Journal of the
Korean Mathematical Society JKMS

We present a unified local and semilocal convergence analysis for secant-type methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost our semilocal convergence criteria can be weaker; the error bounds more precise and in the local case the convergence balls can be larger and the error bounds tighter than in earlier studies such as \cite{1,2,4,6,7,8,9,10,te,11,12,14,18,19} at least for the cases of Newton's method and the secant method. Numerical examples are also presented to illustrate the theoretical results obtained in this study.

Keywords: secant-type method, Banach space, majorizing sequence, divided difference, local convergence, semilocal convergence

MSC numbers: 65H10, 65G99, 65B05,65N30, 47H17, 49M15