Combining binary search and newton’s method to compute real roots for a class of real functions

Abstract

We generalize a hybrid algorithm of binary search and Newton′s method to compute real roots for a class of real functions. We show that the algorithm computes a root inside (0, R] with error ε(lunate) in O(log log(R/ε(lunate))) time, where one function evaluation or one arithmetic operation counts for one unit of time. This work is based on Smale′s criterion for using Newton′s method and Renegar′s result of approximating roots of polynomials. © 1994 Academic Press, Inc.