Scaled parallel iterative method for finding real roots of nonlinear equations

Abstract

Given a nonconvex minimization problem where the objective function is nonlinear and twice differentiable. To gain more information about the objective function, it is essential to obtain all its stationary points and study the behaviour of these points. Since many nonlinear functions are expressible as polynomials via interpolation, there is a need to devise fast and accurate algorithms in finding root(s) of the interpolating polynomial. Through interval computation, the Weierstrass-like parallel iterative methods are known for their efficiency in finding polynomial zeros. However, these schemes are highly dependent on the midpoints of each interval in generating successive intervals. In this study, we propose a scaling function on some Weierstrass-like parallel iterative methods such that the procedures are less dependent on the generated midpoints, hence allowing a more efficient search for the zeros while reducing the width of the intervals. The proposed procedures with the shifted centres of the enclosing intervals are tested on 120 problems and we compare their efficiency with the existing Weierstrass-like methods in terms of the number of iterations and largest final interval width. The results indicate that the proposed procedures outperform the original procedures, giving more reduction on the final interval width with a lesser number of iterations.