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

Noor, Muhammad Aslam; Noor, Khalida Inayat

Appl. Math. Comput. 183, No. 1, 322-327 (2006).

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.

Reviewer: Jiří Vaníček (Praha)

MSC:

65H05

Numerical computation of solutions to single equations

Keywords:

numerical examples; nonlinear equations; three step iterative method; algorithm; fourth-order of convergence

References:

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