On Some Efficient Techniques for Solving Systems of Nonlinear Equations

We present iterative methods of convergence order three, five, and six for solving systems of nonlinear equations. Third-order method is composed of two steps, namely, Newton iteration as the first step and weighted-Newton iteration as the second step. Fifth and sixth-order methods are composed of three steps of which the first two steps are same as that of the third-order method whereas the third is again a weighted-Newton step. Computational efficiency in its general form is discussed and a comparison between the efficiencies of proposed techniques with existing ones is made. The performance is tested through numerical examples. Moreover, theoretical results concerning order of convergence and computational efficiency are verified in the examples. It is shown that the present methods have an edge over similar existing methods, particularly when applied to large systems of equations. 1. Introduction Solving the system of nonlinear equations is a common and important problem in various disciplines of science and engineering [1–4]. This problem is precisely stated as follows: For a given nonlinear function , where and , to find a vector such that . The solution vector can be obtained as a fixed point of some function by means of fixed point iteration One of the basic procedures for solving systems of nonlinear equations is the classical Newton’s method [4, 5] which converges quadratically under the conditions that the function is continuously differentiable and a good initial approximation is given. It is defined by where is the inverse of first Fréchet derivative of the function . Note that this method uses one function, one first derivative, and one matrix inversion evaluations per iteration. In order to improve the order of convergence of Newton’s method, many modifications have been proposed in literature, for example, see [6–18] and references therein. For a system of equations in unknowns, the first Fréchet derivative is a matrix with evaluations whereas the second Fréchet derivative has evaluations. Thus, the methods such as those developed in [6–8] with second derivative are considered less efficient from a computational point of view. In quest of efficient methods without using second Fréchet derivative, a variety of third and higher order methods have been proposed in recent years. For example, Frontini and Sormani in [9] and Homeier in [10] developed some third order methods each requiring the evaluations of one function, two first-order derivatives, and two matrix inversions per iteration. Darvishi and Barati [11] presented a fourth-order