Local Convergence Analysis of Inexact Newton-Like Methods

Abstract

Abstract. We provide a local convergence analysis of inexact Newton-like methods in a Banach space setting under flexible majorant conditions. By introducing center-Lipschitz-type condition, we provide (under the same computational cost) a convergence analysis with the following advantages over earlier work [9]: finer error bounds on the distances involved, and a larger radius of convergence.Special cases and applications are also provided in this study.

Improved local convergence analysis of inexact Gauss–Newton like methods under the majorant condition in Banach spaces

“…(a) Note that γ 0 ≤γ holds in general and γ=γ 0 can be arbitrarily large [4][5][6][7][8][9]. (b) If F is an analytic function, Smale [21] used the following choice:…”

“…The semilocal convergence matter is, based on the information around an initial point, to give criteria ensuring the convergence of iterative procedures; while the local one is, based on the information around a solution, to find estimates of the radii of convergence balls. A plethora of sufficient conditions for the local as well as the semilocal convergence of Newtontype methods as well as an error analysis for such methods can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][16][17][18][19][20][21][22].…”

“…A survey of convergence results under various Lipschitz-type conditions for Gauss-Newton-type methods can be found in [4]. The convergence of these methods requires among other hypotheses that F′ satisfies a Lipschitz condition or F″ is bounded in D. Several authors have relaxed these hypotheses [5][6][7][8]. In particular, Ferreira et al [13][14][15][16] have used the majorant condition in the local as well as semilocal convergence of Newton-type method.…”

Inexact Gauss-Newton like methods for injective-overdetermined systems of equations under a majorant condition

“…This paper is focused on the case in which the least-squares solutions of (1) also solve (1). In the theory of nonlinear least squares problems, this case is called the zero-residual case.…”

“…It is also worth to point out that if F ′ (x) is invertible for all x ∈ Ω, the inexact Gauss-Newton like methods become the inexact Newton-like methods, which in particular include the inexact modified Newton, Newton-like, inexact Newton and Newton methods. Results of convergence for the Gauss-Newton methods have been discussed by many authors, see for example [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,24,25,26]. Recent research attempts to alleviate the assumption of Lipschitz continuity on the operator F ′ .…”