On the Picard-Mann approach for hybridizing the double direction method for solving a system of nonlinear equations.

Subject Areas : Non linear Programming

Abubakar  Halilu ¹ (Sule Lamido University, Kafin Hausa)
Aliyu  kiri ² (Department of Mathematics, Bayero University, Kano)
Mohammed  Waziri ³ (Department of Mathematics, Bayero University, Kano)

Received: 2021-12-15 Accepted : 2022-07-06 Published : 2022-03-01

Keywords: Acceleration parameter, Jacobian matrix, Double direction method, Picard-Mann process,

Abstract :

In this article, the improvement of the numerical performance of the iterative scheme presented by Halilu and Waziri in [5] is considered. This is made possible by hybridizing it with Picard-Mann hybrid iterative process. In addition, the step length is calculated using the inexact line search technique. Under the preliminary conditions, the proposed method's global convergence is established. The numerical experiment shown in this paper depicts the efficiency of the proposed method, which improved the results than the double direction method [5], existing in the literature.

References:

Abdullahi, H., Halilu, A. S., & Waziri, M. Y. (2018). A modified conjugate gradient method via a double direction approach for solving large-scale symmetric nonlinear systems. Journal of Numerical Mathematics and Stochastics, 10(1), 32-44.

Dennis Jr, J. E., & Schnabel, R. B. (1983). Numerical methods for unconstrained optimization and nonlinear equations prentice-hall series in computation. Mathematics, Englewood Cliffs, NJ.

Dolan, E. D., & Moré, J. J. (2002). Benchmarking optimization software with performance profiles. Mathematical programming, 91(2), 201-213.

Halilu A.S. and Waziri M.Y.: Enhanced matrix-free method via double step length approach for solving systems of nonlinear equations. Int J app Math Res. 2017; 6:147–156 .

Halilu, A. S., & Waziri, M. Y. (2017). A transformed double step length method for solving large-scale systems of nonlinear equations. Journal of Numerical Mathematics and Stochastics, 9(1), 20-23.

Halilu, A. S., & Waziri, M. Y. (2018). An improved derivative-free method via double direction approach for solving systems of nonlinear equations. Journal of the Ramanujan Mathematical Society, 33(1), 75-89.

Halilu, A. S., & Waziri, M. Y. (2020). Solving systems of nonlinear equations using improved double direction method. Journal of the Nigerian Mathematical Society, 39(2), 287-301.

Halilu, A. S., Waziri, M. Y., & Musa, Y. B. (2020). Inexact double step length method for solving systems of nonlinear equations. Statistics, Optimization & Information Computing, 8(1), 165-174.

Khan, S. H. (2013). A Picard-Mann hybrid iterative process. Fixed Point Theory and Applications, 2013(1), 1-10.

Li, D., & Fukushima, M. (1999). A globally and superlinearly convergent gauss--Newton-based BFGS method for symmetric nonlinear equations. SIAM Journal on numerical Analysis, 37(1), 152-172.

Petrović, M. J. (2015). An accelerated double step size model in unconstrained optimization. Applied Mathematics and Computation, 250, 309-319.

Petrović, M. J., & Stanimirović, P. S. (2014). Accelerated double direction method for solving unconstrained optimization problems. Mathematical Problems in Engineering, 2014.

Petrovic´, M.J., Stanimirovic´, P.S., Kontrec, N., Mladenovic´, J. Hybrid Modification of accelerated double direction method. Math. Probl. Eng. 2018; Article D 1523267: 8 pages. doi.org/10.1155/2018/1523267.

Yuan, G., & Lu, X. (2008). A new backtracking inexact BFGS method for symmetric nonlinear equations. Computers & Mathematics with Applications, 55(1), 116-129.

Yusuf, M. W., June, L. W., & Hassan, M. A. (2011). Jacobian-free diagonal Newton’s method for solving nonlinear systems with singular Jacobian. Malaysian Journal of Mathematical Sciences, 5(2), 241-255.