An improved regula-falsi method for enclosing simple zeros of nonlinear equations

In this paper, an improved regula-falsi method of order three for finding zeros of nonlinear equations f(x) = 0, where f : [a, b] ⊂ R → R is a continuously differentiable function, is proposed. The proposed method consists of a combination of usual regula-falsi method and a Newton-like method to solve f(x) = 0. It starts with a suitably chosen x0 (generally near to the zero r) and generates a sequence of successive iterates xn, n = 0, 1, … which converges cubically to the zero r. If for an interval [a, b], the diameter of [a, b] be defined as (b − a), then the proposed method generates a sequence of diameters {(bn − an)} for the sequence of intervals {[an, bn]}, each enclosing the zero r and converges cubically to 0. The method is tested on a number of numerical examples and results obtained show that the proposed method is very effective when compared with some existing methods used to solve same problems.