Approximating Solutions of Equations Using Two-Point Newton Methods

We provide a local as well as a semilocal convergence analysis for two-point Newton methods under very general conditions. Our equation contains a Fréchet differentiable operator F and another operator G whose differentiability is not assumed. Using more precise majorizing sequences than before we provide sufficient convergence conditions for Newton methods to a locally unique solution of equation F (x) + G(x) = 0. In the semilocal case we show under weaker conditions that our error estimates on the distances involved are finer and the information on the location of the solution at least as precise as in earlier results [17]. In the local case a larger radius of convergence is obtained [21]. Several applications are provided to show that our results compare favorably with earlier ones [11]–[14], [17], [21], [23], [25]. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)