Concerning the convergence of Newton's method and quadratic majorants (Q1032560)

The paper deals with the calculation of the solutions \(x\) of an equation \(F(x)= 0\) in a Banach space \(X\), where \(F\) is a Fréchet-differentiable operator. Assuming \(F'(x_0)^{-1}\in L(X,X)\) and some strong estimations, then the famous convergence theorem by Kantorovich is valid. It guarantees a unique solution in a certain neighborhood of \(x_0\). A scalar majorant is an important tool in the construction. \textit{L. V. Kantorovich}'s and \textit{G. P. Akilov}'s textbook has been published in 1959 [Functional analysis in normed spaces (Russian). Moscow: Fizmatliz (1959)] and in 1964 (in German translation) [cf. also English translation Oxford etc.: Pergamon Press (1964; Zbl 0127.06104)]; earlier papers of Kantorovich dealing with this topic have been printed 1949 and 1951. The author used an English edition printed in 1982. \textit{O. P. Ferreira} and \textit{B. F. Svaiter} [Comput. Optim. Appl. 42, No. 2, 213--229 (2009; Zbl 1191.90095)] found domains where the scalar Newton sequence is well behaved. The contribution of the present paper is to prove semi-local convergence for a larger convergence region and weaker conditions. It gives more precise information on the location of the solution. A (simple) numerical example in \(R\) completes this study.