Asymptotic error estimates for the method of simple iteration and for the modified and generalized Newton methods (Q1280661)

For some usual iterative methods (method of simple iteration, Newton's method, modified Newton's method and generalized Newton's method) explicit error estimates are given, instead of implicit error estimates as in classical literature [cf. \textit{J.M. Ortega} and \textit{W. C. Rheinboldt}, Iterative solution of nonlinear equations in several variables (1970; Zbl 0241.65046) and \textit{L. V. Kantorovich} and \textit{G. P. Akilov}, Functional analysis (1959; Zbl 0127.06102)]) . These asymptotic error estimates are given as follows: 1) for the method of simple iteration, by Theorems 2 and 3; 2) for the modified Newton's method, by Theorems 5 and 6 and 3) for the generalized Newton's method, by Theorem 8. Although these asymptotic estimates are weaker than the implicit ones (from which they are deduced), they are however very important from a practical point of view.