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Zbl 1113.65052
Noor, Muhammad Aslam; Noor, Khalida Inayat; Mohyud-Din, Syed Tauseef; Shabbir, Asim
An iterative method with cubic convergence for nonlinear equations.
(English)
[J] Appl. Math. Comput. 183, No. 2, 1249-1255 (2006). ISSN 0096-3003

A new three step iterative method for solving a nonlinear equation $f(x)=0$ is introduced based on the following scheme: Let $x_0$ is an initial guess sufficiently close to simple root of the equation $f(x)=0$. The iterative step consists from two predictor steps: $$y_n=x_n-f(x_n)/ f' (x_n),\quad f'(x_n)\ne 0;\quad z_n=-(y_n-x_n)^2\cdot f''(x_n)/2f'(x_n),$$ and one corrector step: $$x_{n+1}=x_n-f(x_n)/f'(x_n)-(y_n+z_n-z_n)^2\cdot f'(x_n)/2\cdot f'(x_n),\quad n=0, 1,2,\dots.$$ The authors show that if the function $f$ is sufficiently differentiable in the open interval, which contain a simple root of the equation $f(x)=0$ and if $x_0$ is sufficiently close to this root, then the proposed iterative algorithm has the order of convergence equal to three. Several numerical examples are given to illustrate the efficiency and performance of the new method.
[Jiří Vaníček (Praha)]
MSC 2000:
*65H05 Single nonlinear equations (numerical methods)

Keywords: third order of convergence; numerical examples; three step iterative method

Cited in: Zbl 1159.65049 Zbl 1153.65047

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