ON COMPARISONS OF CHEBYSHEV-HALLEY ITERATION FUNCTIONS BASED ON THEIR ASYMPTOTIC CONSTANTS
International Journal of Pure and Applied Mathematics
Methods for solving a nonlinear equation are classified by their order of convergence p and the number d of function (and derivatives) evaluations per step. Based on p and d, there are two efficiency measures defined by I = p/d (informational efficiency) and E = p 1/d (efficiency index)  . These measures do not depend on the function f (x) nor on the number of steps required to solve the problem within a given precision. Unfortunately, for methods of the same order p and demanding the same
... demanding the same number of function evaluations d, these two measures are the same for these methods. For example, each iteration function (IF) of the Chebyshev-Halley family is of order p = 3 and requires d = 3 evaluations per step (evaluation of f (x), f (1) (x), and f (2) (x)). Another measure introduced recently is the basin of attraction of a given IF ( and the references therein). The basin of attraction depends on f (x) and, for a given method of order p, the "local order" of convergence in the basin is not necessarily p for the first steps of the method. Moreover, since the number of steps required to reach a given precision is not known, it is difficult to classify methods with respect to the efficiency index, the informational efficiency, and even their basins of attraction (which depend on f (x)). In many papers, to compare methods authors take one arbitrary initial point and study the error after the same number of steps or the number of steps required to get a given error bounds (see for examples [2, 11] ). This number of step depends on how far the initial trial x 0 is from the solution α, and the value of the asymptotic constant. For two methods of the same order p, the method having the smallest asymptotic constant will converge faster than the second method having the higher asymtotic constant, for a starting point x 0 sufficiently close to the solution α. Unfortunately, as for the basin of attraction, the asymtotic constant depends on the function f (x).