A three-free-parameter class of power series based iterative method for approximation of nonlinear equations solution

Document Type : Research Article

Authors

• O. Ogbereyivwe ¹
• O. Izevbizua ²

¹ Department of Mathematics and Statistics, Delta State University of Science and Technology, Delta State, Nigeria.

² Department of Mathematics, University of Benin, Benin City, Nigeria.

10.22067/ijnao.2022.74901.1095

Abstract

In this manuscript, for approximation of solutions to equations that are nonlinear, a new class of two-point iterative structure that is based on a weight function involving two converging power series, is developed. For any method constructed from the developed class of methods, it requires three separate functions evaluation in a complete iteration cycle that is of order four convergence. Also, some well-known existing methods are typical members of the new class of methods. The numerical test on some concrete methods derived from the class of methods indicates that they are effective and competitive when employed in solving a nonlinear equation.

Keywords

• Newton method
• Iterative method
• Power series
• Weight function

20.1001.1.24236977.2023.13.2.1.4

Main Subjects

• Numerical Analysis

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References

[1] Ahmad, F., Comment on: On the Kung-Traub conjecture for iterative methods for solving quadratic equations, Algorithms 2016, 9, 1. Algo-rithms (Basel) 9(2) (2016), Paper No. 30, 11 pp.
[2] Babajee, D.K.R., On the Kung-Traub conjecture for iterative methods for solving quadratic equations, Algorithms (Basel) 9(1) (2016), Paper No. 1, 16 pp.
[3] Chun, C., Some variants of King’s fourth-order family of methods for nonlinear equations, Appl. Math. Comput. 190, (2007), 57–62.
[4] Chun, C., Some fourth-order iterative methods for solving nonlinear equations, Appl. Math. Comput. 195(2) (2008) 454–459.
[5] Chun, C. and Ham, Y.M., Some fourth-order modifications of Newtons method, Appl. Math. Comput. 197, (2008), 654–658.
[6] Ghanbari, B., A new general fourth-order family of methods for finding simple roots of nonlinear equations, J. King Saud Univ. Sci. 23(4), (2011)395–398.
[7] Hafiz, M.A. and Khirallah, M.Q., An optimal fourth order method for solving nonlinear equations, J. Math. and Computer Sc., 23,(2021), 86–97.
[8] Jarratt, P., A review of methods for solving nonlinear algebraic equa-tions in one variable, Numerical methods for nonlinear algebraic equa-tions (Proc. Conf., Univ. Essex, Colchester, 1969), pp. 1–26. Gordon and Breach, London, 1970.
[9] Khattri, S.K. and Abbasbandy, S., Optimal fourth order family of iter-ative methods, Mat. Vesnik 63(1) (2011) 67–72.

[10] Kou, J., Li, Y. and Wang, X., A composite fourth-order iterative method for solving non-linear equations, Appl. Math. Comput. 184(2) (2007)471–475.
[11] Mahdu, K., Two-point iterative methods for solving quadratic equations and its applications, Math. Sci. Appl. E-Notes 6(2) (2018) 66–80.
[12] Ogbereyivwe, O. High order quadrature based iterative method for ap-proximating the solution of nonlinear equations, Caspian Journal of Mathematical Sciences, 9 (2), (2020), 243–255.
[13] Ogbereyivwe, O. and Ojo-Orobosa, V., Family of optimal two-step fourth order iterative method and its extension for solving nonlinear equations, Journal of Interdisciplinary Mathematics, 24 (5), (2021), 1347–1365.
[14] Ostrowski, A.M., Solution of equations in Euclidean and Banach spaces, Third edition of Solution of equations and systems of equations. Pure and Applied Mathematics, Vol. 9. Academic Press [Harcourt Brace Jo-vanovich, Publishers], New York-London, 1973.
[15] Shams, M., Mir, N. A., Rafiq, N., Almatroud, A. O., and Akram, S., On dynamics of iterative techniques for nonlinear equation with applications in engineering, Math. Probl. Eng. 2020, Art. ID 5853296, 17 pp.
[16] Sharma, R., and Bahl, A., An optimal fourth order iterative method for solving nonlinear equations and its dynamics, J. Complex Anal. 2015, Art. ID 259167, 9 pp.
[17] Soleymani F., Khattri S.K. and Karimi V. S., Two new classes of optimal Jarrat type fourth-order methods, Appl. Math. Lett. 25, (2012), 847–853.
[18] Traub, J.F., Iterative methods for the solution of equations, Prentice Hall, New York, 1964.