Home / Regular Issue / JST Vol. 30 (3) Jul. 2022 / JST-3275-2021

 

Spectral Gradient Method with Log-determinant Norm for Solving Non-Linear System of Equations

Yeong Lin Koay, Hong Seng Sim, Yong Kheng Goh and Sing Yee Chua

Pertanika Journal of Science & Technology, Volume 30, Issue 3, July 2022

DOI: https://doi.org/10.47836/pjst.30.3.05

Keywords: Jacobian, log-determinant norm, nonlinear systems, optimization, spectral gradient method

Published on: 25 May 2022

Solving a system of non-linear equations has always been a complex issue whereby various methods were carried out. However, most of the methods used are optimization-based methods. This paper has modified the spectral gradient method with the backtracking line search technique to solve the non-linear systems. The efficiency of the modified spectral gradient method is tested by comparing the number of iterations, the number of function calls, and computational time with some existing methods. As a result, the proposed method shows better performance and gives more stable results than some existing methods. Moreover, it can be useful in solving some non-linear application problems. Therefore, the proposed method can be considered an alternative for solving non-linear systems.

  • Abubakar, A. B., Kumam, P., & Mohammad, H. (2020). A note on the spectral gradient projection method for nonlinear monotone equations with applications. Computational and Applied Mathematics, 39(2), 1-35. https://doi.org/10.1007/s40314-020-01151-5

  • Andrei, N. (2008). An unconstrained optimization test functions collection. Advanced Modeling and Optimization, 10(1), 147-161.

  • Antonelli, L., De Simone, V., & Di Serafino, D. (2016). On the application of the spectral projected gradient method in image segmentation. Journal of Mathematical Imaging and Vision, 54(1), 106-116. https://doi.org/10.1007/s10851-015-0591-y

  • Barzilai, J., & Borwein, J. M. (1988). Two-point step size gradient methods. IMA Journal of Numerical Analysis, 8(1), 141-148. https://doi.org/10.1093/imanum/8.1.141

  • Biglari, F., & Solimanpur, M. (2013). Scaling on the spectral gradient method. Journal of Optimization Theory and Applications, 158(2), 626-635. https://doi.org/10.1007/s10957-012-0265-5

  • Broyden, C. G. (1965). A class of methods for solving nonlinear simultaneous equations. Mathematics of Computation, 19(92), 577-593.

  • Buzzi-Ferraris, G., & Manenti, F. (2013). Nonlinear systems and optimization for the chemical engineer: Solving numerical problems. John Wiley & Sons.

  • Byrd, R. H., & Nocedal, J. (1989). A tool for the analysis of quasi-Newton methods with application to unconstrained minimization. SIAM Journal of Numerical Analysis, 26, 727-739. https://doi.org/10.1137/0726042

  • Cauchy, A. (1847). Méthode générale pour la résolution des systemes d’équations simultanées [General method for solving systems of simultaneous equations]. Comptes rendus de l’Académie des Sciences, 25(1847), 536-538.

  • Chen, X., Liu, Y., Zhou, W., & Peng, X. (2017). Simplex-fruit fly optimization algorithm for solving systems of non-linear equations. In 2017 13th International Conference on Natural Computation, Fuzzy Systems and Knowledge Discovery (ICNC-FSKD) (pp. 615-620). IEEE Publishing. https://doi.org/10.1109/FSKD.2017.8393341

  • Cheng, W. Y., & Li, D. H. (2010). Spectral scaling BFGS method. Journal of Optimization Theory and Applications, 146(2), 305-319. https://doi.org/10.1007/s10957-010-9652-y

  • Cruz, W. L., & Raydan, M. (2003). Nonmonotone spectral methods for large-scale nonlinear systems. Optimization Methods and Software, 18(5), 583-599. https://doi.org/10.1080/10556780310001610493

  • Dai, Y. H., Hager, W. W., Schittkowski, K., & Zhang, H. (2006). The cyclic Barzilai-Borwein method for unconstrained optimization. IMA Journal of Numerical Analysis, 26(3), 604-627. https://doi.org/10.1093/imanum/drl006

  • De Asmundis, R., di Serafino, D., Riccio, F., & Toraldo, G. (2013). On spectral properties of steepest descent methods. IMA Journal of Numerical Analysis, 33(4), 1416-1435. https://doi.org/10.1093/imanum/drs056

