An adaptive continuation process for solving systems of nonlinear equations

Abstract. We present a local convergence analysis of an eighthorder method for approximating a locally unique solution of a nonlinear equation. Earlier studies such as have shown convergence of these methods under hypotheses up to the seventh derivative of the function although only the first derivative appears in the method. In this study, we expand the applicability of these methods using only hypotheses up to the first derivative of the function. This way the applicability of these methods is extended under weaker hypotheses. Moreover, the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

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“…As already noted in [7],r 1 is at least as large as the convergence ball given by Rheinboldt [22] r R = 2 3L 1 .…”

Section: )mentioning

confidence: 61%

“…It follows from (2.22) and the Banach lemma on invertible functions [7], [10], [21], [22], [24] that f (x 0 ) = 0 and…”

Section: )mentioning

confidence: 99%

On the Local Convergence of an Eighth-order Method for Solving Nonlinear Equations

Argyros

Kansal

Kanwar

2016

Annals of West University of Timisoara - Mathematics and Computer Science

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“…However "... effective, computable estimates for convergence radii are rarely available" [1]. Similar remarks were made in more recent papers: "... no estimate for the size of an attraction ball is known" [2], "The location of starting approximations, from which the iterative methods converge to a solution of the equation, is a difficult problem to solve" [3].…”

Section: Introductionmentioning

confidence: 78%

Sharp estimation of local convergence radius for the Picard iteration

Maruster

Măruşter

2018

J. Fixed Point Theory Appl.

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Abstract. We investigate the local convergence radius of a general Picard iteration in the frame of a real Hilbert space. We propose a new algorithm to estimate the local convergence radius. Numerical experiments show that the proposed procedure gives sharp estimation (i.e., close to or even identical with the best one) for several well known or recent iterative methods and for various nonlinear mappings. Particularly, we applied the proposed algorithm for classical Newton method, for multistep Newton method (in particular for third-order Potra-Ptak method) and for fifth-order "M5" method. We present also a new formula to estimate the local convergence radius for multi-step Newton method.Mathematics Subject Classification. Primary 26A18, 47J25, 65J15, Secondary 34K28.

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“…Let R m×n denote the set of all m × n matrices A. We denote by A † the Moore-Penrose inverse of matrix A [13,14]. If A has full rank, i.e., if rank(A) = min{m, n} = n, then A † = (A T A) −1 A T .…”

Section: Auxiliary Resultsmentioning

confidence: 99%

Extended Local Analysis of Inexact Gauss-Newton-like Method for Least Square Problems using Restricted Convergence Domains

Argyros

George²

2016

Annals of West University of Timisoara - Mathematics and Computer Science

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Abstract. We present a local convergence analysis of inexact Gauss-Newton-like method (IGNLM) for solving nonlinear leastsquares problems in a Euclidean space setting. The convergence analysis is based on our new idea of restricted convergence domains. Using this idea, we obtain a more precise information on the location of the iterates than in earlier studies leading to smaller majorizing functions. This way, our approach has the following advantages and under the same computational cost as in earlier studies: A large radius of convergence and more precise estimates on the distances involved to obtain a desired error tolerance. That is, we have a larger choice of initial points and fewer iterations are also needed to achieve the error tolerance. Special cases and numerical examples are also presented to show these advantages.

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An adaptive continuation process for solving systems of nonlinear equations

Cited by 198 publications

References 0 publications

On the Local Convergence of an Eighth-order Method for Solving Nonlinear Equations

On the Local Convergence of an Eighth-order Method for Solving Nonlinear Equations

Sharp estimation of local convergence radius for the Picard iteration

Extended Local Analysis of Inexact Gauss-Newton-like Method for Least Square Problems using Restricted Convergence Domains

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