Kantorovich's type theorems for systems of equations with constant rank derivatives

Abstract

The famous Newton-Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton's method to a solution of an equation. Here we present a "Kantorovich type" convergence analysis for the Gauss-Newton's method which improves the result in [W.M. Häußler, A Kantorovich-type convergence analysis for the Gauss-Newton-method, Numer. Math. 48 (1986) 119-125.] and extends the main theorem in [I.K. Argyros, On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math. 169 (2004) 315-332]. Furthermore, the radius of convergence ball is also obtained.

On the solution of systems of equations with constant rank derivatives

“…Let us compare our results with the corresponding ones in [19] (see also [22,Section 5.1]) which in turn have improved earlier results by Häubler [17]. We need to define scalar sequence {s n } by…”

“…Moreover, under the additional hypotheses (2.36)-(2.43), the conclusions of Theorem 2.8 hold with {s n }, s replacing {t n }, t , respectively [19].…”

“…It then follows from the definitions of sequences {t n }, {s n }, (3.6), and a simple inductive argument that under the hypotheses of our Theorems 2.8 and 3.1 in [19], we have in turn:…”

“…That is {t n } is an at least as tight majorizing sequence as {s n }. We provide a numerical example, where the hypotheses of Theorem 2.8 and Lemma 2.3 are satisfied, but not earlier ones [17,19].…”

“…He provided a Kantorovich-type semilocal convergence analysis using Lipschitz continuity of F . Hu, Shen and Li in the elegant study [19] used an idea by Argyros [1][2][3][4][5], where the center Lipschitz condition is used to provide upper bounds on the inverses of the linear operators involved. This way they provided on the one hand a finer convergence analysis than in [17], and on the other hand they extended the result of Argyros [3].…”