 
Summary: MATHEMATICS OF COMPUTATION
VOLUME 61, NUMBER 204
OCTOBER 1993, PAGES 733744
ON ENCLOSING SIMPLE ROOTS OF NONLINEAR EQUATIONS
G. ALEFELD, F. A. POTRA, AND YIXUN SHI
ABSTRACT.In this paper we present two efficient algorithms for endosing a
simple root of the nonlinear equation f(x) = 0 in the interval [a, b]  They
improve recent methods of Alefeld and Potra by achieving higher efficiency
indices and avoiding the solution of a quadratic equation per iteration. The
efficiency indices of our methods are 1.5537... and 1.618... , respectively.
We show that our second method is an optimal algorithm in some sense. Our
numerical experiments show that the two methods of the present paper compare
well with the above methods of Alefeld and Potra as well as efficient solvers of
Dekker, Brent, and Le. The second method in this paper has the best behavior
of all, especially when the termination tolerance is small.
1. INTRODUCTION
In arecent paper, Alefeld and Potra [2] proposed three efficient methods
for enc10sing a simple zero X* of a continuous function fex) in the interval
[a, b] provided that f(a)f(b) < O. Starting with the initial enclosing interval
[al, bd = [a, b], the methods produce a sequence of intervals {[an, bn]}~l
