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Nondiscrete induction and iterative processes
01 Jan 1984-
About: The article was published on 1984-01-01 and is currently open access. It has received 374 citations till now.
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TL;DR: In this paper, the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation is presented, and the extension to general Banach spaces and some computational aspects of these methods are discussed.
284 citations
TL;DR: This paper introduces the concept of strong approximation of functions, and a related concept ofstrong Bouligand (B-) derivative, and employs these ideas to prove an implicit-function theorem for nonsmooth functions, applicable to a considerably wider class of functions than is the classical theorem.
Abstract: In this paper we introduce the concept of strong approximation of functions, and a related concept of strong Bouligand (B-) derivative, and we employ these ideas to prove an implicit-function theorem for nonsmooth functions. This theorem provides the same kinds of information as does the classical implicit-function theorem, but with the classical hypothesis of strong Frechet differentiability replaced by strong approximation, and with Lipschitz continuity replacing Frechet differentiability of the implicit function. Therefore it is applicable to a considerably wider class of functions than is the classical theorem. In the last part of the paper we apply this implicit function result to analyze local solvability and stability of perturbed generalized equations.
239 citations
TL;DR: A system of a priori error bounds for the Chebyshev method in Banach spaces is obtained through a local convergence theorem that provides sufficient conditions on the initial point in order to ensure the convergence of ChebysHEv iterates.
Abstract: We continue the analysis of rational cubic methods, initiated in [7]. In this paper, we obtain a system of a priori error bounds for the Chebyshev method in Banach spaces through a local convergence theorem that provides sufficient conditions on the initial point in order to ensure the convergence of Chebyshev iterates. The error estimates are exact for second degree polynomials. We also discuss some applications.
211 citations
TL;DR: The theorem supplies sufficient conditions on the initial point to ensure the convergence of Halley iterates, by means of a system of “recurrence relations” analogous to those given for the Newton method by Kantorovich, improving previous results by Döring.
Abstract: In this paper we present a system of a priori error bounds for the Halley method in Banach spaces. Our theorem supplies sufficient conditions on the initial point to ensure the convergence of Halley iterates, by means of a system of “recurrence relations”, analogous to those given for the Newton method by Kantorovich, improving previous results by Doring [4]. The error bounds presented are optimal for second degree polynomials. Other rational cubic methods, as the Chebyshev method, will be treated in a subsequent paper.
186 citations
01 Jan 2004
TL;DR: In this article, the root-finding iterative methods of Chebyshev, Stirling, and Steffensen were studied from a dynamical point of view applied to complex polynomials.
Abstract: From a dynamical point of view applied to complex polynomials, we study a number of root–finding iterative methods. We consider Newton’s method, Newton’s method for multiple roots, Jarratt’s method, the super–Halley method, the convex as well as the double convex acceleration of Whittaker’s method, the methods of Chebyshev, Stirling, and Steffensen, among others. Since all of the iterative root–finding methods we study satisfy the Scaling Theorem, except for Stirling’s method and that of Steffensen, we obtain their conjugacy classes.
168 citations
Cites methods from "Nondiscrete induction and iterative..."
...These three third–order root–finding iterative methods are studied in [25], [41], [45], [59]....
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