Contrasts in the basins of attraction of structurally identical iterative root finding methods

Keywords: Nonlinear equations Iterative methods Order of convergence Basins of attraction a b s t r a c t A numerical comparison is performed between three methods of third order with the same structure, namely BSC, Halley's and Euler–Chebyshev's methods. As the behavior of an iterative method applied to a nonlinear equation can be highly sensitive to the starting points, the numerical comparison is carried out, allowing for complex starting points and for complex roots, on the basins of attraction in the complex plane. Several examples of algebraic and transcendental equations are presented. The construction of mathematical models is a common scientific activity viewing the development of necessary tools capable of answering questions about real-world phenomena. Frequently, solving these models, involves finding roots of nonlinear equations. If exact methods are not available, which is the general rule, numerical methods are necessary to find the approximate solutions. A great variety of iterative methods have been developed and are described in the literature [1–7]. Several measures have been used by different authors to evaluate and compare different methods and to claim the superiority or inferiority of some of them over others. Some measures involve the choice of nonlinear equations and, for each equation, the choice of one or more starting points (that can be chosen randomly), and then count the number of iterations required for the convergence or until a given tolerance is achieved, count the number of functions evaluations per step or evaluate the amount of CPU time needed. Although a starting point can been chosen at random, it represents only one of an infinite number of other choices [8]. In order to improve this, some authors [8–13] studied the basins of attraction of different methods, to visually understand how an algorithm behaves as a function of the various starting points. In the work of Basto et al. [4], the authors proposed a new iterative root finding method of third order. In another work of a different author [5], this iterative method (the author called that method, the BSC root finding method) is compared with competitive tested methods. The similarity of the BSC method to Euler–Chebyshev's root finding method is also pointed out in [5], and there is a discussion of the efficiency of the BSC method in a numerical comparison. The authors of the works [8–10], discuss several methods with different orders of convergence and present the basins of attraction …