Tugal ZHANLAV, R. MİJİDDORJ, Otgondorj KHUDER

A family of Newton-type methods with seventh and eighth-order of convergence for solving systems of nonlinear equations

In this work, we first develop a new family of three-step seventh- and eighth-order Newton-type iterative methods for solving systems of nonlinear equations. We also propose some different choices of parameter matrices that ensure the convergence order. The proposed family includes some known methods as special cases. The computational cost and efficiency index of our methods are discussed. Numerical experiments are conducted to support the theoretical results.

Keywords:

Nonlinear system Newton-type method, seventh- and eighth-order convergence, computational efficiency,

PDF

___

[1] M.F. Abad, A. Cordero and J.R. Torregrosa, A family of seventh-order schemes for solving nonlinear systems, Bull. Math. Soc. Sci. Math. Roum. 57, 133-145, 2014.
[2] F. Ahmad, F. Soleymani, F. Khaksar Haghani and S. Serra-Capizzano, Higher order derivative-free iterative methods with and without memory for systems of nonlinear equations, Appl. Math. Comput. 314, 199-211, 2017.
[3] A.R. Amiri, A. Cordero, M.T. Darvishi and J.R. Torregrosa, Preserving the order of convergence: Low-complexity Jacobian-free iterative schemes for solving nonlinear systems, Appl. Math. Comput. 337, 87-97, 2018.
[4] R. Behl and H. Arora, CMMSE: A novel scheme having seventh-order convergence for nonlinear systems, J. Comput. Appl. Math. 404, 113301, 2022.
[5] A. Cordero, E. Gómez and J.R. Torregrosa, Efficient High-Order Iterative Methods for Solving Nonlinear Systems and Their Application on Heat Conduction Problems, Complexity, Article ID 6457532, 11 pages, 2017.
[6] A. Cordero, F. Soleymani, J.R. Torregrosa and M.Z. Ullah, Numerically stable improved ChebyshevHalley type schemes for matrix sign function, J. Comput. Appl. Math. 318, 189-198, 2017.
[7] E. Cătinaş, A survey on the high convergence orders and computational convergence orders of sequences, Appl. Math. Comput. 343, 1-20, 2019.
[8] L.O. Jay, A note on Q-order of convergence, BIT 41, 422-429, 2001.
[9] M. Kansal, A. Cordero, S. Bhalla and J.R. Torregrosa, New fourth- and sixth-order classes of iterative methods for solving systems of nonlinear equations and their stability analysis, Numer. Algor. 87, 1017-1060, 2021.
[10] H.T. Kung and J.F. Traub, Optimal order of one-point and multi-point iteration, J. Assoc. Comput. Math. 21, 643-651, 1974.
[11] T. Lotfi, P. Bakhtiari, A. Cordero, K. Mahdiani and J.R. Torregrosa, Some new efficient multipoint iterative methods for solving nonlinear systems of equations, Int. J. Comput. Math. 92, 1921-1934, 2015.
[12] K.Madhu, D.K.R. Babajee and J. Jayaraman, An improvement to double-step Newton method and its multi-step version for solving system of nonlinear equations and its applications, Numer. Algor. 74, 593-607, 2017.
[13] K. Madhu, A. Elango, R. Jr. Landry and M. Al-arydah, New multi-step iterative methods for solving systems of nonlinear equations and their application on GNSS pseudorange equations, Sensors 20, 5976, 2020.
[14] R. Meersman, Survey of techniques in applied computational complexity, J. Comput. Appl. Math. 1, 39-46, 1975.
[15] M. Narang, S. Bhatia and V. Kanwar, New efficient derivative free family of seventhorder methods for solving systems of nonlinear equations, Numer. Algor. 76, 283-307, 2017.
[16] J.M. Ortega and W.C. Rheinboldt, Iterative solutions of nonlinear equations in several variables, Academic Press, New York, 1970.
[17] M.S. Petković, B. Neta, L.D. Petković and J. Dzˇunić, Multipoint methods for solving nonlinear equations, Elsevier, 2013.
[18] M.S. Petković, B. Neta, L.D. Petković and J. Džunić, Multipoint methods for solving nonlinear equations, Appl. Math. Comput. 226, 635-660, 2014.
[19] J.R. Sharma and H. Arora, A novel derivative free algorithm with seventh order convergence for solving systems of nonlinear equations, Numer. Algor. 67, 917-933, 2014.
[20] J.R. Sharma and H. Arora, A simple yet efficient derivative free family of seventh order methods for systems of nonlinear equations, SeMA Journal 73, 59-75, 2016.
[21] J.R. Sharma and H. Arora, Improved Newton-like methods for solving systems of nonlinear equations, SeMA Journal 74, 147-163, 2017.
[22] J.F. Traub, Iterative methods for the solution of equations, Prentice-Hall, Englewood Cliffs, 1964.
[23] X. Wang, Fixed-point iterative method with eighth-order constructed by undetermined parameter technique for solving nonlinear systems, Symmetry 13, 863, 2021.
[24] X. Wang, T. Zhang, W. Qian and M. Teng, Seventh-order derivative-free iterative method for solving nonlinear systems, Numer. Algor. 70, 545-558, 2015.
[25] T. Zhanlav, Changbum Chun, Kh. Otgondorj and V. Ulziibayar, High-order iterations for systems of nonlinear equations, Int. J. Comput. Math. 97, 1704-1724, 2020.
[26] T. Zhanlav, O. Chuluunbaatar and V. Ulziibayar, Generating functions method for construction new iterations, Appl. Math. and Comput. 315, 414-423, 2017.
[27] T. Zhanlav and Kh. Otgondorj, Higher order Jarratt-like iterations for solving systems of nonlinear equations, Appl. Math. and Comput. 395, 125849, 2021.