BALL CONVERGENCE FOR A NINTH ORDER NEWTON-TYPE METHOD FROM QUADRATURE AND A DOMIAN FORMULAE IN A BANACH SPACE

Abstract

We present a local convergence analysis of a ninth order Newton-type method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. We only use hypotheses on the first Fŕechet-derivative. The local convergence analysis in [7, 11, 12, 24] used hypotheses up to the second Fŕechet derivative although only the first derivative appears in this method. Hence, the application of the methods is extended under less computational cost. This work also provides computable convergence ball and computable error bounds. Numerical examples are also provided in this study.