On Some Efficient Techniques for Solving Systems of Nonlinear Equations

Abstract

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.

An Efficient Fifth Order Method for Systems of Nonlinear Equations

“…The numerical results in Table II, Table III and Table IV show that the newly presented method M 5 in this paper as defined by (2) outperforms NM, the third order method of Sharma J R et al mentioned in [5](denoted by S 3 ) and the fourth order method of Jarratt P (see12)(denoted by M 4 ). The numerical results have been given to conform validity of theoretical results.…”

“…In this section, we present a new method based on the two-step iterative method by Sharma and Gupta in [5] and the multi-step iterative method by Soleymani et al in [6]:…”

“…In this section we present some examples to show the efficiency of four different methods, including the Newton method (denoted by NM),the third order method in [5] by Sharma et al (denoted by S 3 ), the fourth order method in [12] by Jarratt (denoted by M 4 ) and the method M 5 presented in this paper.…”

“…6.7973e-006 5.5869e-013 5.5869e-013 4.7592e-007 S 3 3.9134e-005 3.7315e-011 3.7315e-011 3.0436e-006 M 4 1.1791e-005 2.1042e-012 2.1042e-012 8.5821e-007 M 5 9.8149e-007 1.0049e-014 1.0049e-14 9.7125e-008 …”

“…The new method is based on the two-step iterative method of order three proposed by Sharma and Gupta in [5]. The efficiency index is used to compare the efficiency of the new method with those of other ones.…”