Inexact Double Step Length Method For Solving Systems Of Nonlinear Equations

  • Abubakar Sani Halilu Department of Mathematics and Computer Science, Sule Lamido University, Kafin Hausa, Nigeria
  • Mohammed Yusuf Waziri Department of Mathematical Sciences, Bayero University, Kano, Nigeria
  • Yau Balarabe Musa Department of Mathematics and Computer Science, Sule Lamido University, Kafin Hausa, Nigeria
DOI: https://doi.org/10.19139/soic-2310-5070-532
Keywords: Acceleration parameter, Double step length, Global Convergent, Inexact line Search

Abstract

In this paper, a single direction with double step length method for solving systems of nonlinear equations is presented. Main idea used in the algorithm is to approximate the Jacobian via acceleration parameter. Furthermore, the two step lengths are calculated using inexact line search procedure. This method is matrix-free, and so is advantageous when solving large-scale problems. The proposed method is proven to be globally convergent under appropriate conditions. The preliminary numerical results reported in this paper using a large-scale benchmark test problems show that the proposed method is practically quite effective.

Author Biographies

Abubakar Sani Halilu, Department of Mathematics and Computer Science, Sule Lamido University, Kafin Hausa, Nigeria
Department of Mathematics and Computer Science, Sule Lamido University, Kafin Hausa, Nigeria
Mohammed Yusuf Waziri, Department of Mathematical Sciences, Bayero University, Kano, Nigeria
Department of Mathematical Sciences, Bayero University, Kano, Nigeria
Yau Balarabe Musa, Department of Mathematics and Computer Science, Sule Lamido University, Kafin Hausa, Nigeria
Department of Mathematics and Computer Science, Sule Lamido University, Kafin Hausa, Nigeria

References

Zhen-Jun Shi and Jie Shen, Convergence of nonmonotone line search method, Journal of Computational and Applied Mathematics, vol.193, pp. 397-412, 2006.

M.Y Waziri and Jamilu Sabiu, A derivative-free conjugate gradient method and its global convergence for symmetric nonlinear equation, Journal of mathematics and mathematical sciences, vol. 2015, pp. 8, 2015.

W. Cheng and Q. Hu, A Nonmonotone Spectral Gradient Method for l1-regularized Least Squares, statistics, optimization and information computing, vol.4, pp.265-277, 2016.

M.Y Waziri and A.S. Halilu, An improved derivative-free method via double direction approach for solving systems of nonlinear equations, Journal of the Ramanujan Mathematical Society, vol. 33, no. 1, pp. 75-89, 2018.

P. N. Brown and Y. Saad, Convergence theory of nonlinear Newton-Krylov algorithms search, SIAM Journal on Optimization, vol.4, no.2, pp.297-330, 1994.

M. Y. Waziri, K. Ahmed and J. Sabiu, A Dai-Liao conjugate gradient method via modified secant equation for system of nonlinear equations, Arabian Journal of Mathematics, pp.15, 2019.

A.S Halilu and M.Y Waziri, A transformed double steplength method for for solving large-scale systems of nonlinear equations, Journal of Numerical Mathematics and Stochastics, vol.9, no. 1, pp. 20–23, 2017.

M. Y. Waziri, K. Ahmed and J. Sabiu, A family of Hager-Zhang conjugate gradient methods for system of monotone nonlinear equations, Applied Mathematics and Computation, vol.361, pp. 645-660, 2019.

M.Raydan, On Barzilai and Borwein choice of step length for the gradient method, IMA Journal of Numerical Analysis, vol.13, pp.321–326.

D.Li and M.Fukushima, A global and superlinear convergent Gauss-Newton based BFGS method for symmetric nonlinear equations,SIAM Journal on numerical Analysis, vol.37, pp. 152–172, 1999.

I. A. Osinuga and S. O. Yusuff, Quadrature based Broyden-like method for systems of nonlinear equations, statistics, optimization and information computing, vol.6, pp. 130-?38, 2018.

Gongli Yuan∗, Xiwen Lu, A new backtracking inexact BFGS method for symmetric nonlinear equations, Journal of computers and mathematics with applications, vol.55, pp. 116–129, 2008.

M.J petrovic, An accelerated double step size model in unconstrained optimization, Journal of mathematics and computation, vol.250, pp. 309–319, 2015.

J.E Dennis, Jr and R.B Schnabel, Numerical method for unconstrained optimization and non-linear equations, practice Hall, Englewood cliffs, NJ, USA, 1983.

E.Dolan and J.More, Benchmarking optimization software with performance profiles, Mathematical Programming, vol. 91, no. 2,pp. 201–213, 2002.

A.S Halilu and M.Y Waziri, Enhanced matrix-free method via double step length approach for solving systems of nonlinear equations, International journal of applied mathematical research, vol.6, no. 4, pp. 147–156, 2017.

Published
2020-02-18
How to Cite
Halilu, A. S., Yusuf Waziri, M., & Balarabe Musa, Y. (2020). Inexact Double Step Length Method For Solving Systems Of Nonlinear Equations. Statistics, Optimization & Information Computing, 8(1), 165-174. https://doi.org/10.19139/soic-2310-5070-532
Issue
Vol 8 No 1 (2020)
Section
Research Articles
Authors who publish with this journal agree to the following terms:

    1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
    2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
    3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).