. The error terms satisfyf 2m+1≍(ƒ2m)γ,f 2m+2≍(ƒ2m+1)2γ,m≥0, with 1<gg<2, andγ=γ(x 1,x 0). According to our main result we have limn→∞{‖x n+2‖/(‖x n‖)2}=Q, 0<Q<∞.
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This paper constitutes an extension of a part of the author’s doctoral thesis realized under the direction of Prof. E. Wirsing and Prof. A. Peyerimhoff, University of Ulm (Germany).
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Rocha, L. On the rate of convergence of 2-term recursions in ℝd . Computing 59, 187–207 (1997). https://doi.org/10.1007/BF02684440
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DOI: https://doi.org/10.1007/BF02684440