Abstract
We present a new fourth order method for finding simple roots of a nonlinear equation f(x)=0. In terms of computational cost, per iteration the method uses one evaluation of the function and two evaluations of its first derivative. Therefore, the method has optimal order with efficiency index 1.587 which is better than efficiency index 1.414 of Newton method and the same with Jarratt method and King’s family. Numerical examples are given to support that the method thus obtained is competitive with other similar robust methods. The conjugacy maps and extraneous fixed points of the presented method and other existing fourth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane.
Highlights
Solving nonlinear equations is a common and important problem in science and engineering [1, 2]
We consider the problem of finding simple root α of a nonlinear equation f(x) = 0, where f(x) is a continuously differentiable function
Newton method is probably the most widely used algorithm for finding simple roots, which starts with an initial approximation x0 closer to the root and generates a sequence of successive iterates {xk}∞ 0 converging quadratically to simple roots
Summary
Solving nonlinear equations is a common and important problem in science and engineering [1, 2]. The method requires two f and one f evaluations per step and is seen to be efficient compared to classical Newton method Another well-known example of fourth order multipoint methods with same number of evaluations is King’s family of methods [10], which contains Ostrowski’s method as a special case. Journal of Complex Analysis proposed fourth order methods requiring one f evaluation and two f evaluations per iteration All of these methods are classified as multistep methods in which a Newton or weighted-Newton step is followed by a faster Newton-like step. The complex dynamics of various other known iterative methods, such as King’s and Chebyshev-Halley’s families, Jarratt method, has been analyzed by various researchers; for example, see [13, 21,22,23,24,25,26].
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