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Some iterative methods for solving a system of nonlinear equations. (English) Zbl 1165.65349

Summary: We suggest and analyze two new two-step iterative methods for solving the system of nonlinear equations using quadrature formulas. We prove that these new methods have cubic convergence. Several numerical examples are given to illustrate the efficiency and the performance of the new iterative methods. These new iterative methods may be viewed as an extension and generalizations of the existing methods for solving the system of nonlinear equations.

MSC:

65H10 Numerical computation of solutions to systems of equations
65J15 Numerical solutions to equations with nonlinear operators
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[1] Burden, R.L.; Faires, J.D., Numerical analysis, (2001), PWS Publishing Company Boston
[2] Ortega, J.M.; Rheinboldt, W.C., Iterative solution of nonlinear equations in several variables, (1970), Academic Press · Zbl 0241.65046
[3] Abbasbandy, S., Extended newton’s method for a system of nonlinear equations by modified Adomian decomposition method, Appl. math. comput., 170, 648-656, (2005) · Zbl 1082.65531
[4] Babolian, E.; Biazar, J.; Vahidi, A.R., Solution of a system of nonlinear equations by Adimian decomposition method, Appl. math. comput., 150, 847-854, (2004) · Zbl 1075.65073
[5] Darvishi, M.T.; Barati, A., A third-order Newton-type method to solve systems of nonlinear equations, Appl. math. comput., 187, 630-635, (2007) · Zbl 1116.65060
[6] Darvishi, M.T.; Barati, A., Super cubic iterative methods to solve systems of nonlinear equations, Appl. math. comput., 188, 1678-1685, (2007) · Zbl 1119.65045
[7] Golbabai, A.; Javidi, M., A new family of iterative methods for solving system of nonlinear algebric equations, Appl. math. comput., 190, 1717-1722, (2007) · Zbl 1227.65046
[8] Babajee, D.K.R.; Dauhoo, M.Z.; Darvishi, M.T.; Barati, A., A note on the local convergence of iterative methods based on Adomian decomposition method and 3-node quadrature rule, Appl. math. comput., 200, 452-458, (2008) · Zbl 1160.65018
[9] Cordero, A.; Torregrosa, J.R., Variants of newton’s method for functions of several variables, Appl. math. comput., 183, 199-208, (2006) · Zbl 1123.65042
[10] Cordero, A.; Torregrosa, J.R., Variants of newton’s method using fifth-order quadrature formulas, Appl. math. comput., 190, 686-698, (2007) · Zbl 1122.65350
[11] Darvishi, M.T.; Barati, A., A fourth-order method from quadrature formulae to solve systems of nonlinear equations, Appl. math. comput., 188, 257-261, (2007) · Zbl 1118.65045
[12] Frontini, M.; Sormani, E., Third-order methods from quadrature formulae for solving systems of nonlinear equations, Appl. math. comput., 149, 771-782, (2004) · Zbl 1050.65055
[13] Atkinson, K.E., An introduction to numerical analysis, (1987), John Wiley and Sons New York
[14] Babajee, D.K.R.; Dauhoo, M.Z., Analysis of the properties of the variants of newton’s method with third order convergence, Appl. math. comput., 183, 659-684, (2006) · Zbl 1123.65036
[15] Freudensten, F.; Roth, B., Numerical solution of systems of nonlinear equations, J. ACM, 10, 550-556, (1963) · Zbl 0131.33703
[16] Grau-Sánchez, M.; Peris, J.M.; Gutiérrez, J.M., Accelerated iterative methods for finding solutions of a system of nonlinear equations, Appl. math. comput., 190, 1815-1823, (2007) · Zbl 1122.65351
[17] Homeier, H.H.H., A modified Newton method with cubic convergence: the multivariate case, J. comput. appl. math., 169, 161-169, (2004) · Zbl 1059.65044
[18] Kou, J., A third-order modification of Newton method for systems of nonlinear equations, Appl. math. comput., 191, 117-121, (2007) · Zbl 1193.65077
[19] M. Aslam Noor, Numerical Analysis and Optimization, Lecture Notes, Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan, 2007 · Zbl 1112.49013
[20] Shampine, L.F.; Allen, R.C.; Pruess, S., Fundamentals of numerical computing, (1997), John Wiley and Sons New York · Zbl 0904.65001
[21] Varga, R.S., Matrix iterative analysis, (2000), Springer Berlin · Zbl 0133.08602
[22] Aslam Noor, M., Some applications of quadrature formulas for solving nonlinear equations, Nonlinear anal. forum, 12, 1, 91-96, (2007) · Zbl 1146.65316
[23] Aslam Noor, M., Fifth-order convergent iterative method for solving nonlinear equations using quadrature formula, J. math. control sci. appl., 1, 241-249, (2007)
[24] Podisuk, M.; Chundang, U.; Sanprasert, W., Single-step formulas and multi-step formulas of the integration method for solving the initial value problem of ordinary differential equation, Appl. math. comput., 190, 1438-1444, (2007) · Zbl 1122.65367
[25] Gautschi, W., Numerical analysis: an introduction, (1997), Birkhäuser · Zbl 0877.65001
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