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Three-step iterative methods for nonlinear equations. (English) Zbl 1113.65050
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.

MSC:
65H05Single nonlinear equations (numerical methods)
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References:
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