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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: \[ y_n=x_n-f(x_n)/ f(x_n),\quad f'(x_n)\neq 0; \quad z_n=-(y_n-x_n)^2\cdot f''(x_n)/2\cdot f'(x_n) \] 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:

65H05 Numerical computation of solutions to single equations
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