On enclosing simple roots of nonlinear equations

Abstract

In this paper we present two efficient algorithms for enclosing a simple root of the nonlinear equation f ( x ) = 0 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.