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.polynomial equations, nonlinear equations. 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 two-point 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 …