Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations. (English) Zbl 1204.65050

The authors consider two third order Chebyshev-like methods for solving systems of nonlinear equations. It is proved that the methods can be obtained by approximating the second derivatives in the Chebyshev methods. In addition, the authors show the local and cubic convergence of both methods using point attraction theory. Finally, they also compare the computational cost of the two methods with the classical Newton method, and they apply the methods to solve some systems of nonlinear equations including an application to the Chandrasekhar integral equations.


65H10 Numerical computation of solutions to systems of equations
65R20 Numerical methods for integral equations
45G10 Other nonlinear integral equations
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[1] M. Drexler, Newton method as a global solver for non-linear problems, Ph.D. Thesis, University of Oxford, 1997
[2] Grau-Sanchez, M., Improvements of the efficiency of some three-step iterative like-Newton methods, Numer. math., 107, 131-146, (2007) · Zbl 1123.65037
[3] Amat, S.; Busquier, S.; Gutierrez, J.M., Geometric constructions of iterative methods to solve nonlinear equations, Comput. appl. math., 157, 197-205, (2003) · Zbl 1024.65040
[4] Gutierrez, J.M.; Hernandez, M.A., A family of chebyshev – halley type methods in Banach spaces, Bull. austral. math. soc., 55, 113-130, (1997) · Zbl 0893.47043
[5] Babajee, D.K.R.; Dauhoo, M.Z., An 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
[6] Babajee, D.K.R.; Dauhoo, M.Z., Analysis of a family of two-point iterative methods with third order convergence, (), 658-661 · Zbl 1123.65036
[7] Cordero, A.; Torregrosa, J.R., Variants of newton’s method for functions of several functions, Appl. math. comput., 183, 199-208, (2006) · Zbl 1123.65042
[8] 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
[9] 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
[10] Ezquerro, J.A.; Hernandez, M.A., A uniparametric Halley-type iteration with free second derivative, Int. J. pure appl. math., 6, 103-114, (2003) · Zbl 1026.47056
[11] Ezquerro, J.A.; Hernandez, M.A., On Halley-type iterations with free second derivative, J. comput. appl. math., 170, 455-459, (2004) · Zbl 1053.65042
[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] Hernandez, M.A., Second-derivative-free variant of the Chebyshev method for nonlinear equations, J. optim. theory appl., 104, 501-515, (2000) · Zbl 1034.65039
[14] Homeier, H.H.H., Modified Newton method with cubic convergence: the multivariate case, J. comput. appl. math., 169, 161-169, (2004) · Zbl 1059.65044
[15] Varmann, O., High order iterative methods for decomposition – coordination problems, Okio technol. IR ekon. vystymas technol. econ. dev. econ., 7, 56-61, (2006)
[16] Traub, J.F., Iterative methods for the solution of equations, (1982), Chelsea New York · Zbl 0472.65040
[17] Ortega, J.M.; Rheinboldt, W.C., Iterative solution of nonlinear equations in several variables, (1970), Academic Press New York · Zbl 0241.65046
[18] Potra, F.A.; Ptak, V., Nondiscrete induction and iterative processes, () · Zbl 0549.41001
[19] 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 formula, Appl. math. comput., 200, 452-458, (2008) · Zbl 1160.65018
[20] Ortega, J.M., Numerical analysis, (1990), SIAM, A Second Course
[21] Kelley, C.T., Solution of the Chandrasekhar \(H\)-equation by newton’s method, J. math. phys., 21, 1625-1628, (1980) · Zbl 0439.45020
[22] Krejić, N.; Luz˘anin, Z., Newton-like method with modification of the right-hand-side vector, Math. comp., 71, 237-250, (2000) · Zbl 0985.65050
[23] Ostrowski, A., LES points d’attraction et de repulsions pour l’iteration dans l’espace à \(n\) dimensions, C.R. acad. sci. Paris, 244, 288-289, (1957)
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