Computing iterative roots of polygonal functions

Zhang, Wanxiong; Zhang, Weinian

doi:10.1016/j.cam.2006.05.026

Cited by 17 publications

(7 citation statements)

References 7 publications

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“…The Hyers-Ulam stability of continuous roots, provided by Theorem 3.1, cannot fully support the algorithm given in [70] for the following two reasons:…”

Section: Stability Problems For Iterative Roots Approximate Rootsmentioning

confidence: 99%

“…An algorithm for computing square roots of increasing polygonal mappings was given by J. Kobza in [16]. Then it was generalized by Wanxiong Zhang and Weinian Zhang [70] to not only increasing polygons, but also decreasing polygons and some non-monotonic ones, and to roots of an arbitrary order.…”

Section: Stability Problems For Iterative Roots Approximate Rootsmentioning

confidence: 99%

“…it is hard to construct a polygonal solution, following the algorithm presented in [70], satisfying ϑ < 1, where ϑ = n−1 i=1 c i as in Theorem 3.1; the assumption ϑ < 1 obstructs obtaining iterative roots with more than one xed point.…”

Section: Stability Problems For Iterative Roots Approximate Rootsmentioning

confidence: 99%

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Recent results on iterative roots

Jarczyk

2014

ESAIM: Proc.

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Abstract. This is a rough survey of some results on iterative roots (fractional iterates) published recently. Also some historical information to clear the connection to previous results has been given.The main topics are: the conjugacy of piecewise monotonic functions and their iterative roots, stability of iterative roots, some substitutes and generalizations of the notion of iterative root using set-valued functions.Résumé. Le présent article de revue approximative contient de certains résultats sur les racines itératives (itérées fractionnaires) publiés récemment. Une information historique est aussi donnée pour montrer la liason avec des résultats précédents. Les sujets principaux sont: la conjugaison des fonctions monotones par morceaux et leurs racines itératives, stabilité des racines itératives, certains substituts et généralizations de la notion de la racine itérative qui utilisent les fonctions multivaluées.

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“…The Hyers-Ulam stability of continuous roots, provided by Theorem 3.1, cannot fully support the algorithm given in [70] for the following two reasons:…”

Section: Stability Problems For Iterative Roots Approximate Rootsmentioning

confidence: 99%

Section: Stability Problems For Iterative Roots Approximate Rootsmentioning

confidence: 99%

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Recent results on iterative roots

Jarczyk

2014

ESAIM: Proc.

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“…[62]). In [181] Zhang and Zhang gave an algorithm to compute piecewise linear iterative roots of a piecewise linear monotonic mapping with finitely many forts.…”

Section: Theorem 24 ([14]) Let F Be a Strict Horseshoe Map Of Type (mentioning

confidence: 99%

“…This method allows us to find approximate roots. In [181] an algorithm determining the iterative roots of the polygonal functions was given.…”

Section: Stability Of Iterative Rootsmentioning

confidence: 99%

Recent results on iteration theory: iteration groups and semigroups in the real case

Zdun

Solarz

2013

Aequat. Math.

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Abstract. In this survey paper we present some recent results in iteration theory. Mainly, we focus on the problems concerning real iteration groups (flows) and semigroups (semiflows) such as existence, regularity and embeddability. We also discuss some issues associated to the problem of embeddability, i.e. iterative roots and approximate iterability. The topics of this paper are: (1) measurable iteration semigroups; (2) embedding of diffeomorphisms in regular iteration semigroups in the space R n ; (3) iteration groups of fixed point free homeomorphisms on the plane; (4) embedding of interval homeomorphisms with two fixed points in a regular iteration group; (5) commuting functions and embeddability; (6) iterative roots; (7) the structure of iteration groups on an interval; (8) iteration groups of homeomorphisms of the circle; (9) approximately iterable functions; (10) set-valued iteration semigroups; (11) iterations of mean-type mappings; (12) Hayers-Ulam stability of the translation equation. Most of the results presented here were obtained by means of functional equations. We indicate the relations between iteration theory and functional equations.

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