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A new iterative method to compute nonlinear equations. (English) Zbl 1091.65043
Summary: The aim of this paper is to construct a new efficient iterative method to solve nonlinear equations. The new method is based on the proposals of S. Abbasbandy on improving the order of accuracy of Newton-Raphson method [ibid. 145, 887–893 (2003; Zbl 1032.65048)] and on the proposals of E. Babolian and J. Biazar on improving the order of accuracy of Adomian’s decomposition method [ibid. 130, No. 2–3, 383–387 (2002; Zbl 1044.65043)]. The convergence of the new scheme is proved and at least the cubic order of convergence is established.
Several examples are presented and compared to other methods, showing the accuracy and fast convergence of this new method. Also, it is shown in this paper, that the modified Adomian’s method developed by E. Babolian and J. Biazar to solve nonlinear equations [ibid. 132, No. 1, 167–172 (2002; Zbl 1023.65040)] should be slightly modified, due to the fact that convergence of Adomian’s method does not ensure convergence of the modified method. An example illustrates this fact, which, unlike what is claimed by the authors, does not converge with their method, but with a simple different choice of the zero component becomes convergent.

MSC:
65H05 Numerical computation of solutions to single equations
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References:
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