New Two-step Iterative Methods for Solving Nonlinear Equations

In this paper, we have suggested and analyzed a new two-step type iterative methods for solving nonlinear equations of the type. We show that this new two-step method is cubic convergence method. It is proved that this method is better than the Newton method and all results in (Soheili et al., 2008). Several examples are given to illustrate the efficiency of this new method and its comparison with other methods. This method can se considered as a significant improvement of the Newton method and its variant forms.


Introduction
Iterative methods for finding the roots of nonlinear equations f (x) = 0 are common yet important problem in science and engineering.Analytical methods for solving such equations are difficult or almost non-existent.It has been shown in (Noor, 2006b) that these quadrature formulas can be used to develop some iterative methods for solving nonlinear equations, we have suggested and analyzed new iterative method by using the Newton and the Halley (Noor & Inayat Noor, 2007) methods and some newly developed method by (Noor, 2006a;Noor & Inayat Noor, 2007;Noor & Ahmad, 2006a) as predictor method and then use this new method as a corrector method.We prove that this new method has a cubic convergence.Several examples are given to illustrate the efficiency and performance of this new method and its comparison with other methods.All test problems reveals a good accuracy and fast convergence of the new method.

Iterative Method and Convergence
Consider the nonlinear equation of the type f (x) = 0.For simplicity, assume that r is a simple root zero and γ is an initial guess sufficiently close to r.Using the Taylor's series expansion of the function f (x), we have This formulation allows us to suggest the following iterative methods for solving the nonlinear equations.
Algorithm 2.1 For a given x 0 find the approximate solution x n+1 by the iterative scheme which is an implicit method, since x n+1 occurs on both sides of the equation, which it itself a difficult problem.We remark that f (x n ) = 0 , then algorithm 2.1 reduces to the well known Newton Method, that is.
Algorithm 2.2 For a given x 0 , find the approximate solution x n+1 by the iterative scheme It is well known that algorithm 2.2 has a quadratic convergence, see (Burden & Douglas Faires, 2001).Using this algorithm as a predictor, in (Noor, 2006a) has suggested and analyzed the following two-step iterative method.
Algorithm 2.3 For a given x 0 , find the approximate solution x n+1 by the iterative schemes It has been shown (Noor, 2006a) that algorithm has a cubic convergence and is called the two-step iterative method.Now using algorithm 2.3 to suggest and analyze a new two-step iterative method for solving the nonlinear equation, which is the tain motivation of this paper.We use in the following algorithms as predictor in (Nooc & Inayat Noor, 2007).
We point out that algorithm 2.4 can be written in the following equivalent form.
Algorithm 2.5 For a given x 0 find the approximate solution x n+1 by the iterative scheme Algorithm 2.6 For a given x 0 find the approximate solution x n+1 by the iterative scheme We now study the convergence of algorithm 2.4.In similar way, one can prove the convergence of other two-step algorithm 2.6.
Theorem 1 Let r ∈ I be a simple zero of sufficiently differentiable function f : ⊆ R → R for an open interval I.If x 0 is sufficiently close to r, then the two-step method defined by the algorithm 2.4 has third-order convergence.
From ( 4), ( 5) and ( 7) I can get Using ( 6) and ( 8) in ( 2), we have: From ( 9), e n+1 = x n+1 − r and e n = x n − r we have ) Which shows that the algorithm 2.4 has third-order convergence.Since asymptotic convergence of Newton method is c 2 and from Theorem 1, We deduced that the convergence rate of algorithm 2.4 is better than the Newton method and the method of Soheili (SM) (Soheili et al., 2008).We know that the cubic convergent method is vastly superior to the linear and the quadratically convergent methods.

Numerical Results
We present some examples to illustrate the efficiency of the new developed two-step methods, compare the Newton method (NM), the method of Soheili (SM), and the methods (BM), introduced in this present paper (algorithms 2.4 and 2.6).We take ε = 10 −15 as tolerance.The following criteria is used for estimating the zero: Table 1 presents iteration number comparison of algorithms 2.4, and 2.6 with (NM) and (SM), in given precision.In Table 2, the CPU time (per second) of the new algorithms, (NM) and (SM) are compared.The numerical computations listed in tables are performed using Fortran programs with double precision.

Conclusions
The present two-step methods is generalized and applied for solving the nonlinear algebraic equations.The numerical results in the tables 1,2 show that the new method is very effective and provide highly accurate results in a less number of iterations as compared with Newton method (NM) and (SM) .

Table 1 .
Examples and comparison between other methods