Jarratt and Jarratt-variant families of iterative schemes for scalar and system of nonlinear equations

Document Type : Research Article

Authors

• O. Ogbereyivwe ¹
• E. J. Atajeromavwo ²
• S. S. Umar ³

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

² Department of Software Engineering, Delta State University of Science and Technology, Delta State, Nigeria.

³ Department of Statistics, Federal Polytechnic, Auchi, Nigeria.

10.22067/ijnao.2024.84415.1314

Abstract

This manuscript puts forward two new generalized families of Jarratt’s iterative schemes for deciding the solution of scalar and systems of non-linear equations. The schemes involve weight functions that are based on bi-variate rational approximation polynomial of degree two in both its numerator and denominator. The convergence study conducted on the schemes, indicated that they have convergence order (CO) four in scalar space and retain the same number of CO in vector space. The numerical experiments conducted on the schemes when used to decide the solutions of some real-life nonlinear models show that they are good challengers of some well-known and robust existing iterative schemes.

Keywords

• Jarratt scheme
• Iterative scheme
• Rational approximation poly-nomial
• Nonlinear equation
• System of nonlinear equation

Main Subjects

• Numerical Analysis

---

References

[1] Behl, R., Kanwar, V. and Sharma, K.K. Optimal equi-scaled familie of Jarratt’s method for solving equations, Int. J. Comput. Math. 90 (2013) 408–422.
[2] Cordero, A., Segura, E. and Terregrosa, J.R. Behind Jarratt’s Steps: Is Jarratt’s scheme the Best version of itself?, Discrete Dynamics in Nature and Society, 2013.
[3] Grosan, C. and Abraham, A. A new approach for solving nonlinear equations systems, IEEE Trans. Syst. Man Cybern. A Syst. Hum. 38 (2008) 698–714.

[4] Jarratt, P. Some fourth-order multipoint iterative methods for solving equations, Math. Comput. 20 (1966) 434–437.
[5] Kanwar, V., Kumar, S. and Behl, R. Several new families of Jarratt’s method for for solving systems of nonlinear equations, Appl. Appl. Math. 8 (2013) 701–716.
[6] Lin, Y., Bao, L. and Jia, X. Convergence analysis of a variant of the Newton method for solving nonlinear equations, Comput. Math. Appl. 59 (2010) 2121–2127.
[7] Meintjes, K. and Morgan, A.P. Chemical equilibrium systems as numer-ical test problems, ACM Trans. Math. Softw. 16 (1990) 143–151.
[8] More, J.J. A collection of nonlinear model problems, Computational solution of nonlinear systems of equations (Fort Collins, CO, 1988), 723–762, Lectures in Appl. Math., 26, Amer. Math. Soc., Providence, RI, 1990.
[9] Morgan, A.P. Solving polynomial systems using continuation for scien-tific and engineering problems, Englewood Cliffs, Prentice-Hall, 1987.
[10] Ogbereyivwe, O. and Izevbizua, O. A three-parameter class of power series based iterative method for approximation of nonlinear equations, Iran. J. Numer. Anal. Optim. 13(2) (2023) 157–169.
[11] Ogbereyivwe, O. and Ojo-Orobosa, V. Family of optimal two-step fourth order iterative method and its extension for solving nonlinear equations, J. Interdiscip. Math. 24(5) (2022) 1347–1365.
[12] Ogbereyivwe, O., Umar, S.S. and Izevbizua, O. Some high-order conver-gence modifications of the Housedholder method for nonlinear equations, Commun. Nonlinear Anal. 5 (2023) 1–16.
[13] Ortega, J.M. and Rheinboldt, W.C. Iterative solution of nonlinear equa-tions in several variables, Academic Press, New York, 1970.
[14] Petkovic, M.S. Remarks on “On a general class of multipoint root-finding methods of high computational efficiency” , SIAM J. Numer. Anal. 49(3) (2011) 1317–1319.

[15] Shacham, M. and Kehat, E. An iteration method with memory for the solution of a non-linear equation, Chem. Eng. Sci. 27(11) (1972) 2099–2101.
[16] Sharma, R., Kumar, S. and Singh, H. A new class of derivative-free root solvers with increasing optimal convergence order and their complex dynamics, SeMA J. 80 (2) (2023) 333–352.
[17] Sivakumar, P. and Jayaraman, J. Some new higher order weighted New-ton methods for solving nonlinear equations with applications, Comput. Math. Appl. 24(2) (2023) 59.
[18] Traub. J.F. Iterative methods for the solution of equations, Prentice-Hall, New Jersey, 1964.
[19] Yaseen, S., Zafar, F. and Alsulami, H.A. An efficient Jarratt-type iter-ative method for solving nonlinear Global Positioning System problem, Axioms, 12(6) (2023) 562.