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Zbl 1113.65050
Noor, Muhammad Aslam; Noor, Khalida Inayat
Three-step iterative methods for nonlinear equations.
(English)
[J] Appl. Math. Comput. 183, No. 1, 322-327 (2006). ISSN 0096-3003

A new three step iterative method for solving nonlinear equations $f(x)=0$ is introduced based on the following scheme: Let $x_0$ be an initial guess sufficiently close to a simple root of the equation $f(x)=0$. The iterative step consists two predictor steps: $$y_n=x_n-f(x_n)/ f(x_n),\quad f'(x_n)\ne 0; \quad z_n=-(y_n-x_n)^2\cdot f''(x_n)/2\cdot f'(x_n)$$ and one corrector step: $$x_{n+1}=x_n-f(x_n)f'(x_n)-(y_n+ x_n)^2\cdot f''(x_n)/2\cdot f'(x_n)-(y_n+z_n-x_n)^2\cdot f''(x_n)/2\cdot f'(x_n),$$ $n=0,1,2,\dots$. The authors show that if the function $f$ is sufficiently differentiable on an open interval which contains a single root, and if $x_0$ is sufficiently close to this root, then the proposed iterative algorithm has the fourth-order of convergence. 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: numerical examples; nonlinear equations; three step iterative method; algorithm; fourth-order of convergence

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