## 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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### References:

  Abbasbandy, S., Improving newton – raphson method for nonlinear equations by modified Adomian decomposition method, Appl. math. comput., 145, 887-893, (2003) · Zbl 1032.65048  Adomian, G., Nonlinear stochastic systems and applications to physics, (1989), Kluwer Academic Publishers Dordrecht · Zbl 0698.35099  Chun, C., Iterative methods improving newton’s method by the decomposition method, Comput. math. appl., 50, 1559-1568, (2005) · Zbl 1086.65048  Daftardar-Gejji, V.; Jafari, H., An iterative method for solving nonlinear functional equations, J. math. anal. appl., 316, 753-763, (2006) · Zbl 1087.65055  He, J.H., A new iteration method for solving algebraic equations, Appl. math. comput., 135, 81-84, (2003) · Zbl 1023.65039  Homeier, H.H., On Newton-type methods with cubic convergence, J. comput. appl. math., 176, 425-432, (2005) · Zbl 1063.65037  Luo, X., A note on the new iteration for solving algebraic equations, Appl. math. comput., 171, 1177-1183, (2005) · Zbl 1091.65044  Aslam Noor, M., Numerical analysis and optimization, lecture notes, (2006), COMSATS Institute of Information Technology Islamabad, Pakistan
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