Optimal Parameterised Families of Modified Householder’s Method with and without Restraint on Function Derivative

Document Type : Research Paper

Authors

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

2 Department of Statistics, Federal Polytechnic Auchi, Nigeria.

Abstract

This paper introduces two families of modified Householder’s method (HM) that are optimal in line with Kung-Traub conjecture given in [4]. The modification techniques employed involved approximation of the function derivatives in the HM with divided difference operator, a polynomial function approximation and the modified Wu function approximation in [17]. These informed the formation of two families of methods that that are optimal and do not or require function derivative evaluation. The both families do not breakdown when f(·) ≈ 0 as in the case with the HM and many existing modified HM. From the convergence investigation carried out on the methods, the sequence of approximations produced by the methods, converged to solution of nonlinear equation with order four. The implementation of the methods was illustrated and numerical results obtained were compared with that of some recently developed methods.

Keywords

References

 1. M. A. Hafiz, S. M. H. Al-Goria, Solving nonlinear equations using a new tenth-and seventh-order methods free from second derivatives, Int. J. of Differential Equations and Applications, 12, 169–183 (2013). 
2. A. S. Householder, The numerical treatment of a single nonlinear equation, McGrawHill, New York, 1970.
3. L, O. Jay, A note on Q-order of convergence BIT Numerical Math., 41, 422–429 (2001).
4. H. Kung, J. F. Traub, Optimal order of one-point and multi-point iteration, Applied Mathematics and Computation, 21, 643–651 (1974).
5. G. A. Nadeem, W. Aslam, F. Ali, An optimal fourth-order second derivative free iterative method for nonlinear scientific equations, Kuwait J. Sci., 50, 1–15 (2023).
6. W. Nazeer, A. R. Nizami, M. Tanveer, I. Sarfraz, A ninth-order iterative method for nonlinear equations along with polynomiography, J. of Prime Research in Mathematics, 13, 41–55 (2017).
7. W. Nazeer, M. Tanveer, S. M. Kang, A. Naseem, A new Householder’s method free from second derivatives for solving nonlinear equations and polynomiography, J. of Nonlinear Science and Applications, 9, 998–1007 (2016).
8. M. A. Noor, V. Gupta, Modified Householder iterative method free from second derivatives for nonlinear equations, Applied Mathematics and Computation, 190, 1701–1706 (2007).
9. O. Ogbereyivwe, J. O. Emunefe, An optimal family of methods for obtaining the zero of nonlinear equation, Mathematics and Computational Sciences , 3, 17–24 (2022).
10. O. Ogbereyivwe, O. Izevbizua, A three-parameter class of power series based iterative method for approximation of nonlinear equations solution, Iran. J. Numer. Anal. Optim., 13, 157–169 (2023).
11. O. Ogbereyivwe, S. S. Umar, O. Izevbizua, Some high-order convergence modifications of the Householder method for nonlinear equations, Communications in Nonlinear Analysis, 5, 1–16 (2023).
12. O. Ogbereyivwe, V. Ojo-Orobosa, Families of Means-Based Modified Newtons Method for Solving Nonlinear Models, Punjab University Journal of Mathematics, 53, 779–791 (2021).
13. O. Ogbereyivwe, S. S. Umar, Modified Householder iterative scheme requiring no function derivative for solving nonlinear equations, Mathematical Anal. and Convex Optimization, 3, 77–82 (2022).
14. A. M. Ostrowski, Solutions of Equations and System of Equations, Academic Press, New York, London, (1966).
15. S. A. Sarima, I. Hashim, New optimal Newton-Householder methods for solving nonlinear equations and their dynamics, Computers, Materials and Continua, 65, 69–85 (2020).
16. M. Tanveer, I. Ahmad, W. Aslam, O. Mehmood, Predictor corrector iterative method for solving nonlinear equations, Sci. Int. (Lahore), 28, 891–898 (2016).
17. X. Wu, A significant improvement on Newton’s iterative method, J. Appl. Math. Mech., 20, 924–927 (1999). 
Analytical and Numerical Solutions for Nonlinear Equations
Volume 8, Issue 1
November 2023
Pages 1-10

Files

History

  • Receive Date: 24 May 2023
  • Revise Date: 17 July 2023
  • Accept Date: 24 July 2023

Share

How to cite

Statistics