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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.

65H10Systems of nonlinear equations (numerical methods)
Full Text: DOI
[1] Cordero, A.; Torregrosa, J. R.: Variants of Newton’s method for functions of several variables, Applied mathematics and computation 183, 199-208 (2006) · Zbl 1123.65042 · doi:10.1016/j.amc.2006.05.062
[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
[3] A. Cordero, J.R. Torregrosa, Two families of multi-point iterative methods for solving nonlinear systems, Mathematics of Computation (in press) · Zbl 1135.65322
[4] Frontini, M.; Sormani, E.: Some variant of Newton’s method with third-order convergence, Applied mathematics and computation 140, 419-426 (2003) · Zbl 1037.65051 · doi:10.1016/S0096-3003(02)00238-2
[5] Traub, J. F.: Iterative methods for the solution of equations, (1982) · Zbl 0472.65040
[6] Adomian, G.: A review of the decomposition method in applied mathematics, Journal of mathematical analysis and applications 135, 501-544 (1988) · Zbl 0671.34053 · doi:10.1016/0022-247X(88)90170-9
[7] Babolian, E.; Biazar, J.; Vahidi, A. R.: Solution of a system of nonlinear equations by Adomian decomposition method, Applied mathematics and computation 150, 847-854 (2004) · Zbl 1075.65073 · doi:10.1016/S0096-3003(03)00313-8
[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
[9] Weerakoon, S.; Fernando, T. G. I.: A variant of Newton’s method with accelerated third-order convergence, Applied mathematics letters 13, No. 8, 87-93 (2000) · Zbl 0973.65037 · doi:10.1016/S0893-9659(00)00100-2
[10] Ostrowski, A. M.: Solutions of equations and systems of equations, (1966) · Zbl 0222.65070