A New Iterative Method for Solving Nonlinear Equations

In this study, a new root-finding method for solving nonlinear equations is proposed. This method requires two starting values that do not necessarily bracketing a root. However, when the starting values are selected to be close to a root, the proposed method converges to the root quicker than the secant method. Another advantage over all iterative methods is that; the proposed method usually converges to two distinct roots when the given function has more than one root, that is, the odd iterations of this new technique converge to a root and the even iterations converge to another root. Some numerical examples, including a sine-polynomial equation, are solved by using the proposed method and compared with results obtained by the secant method; perfect agreements are found. Keywords— Iterative method, root-finding method, sinepolynomial equations, nonlinear equations. I. FORMULATION OF THE PROBLEM NE classical problem in numerical analysis is the solution of nonlinear equations 0 ) ( = x f . To approximate a solution to one of these equations we can use iterative methods. An iterative method starts from two initial guesses 0 x and 1 x , which are then improved by means of a sequence { } ) , ( 1 1 1 k k k k x x x − = + Φ = , 1 ≥ k , is known as a twopoint iterative method. Conditions are imposed on 0 x , 1 x and, eventually, on f or Φ or both, in order to ensure the convergence of the sequence to { } 1 + k x for 1 ≥ k to a solution α of the nonlinear equation 0 ) ( = x f , then proceed to find the order of convergence of the sequence. In this paper, we shall study a new iterative approach that requires two starting values, but the order of convergence and the convergence criterion of the proposed method will be put off to a later extended work. Assume that ) ( ), ( x f x f ′ and ) (x f ′ ′ are continuous near a root α , then the graphs of the functions ) (x f and ) (x f − intersect the x -axis at the point ) 0 , (α . Furthermore, assume that initial approximations 0 x and 1 x are near the root α and 1 0 x x ≠ , then the points )) ( , ( 0 0 x f x and )) ( , ( 1 1 x f x lie on the curve of ) (x f y = near the point ) 0 , (α , see Figure 1. The information regarding the nature of ) (x f and ) (x f − can Manuscript received on March 29, 2005. I. Abu-Alshaikh is with the Department of Mathematics, Fatih University, Istanbul, 34500 TURKEY, (e-mail: [email protected] ). be used to develop an algorithm that will produce a sequence { } 1 + k x that converges to the root α . We shall first introduce the algorithm of this sequence graphically and then a more rigorous treatment based on the Taylor series will be given. Now, define 2 x to be the x -coordinate of the point where the straight line 1 L intersects the graph of ) (x f y = , where 1 L is the line that emanates from the point ) 0 , ( 0 x and passes through the point )) ( , ( 1 1 x f x − . And 3 x is defined as the x coordinate of the point where the line 2 L intersects the graph of ) (x f y = , where 2 L is the line that emanates from ) 0 , ( 1 x and passes through the point )) ( , ( 2 2 x f x − . Now, referring to figure 1 the slopes 1 m and 2 m of the lines 1 L and 2 L can be written, respectively, as . ) ( ) ( ) ( ) ( tan , ) ( ) ( ) ( ) ( tan