Comparing two techniques for developing higher order two-point iterative methods for solving quadratic equations

Abstract

Kung–Traub conjecture states that an iterative method without memory for finding the simple zero of a scalar equation could achieve convergence order $2^{d-1}$, where d is the total number of function evaluations. Babajee (Algorithms 9:1, 2016) proposed some higher order two-point methods which fail the conjecture. He developed these methods for solving quadratic equations using weight functions. Recently, Ahmad (Algorithms 9:30, 2016) showed that the proposed method in Babajee (2016) was reported by him (Ahmad, in Researchgate, https://doi.org/10.13140/RG.2.1.1519.2487, 2015) in which he used a loop to develop his methods. He also showed Babajee’s method is a member of his methods developed in Ahmad (2015). In this paper, we compare the two techniques and adopt the technique of weight functions to develop higher order Jarratt and Ostrowski’s methods for solving quadratic equations. We prove the local convergence of the methods by induction. Numerical experiments are carried out to compare the new methods with some existing methods. We apply our methods to find the optimal launch angle in a projectile problem.