Abstract
A new iterative method is described for finding the real roots of nonlinear equations in R. Starting with a suitably chosen x 0, the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. The number of iterations and the total number of function evaluations used to get a simple root are taken as performance measure of our method. The efficacy of the method is tested on a number of numerical examples and the results obtained are summarized in tables. It is observed that our method is superior to Newton's method and other sixth order methods considered.
Highlights
One of the most important and challenging problems in science and engineering is to find real roots of nonlinear equations in R
This paper is concerned with the iterative methods and their convergence analysis for finding a simple real root α; that is, f(α) = 0 and f(α) ≠ 0 of f (x) = 0, (1)
Starting with an initial approximation x0 near to the root α, a one parameter family of sixth order methods proposed in [10] is given for k = 0, 1, 2, . . ., by wk
Summary
One of the most important and challenging problems in science and engineering is to find real roots of nonlinear equations in R. A number of ways are considered by many researchers to improve the local order of convergence of Newton’s method at the expense of additional evaluations of functions, derivatives, and changes in the points of iterations. Ostrowski [2] developed both third and fourth order methods each requiring evaluations of two functions and one derivative per iteration leading to its efficiency index equal to 1.587. Starting with an initial approximation x0 near to the root α, a one parameter family of sixth order methods proposed in [10] is given for k = 0, 1, 2, .
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