Weaker conditions for the convergence of Newton’s method

Annals of West University of Timisoara - Mathematics and Computer Science

George²

2016

Self Cite

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.

“…Estimate (2.9) holds as a strict inequality. Therefore, the new error bounds are more precise than the old ones using only L 1 [7][8][9] or L 0 and L 1 [4,5].…”

Section: Applications and Examplesmentioning

confidence: 90%

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

Annals of West University of Timisoara - Mathematics and Computer Science

George²

2016

Self Cite

“…We state the following modified majorant condition for the convergence of various Newton-type methods in [10,11,12,13]. Definition 2.6.…”

Section: Introductionmentioning

confidence: 99%

Extended local convergence analysis of inexact Gauss-Newton method for singular systems of equations under weak conditions

Stud. Univ. Babes-Bolyai Math.

George

2017

Self Cite

Abstract.A new local convergence analysis of the Gauss-Newton method for solving some optimization problems is presented using restricted convergence domains. The results extend the applicability of the Gauss-Newton method under the same computational cost given in earlier studies. In particular, the advantages are: the error estimates on the distances involved are tighter and the convergence ball is at least as large. Moreover, the majorant function in contrast to earlier studies is not necessarily differentiable. Numerical examples are also provided in this study.Mathematics Subject Classification (2010): 65D10, 65D99, 65G99, 65K10, 90C30.

“…In this study we use only assumptions on the first Fréchet derivative of the operator F in our convergence analysis, so that the method (1) can be applied to solve equations but the earlier results cannot be applied [1,2,3,4,5,18,6,7,8,10,11,12,13,14,15,16,17,9,19,20,21,22] (see Example 3.2).…”

Section: Introductionmentioning

confidence: 99%

Local convergence of a multi-step high order method with divided differences under hypotheses on the first derivative

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

George²

2017

Self Cite

Abstract. This paper is devoted to the study of a multi-step method with divided differences for solving nonlinear equations in Banach spaces. In earlier studies, hypotheses on the Fréchet derivative up to the sixth order of the operator under consideration is used to prove the convergence of the method. That restricts the applicability of the method. In this paper we extended the applicability of the sixth-order multi-step method by using only hypotheses on the first derivative of the operator involved. Our convergence conditions are weaker than the conditions used in earlier studies. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study.