Improved quasi-newton method via SR1 update for solving symmetric systems of nonlinear equations

Muhammad Kabir Dauda, Mustafa Mamat, Mohamad Afendee Mohamed, Mahammad Yusuf Waziri

Abstract


The systems of nonlinear equations emerges from many areas of computing, scientific and engineering research applications. A variety of an iterative methods for solving such systems have been developed, this include the famous Newton method. Unfortunately, the Newton method suffers setback, which includes storing  matrix at each iteration and computing Jacobian matrix, which may be difficult or even impossible to compute. To overcome the drawbacks that bedeviling Newton method, a modification to SR1 update was proposed in this study. With the aid of inexact line search procedure by Li and Fukushima, the modification was achieved by simply approximating the inverse Hessian matrix  with an identity matrix without computing the Jacobian. Unlike the classical SR1 method, the modification neither require storing  matrix at each iteration nor needed to compute the Jacobian matrix. In finding the solution to non-linear problems of the form  40 benchmark test problems were solved. A comparison was made with other two methods based on CPU time and number of iterations. In this study, the proposed method solved 37 problems effectively in terms of number of iterations. In terms of CPU time, the proposed method also outperformed the existing methods. The contribution from the methodology yielded a method that is suitable for solving symmetric systems of nonlinear equations. The derivative-free feature of the proposed method gave its advantage to solve relatively large-scale problems (10,000 variables) compared to the existing methods. From the preliminary numerical results, the proposed method turned out to be significantly faster, effective and suitable for solving large scale symmetric nonlinear equations.


Keywords


SR1, global convergence, nonlinear equations

Full Text:

PDF

References


Dauda, M. K., Mamat, M., Waziri, M. Y., Ahmad, F., Mohamad, F. S. 2016. Inexact cg-method via sr1 update for solving systems of nonlinear equations. Far East Journal of Mathematical Sciences, 100, 11.

Fukushima, M., Li, D.-H. 2000. A derivative-free line search and global convergence of broyden like methods for nonlinear equations. Optimization Methods and Software, 13, 181-201.

Fukushima, M., Li, D.-H. 2000. A globally and superlinearly convergent gauss newton-based bfgs method for symmetric nonlinear equations. SIAM Journal on Numerical Analysis, 37, 152-172.

Li, J. L., Shengjie. 2015. Spectral dy type projection method for nonlinear monotone systems of equations. Journal of Computation of Mathematics, 4, 341-354.

Mamat, M., Dauda, M., Waziri, M., Ahmad, F., Mohamad, F. S. 2016. Improved quasi-newton method via psb update for solving systems of nonlinear equations. AIP Conference Proceedings 1782, 030009.

Morales, J. L. 2008. Variational quasi-newton formulas for systems of nonlinear equations and optimization problems. Retrieved from https://www.researchgate.net/publication/265200200_Variational_quasi-Newton_formulas_for_systems_of_nonlinear_equations_and_optimization_problems

Morťe, J. J., Dolan, E. D. 2002. Benchmarking optimization software with performance profiles. Mathematical Programming, Series A, 91, 201-213.

Waziri, M. Y., Leong, W. J., Mamat, M. 2012. A two-step matrix-free secant method for solving large-scale systems of nonlinear equations. Journal of Applied Mathematics, 2012, 348654.

Wright, S. J., Nocedal, J. 2006. Numerical Optimization (second ed.). New York: Springer.

Zhang, G. Y., Maojun. 2015. A three-terms polak-ribière-polyak conjugate gradient algorithm for large-scale nonlinear equations. Journal of Computational and Applied Mathematics, 286, 186-195.

Zhou, W. 2013. A short note on the global convergence of the unmodified PRP method. Optimization Letter, 7, 1367-1372.




DOI: https://doi.org/10.11113/mjfas.v15n2019.1085

Refbacks

  • There are currently no refbacks.


Copyright (c) 2019 M.K. Dauda, Mustafa Mamat, Mohamad Afendee Mohamed, M.Y. Waziri

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.


Copyright © 2005-2019 Penerbit UTM Press, Universiti Teknologi Malaysia. Disclaimer: This website has been updated to the best of our knowledge to be accurate. However, Universiti Teknologi Malaysia shall not be liable for any loss or damage caused by the usage of any information obtained from this website.