Kantorovich’s majorants principle for Newton’s method

Abstract

We prove Kantorovich's theorem on Newton's method using a convergence analysis which makes clear, with respect to Newton's Method, the relationship of the majorant function and the non-linear operator under consideration. This approach enable us to drop out the assumption of existence of a second root for the majorant function, still guaranteeing Q-quadratic convergence rate and to obtain a new estimate of this rate based on a directional derivative of the derivative of the majorant function. Moreover, the majorant function does not have to be defined beyond its first root for obtaining convergence rate results. AMSC: 49M15, 90C30.

Local Convergence of the Gauss-Newton Method for Injective-Overdetermined Systems

“…We present, under a weak majorant condition, a local convergence analysis for the Gauss-Newton method for injective-overdetermined systems of equations in a Hilbert space setting. Our results provide under the same information a larger radius of convergence and tighter error estimates on the distances involved than in earlier studies such us [10,11,13,14,18]. Special cases and numerical examples are also included in this study.…”

“…It was shown that this condition is better that Wang's condition [18], [22] in some sense. A certain relationship between the majorant function and operator F was established that unifies two previously unrelated results pertaining to inexact Gauss-Newton methods, which are the result for analytical functions and the one for operators with Lipschitz derivative.similar improvements in both the local and semilocal case of the works in [10,11,13,12,22], have already been obtained by us in [3,4,5,6,7,9] under stronger than (2.2) majorant-type conditions. The paper is organized as follows: Section 2 contains the local convergence analysis of the Gauss-Newton method (1.2) under weak majorant conditions, whereas in Section 3 we provide special cases and numerical examples further validating the theoretical results.…”

“…If L = L 0 , our results reduce to the ones in [14,Theorem 13] (see also [4,10,11,12,13,16]). Moreover, if L 0 < L, we have…”

“…That is our results provide a larger convergence radius and tighter error bounds than the ones in [10,11,12,13,14,16,18]. Note also that we have times larger than the one in [10,11,12,13,14,16,18].…”

“…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. In particular, Ferreira et al [10,11,12,13] used the majorant condition in the local as well as semilocal convergence of Newtontype method. Argyros and Hilout [3,4,5,6,7,9] have also used the majorant condition to provide a tighter convergence analysis and weaker convergence criteria for Newton-type method.…”