Local convergence analysis of inexact Newton-like methods under majorant condition

Abstract

We present a local convergence analysis of inexact Newton-like methods for solving nonlinear equations under majorant conditions. This analysis provides an estimate of the convergence radius and a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the nonlinear operator under consideration. It also allow us to obtain some important special cases.

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

“…The results in [6] improved the corresponding ones in [21,22,23,24,25,42,43]. In the present study, we improved the results in [6], since D 1 ⊂ U (x * , r) leading to an at least as tight function h λ,θ than the one used in [6] (see also the Examples).…”

“…The advantages of our analysis over earlier works such as [8,9,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43] are also shown under the same computational cost for the functions and constants involved. These advantages include: a large radius of convergence and more precise error estimates on the distances x n+1 − x * for each n = 0, 1, 2, .…”

“…and a clearer relationship between the majorant function (see (2.8) and the associated least squares problems (1.1)). These advantages are obtained because we use a center-type majorant condition (see (2.11)) for the computation of inverses involved which is more precise that the majorant condition used in [21,22,23,24,25,26,30,31,39,40,41,42,43]. Moreover, these advantages are obtained under the same computational cost, since as we will see in section 3 and section 4, the computation of the majorant function requires the computation of the center-majorant function.…”

Local Convergence Analysis of Inexact Newton-Like Methods

“…In this study, we are motivated by the elegant work in [9], and optimization considerations. We introduce center-Lipschitz-type conditions (see (2.2)), and use it to find upper bounds on the distances…”

“…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. …”