  • Dolan, E. D., & Moré, J. J. (2002). Benchmarking optimization software with performance profiles. Mathematical Programming, 91(2), 201-213. https://doi.org/10.1007/s101070100263

  • Fang, X., Ni, Q., & Zeng, M. (2018). A modified quasi-Newton method for nonlinear equations. Journal of Computational and Applied Mathematics, 328, 44-58. https://doi.org/10.1016/j.cam.2017.06.024

  • Grosan, C., & Abraham, A. (2008). A new approach for solving nonlinear equations systems. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 38(3), 698-714. https://doi.org/10.1109/TSMCA.2008.918599

  • Hestenes, M. R., & Stiefel, E. (1952). Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards, 49(6), 409-436.

  • Ibrahim, S. M., Yakubu, U. A., & Mamat, M. (2020). Application of spectral conjugate gradient methods for solving unconstrained optimization problems. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 10(2), 198-205. https://doi.org/10.11121/ijocta.01.2020.00859

  • Luenberger, D. G., & Ye, Y. (1984). Linear and nonlinear programming (Vol. 2). Addison-Wesley.

  • Marini, F. (2009). Neural networks. In R. Tauler & B. Walczak (Eds.), Comprehensive Chemometrics: Chemical and Biochemical Data Analysis (pp. 477-505). Elsevier. https://doi.org/10.1016/B978-044452701-1.00128-9

  • Martinez, J. M. (2000). Practical quasi-Newton methods for solving nonlinear systems. Journal of Computational and Applied Mathematics, 124(1-2), 97-121. https://doi.org/10.1016/S0377-0427(00)00434-9

  • Raydan, M. (1993). On the Barzilai and Borwein choice of steplength for the gradient method. IMA Journal of Numerical Analysis, 13(3), 321-326. https://doi.org/10.1093/imanum/13.3.321

  • Raydan, M. (1997). The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM Journal on Optimization, 7(1), 26-33. https://doi.org/10.1137/S1052623494266365

  • Raydan, M., & Svaiter, B. F. (2002). Relaxed steepest descent and Cauchy-Barzilai-Borwein method. Computational Optimization and Applications, 21(2), 155-167. https://doi.org/10.1023/A:1013708715892

  • Sim, H. S., Leong, W. J., & Chen, C. Y. (2019). Gradient method with multiple damping for large-scale unconstrained optimization. Optimization Letters, 13(3), 617-632. https://doi.org/10.1007/s11590-018-1247-9

  • Simpson, T. (1740). Essays on several curious and useful subjects, in speculative and mix’d mathematics. London, 1740, Article 81.

  • Solodov, M. V., & Svaiter, B. F. (1998). A globally convergent inexact Newton method for systems of monotone equations. In Reformulation: Nonsmooth, piecewise smooth, semismooth and smoothing methods (pp. 355-369). Springer. https://doi.org/10.1007/978-1-4757-6388-1_18

  • Turgut, O. E., Turgut, M. S., & Coban, M. T. (2014). Chaotic quantum behaved particle swarm optimization algorithm for solving nonlinear system of equations. Computers & Mathematics with Applications, 68(4), 508-530. https://doi.org/10.1016/j.camwa.2014.06.013

  • Wallis, J. (1095). A treatise of algebra, both historical and practical. Philosophical Transactions of the Royal Society of London, 15(173), 1095-1106. https://doi.org/10.3931/e-rara-8842

  • Xiao, Y., Wang, Q., & Wang, D. (2010). Notes on the Dai–Yuan–Yuan modified spectral gradient method. Journal of Computational and Applied Mathematics, 234(10), 2986-2992. https://doi.org/10.1016/j.cam.2010.04.012

  • Yuan, G., & Lu, X. (2008). A new backtracking inexact BFGS method for symmetric nonlinear equations. Computers & Mathematics with Applications, 55(1), 116-129. https://doi.org/10.1016/j.camwa.2006.12.081

  • Zhang, L., & Zhou, W. (2006). Spectral gradient projection method for solving nonlinear monotone equations. Journal of Computational and Applied Mathematics, 196(2), 478-484. https://doi.org/10.1016/j.cam.2005.10.002

ISSN 0128-7680

e-ISSN 2231-8526

Article ID

JST-3275-2021

Download Full Article PDF

Share this article

Recent Articles