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Iterative methods of order four and five for systems of nonlinear equations. (English) Zbl 1173.65034
The authors present new iterative schemes for solving systems of nonlinear equations based on modifications of the classical Newton method which accelerate the convergence. Using Adomian polynomials [see {\it G. Adomian}, J. Math. Anal. Appl. 135, 501--544 (1988; Zbl 0671.34053)]. they obtain a family of multipoint iterative formulas including the Newton and Traub methods as simple special cases. The convergence analysis leads to the conclusion that the order of convergence of the new iterative methods is $p\ge 2$ under the same assumptions as for the classical Newton method. Finally, the results of numerical experiments are given and the new methods are compared with the classical Newton method and the Traub method [see {\it J. F. Traub}, Iterative methods for the solution of equations. 2nd ed. New York, N.Y.: Chelsea Publishing Company (1982; Zbl 0472.65040)] to confirm the theoretical results.

MSC:
65H10Systems of nonlinear equations (numerical methods)
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References:
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[2] Cordero, A.; Torregrosa, J. R.: Variants of Newton’s method using fifth-order quadrature formulas, Applied mathematics and computation 190, 686-698 (2007) · Zbl 1122.65350 · doi:10.1016/j.amc.2007.01.062
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[8] Abbasbandy, S.: Extended Newton’s method for a system of nonlinear equations by modified Adomian decomposition method, Applied mathematics and computation 170, 648-656 (2005) · Zbl 1082.65531 · doi:10.1016/j.amc.2004.12.048
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[10] Ostrowski, A. M.: Solutions of equations and systems of equations, (1966) · Zbl 0222.65070