New Highly Efficient Families of Higher-Order Methods for Simple Roots, Permitting

Construction of higher-order optimal and globally convergent methods for computing simple roots of nonlinear equations is an earliest and challenging problem in numerical analysis. Therefore, the aim of this paper is to present optimal and globally convergent families of King's method and Ostrowski's method having biquadratic and eight-order convergence, respectively, permitting in the vicinity of the required root. Fourth-order King's family and Ostrowski's method can be seen as special cases of our proposed scheme. All the methods considered here are found to be more effective to the similar robust methods available in the literature. In their dynamical study, it has been observed that the proposed methods have equal or better stability and robustness as compared to the other methods. 1. Introduction One topic which has always been of paramount importance in computational mathematics is that of approximating efficiently roots of equations of the form where is a nonlinear continuous function on . The most famous one-point iterative method for solving preceding equation (1) is probably the quadratically convergent Newton's method given by However, a major difficulty in the application of Newton's method is the selection of initial guess such that neither the guess is far from zero nor the derivative is small in the vicinity of the required root; otherwise the method fails miserably. Finding a criterion for choosing initial guess is quite cumbersome and, therefore, more effective globally convergent algorithms are still needed. For resolving this problem, Kumar et al. [1] have proposed the following one-point iterative scheme given by This scheme is derived by implementing approximations through a straight line in the vicinity of required root. This family converges quadratically under the condition , while is permitted at some points. For , we obtain Newton's method. The error equation of scheme (3) is given by where , , , and is the root of nonlinear equation (1). In order to obtain quadratic convergence, the entity in the denominator should be the largest in magnitude. Further, it can be seen that this family of Newton's method gives very good approximation to the root when is small. This is because, for small values of , slope or angle of inclination of straight line with -axis becomes smaller; that is, as , the straight line tends to -axis. Multipoint iterative methods can overcome theoretical limits of one-point methods concerning the convergence order and computational efficiency. Therefore, the convergence order and computational efficiency of