Higher-order iterative methods for approximating zeros of analytic functions

Abstract

Iterative methods with extremely rapid convergence in floating-point arithmetic and circular arithmetic for simultaneously approximating simple zeros of analytic functions (inside a simple smooth closed contour in the complex plane) are presented. The R-order of convergence of the basic total-step and single-step methods, as well as their improvements which use Newton's and Halley's corrections, is given. Some hybrid algorithms that combine the efficiency of ordinary floating-point iterative methods with the accuracy control provided by interval arithmetic are also considered. © 1992.