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Efficient matrix-free direction method with line search for solving large-scale system of nonlinear equations. (English) Zbl 1474.65166

Summary: We proposed a matrix-free direction with an inexact line search technique to solve system of nonlinear equations by using double direction approach. In this article, we approximated the Jacobian matrix by appropriately constructed matrix-free method via acceleration parameter. The global convergence of our method is established under mild conditions. Numerical comparisons reported in this paper are based on a set of large-scale test problems and show that the proposed method is efficient for large-scale problems.

MSC:

65K05 Numerical mathematical programming methods
90C53 Methods of quasi-Newton type
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[1] Broyden, C.G., “A class of methods for solving nonlinear simultaneous equations”,Mathematics of Computation, 19 (92) (1965) 577-593. · Zbl 0131.13905
[2] Duranovic-Milicic, N.I., “A multi step curve search algorithm in nonlinear optimization”, Yugoslav Journal of Operations Research, 18 (1) (2008) 47-52. · Zbl 1274.90409
[3] Dennis, J.E., and Schnabel, R.B.,Numerical Methods for Unconstrained Optimization and Non-Linear Equations, Prentice Hall, Englewood Cliffs, NJ, 1983. · Zbl 0579.65058
[4] Dolan, E., and Mor´e, J., “Benchmarking optimization software with performance profiles”, Journal of Mathematical Programming, 91 (2) (2002) 201-213. · Zbl 1049.90004
[5] Halilu, A.S., and Waziri, M.Y., “A transformed double step length method for solving largescale systems of nonlinear equations”Journal of Numerical Mathematics and Stochastics, 9 (1) (2017) 20-32. · Zbl 1387.65048
[6] Halilu, A.S., and Waziri, M.Y., “En ehanced matrix-free method via double step length approach for solving systems of nonlinear equations”,International Journal of Applied Mathematical Research, 6 (4) (2017) 147-156.
[7] Li, D., and Fukushima, M., “A global and superlinear convergent Gauss-Newton based BFGS method for symmetric nonlinear equation”,SIAM Journal of Numerical Analysis, 37 (1) (2000) 152-172. · Zbl 0946.65031
[8] Petrovic, M.J., and Stanimirovic, P.S., “Accelerated double direction method for solving unconstrained optimization problems”,Mathematical Problems in Engineering, Article ID 965104, (2014) 1-8. · Zbl 1407.90262
[9] Halilu, A.S., and Waziri, M.Y., “An improved derivative-free method via double direction approach for solving systems of nonlinear equations”,Journal of the Ramanujan Mathematical Society, 33 (1) (2018) 75-89. · Zbl 1427.65082
[10] Waziri, M.Y., and Sabiu, J., “A derivative-free conjugate gradient method and its global convergence for symmetric nonlinear equations”,International Journal of Mathematics and Mathematical Science, Article ID 961487, (2015) 1-8. · Zbl 1476.65078
[11] Waziri M.Y., Leong W.J., Hassan, M.A., and Monsi, M., “A new Newtons method with diagonal Jacobian approximation for system of nonlinear equations”,Journal of Mathematics and Statistics, 6 (3) (2010) 246-252. · Zbl 1205.65182
[12] Waziri, M.Y., Leong, W.J., and Hassan, M.A., “Jacobian-Free Diagonal Newtons Method for Solving Nonlinear Systems with Singular Jacobian”,Malasian Journal of Mathematical Science, 5 (2) (2011) 241-255. · Zbl 1244.65072
[13] Abdullahi, H., Halilu, A. S., and Waziri, M. Y., “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) (2018) 32-44. · Zbl 1438.65109
[14] Xiao, Y.H., and Zhu, H., “A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing”,Journal of Mathematical Analysis and Application, 405 (1) (2013) 310319. · Zbl 1316.90050
[15] Yana, Q.R, Penga, X.Z., and Li, D.H., “A globally convergent derivative-free method for solving large-scale nonlinear monotone equations”,Journal of Computational and Applied Mathematics, 234 (3) (2010) 649-657. · Zbl 1189.65102
[16] Marquardt, D.W., “An algorithm for least-squares estimation of nonlinear parameters”, SIAM Journal of Applied Mathematics, 11 (2) (1963) 431-441. · Zbl 0112.10505
[17] Meintjes, K., and Morgan, A.P., “A methodology for solving chemical equilibrium systems”, Applied Mathematics Compututation, 22 (4) (1987) 333361. · Zbl 0616.65057
[18] Yuan, G., and Lu, X., “A new backtracking inexact BFGS method for symmetric nonlinear equations”,Computers and Mathematics with Application, 55 (1) (2008) 116-129. · Zbl 1176.65063
[19] Levenberg, K., “A method for the solution of certain non-linear problems in least squares”, Quarterly Applied Mathematics, 2 (2) (1994) 164-166. · Zbl 0063.03501
[20] Zhou, W., and Shen, D., “An inexact PRP conjugate gradient method for symmetric nonlinear equations”,Numerical Functional Analysis and Optimization, 35 (3) (2014) 370-388. · Zbl 1320.90087
[21] Sun, M., Tian, M.Y., and Wang, Y.J., “Multi-step discrete-time Zhang neural networks with application to time-varying nonlinear optimization”,Discrete Dynamics in Nature and Society Article, Article ID 4745759, (2019) 114.
[22] Fasano, G., Lampariello, F., and Sciandrone, M., “A truncated nonmonotone Gauss-Newton method for large-scale nonlinear least-squares problems”,Computational Optimization and Application, 34 (3) (2006) 343-358. · Zbl 1122.90094
[23] Kanzow, C., Yamashita, N., and Fukushima, M., “Levenberg-Marquardt methods for constrained nonlinear equations with strong local convergence properties”,Journal Computational and Applied Mathematics, 172 (2) (2004) 375-397. · Zbl 1064.65037
[24] Yuan, Y., “Subspace methods for large scale nonlinear equations and nonlinear least squares”,Optimization and Engineering, 10 (2) (2009) 207-218. · Zbl 1171.65040
[25] Halilu, A.S., and Waziri, M.Y., “Inexact Double Step Length Method for Solving Systems of Nonlinear Equations”,Statatistics, Optimization and Information Computing, 8 (1) (2020) 165-174.
[26] Bouaricha, A., and Schnabel, R.B., “Tensor methods for large sparse systems of nonlinear equations”,Mathematical Programming, 82 (1998) 377-400. · Zbl 0951.65046
[27] Waziri, M.Y., Ahmad k., and Sabiu, J., “A family of Hager-Zhang conjugate gradient methods for system of monotone nonlinear equations”,Applied mathematics and Computation, 361 (2019) 645-660. · Zbl 1428.90163
[28] Halilu, A.S., Dauda M.K., Waziri, M.Y., “Mamat M. A derivative-free decent method via acceleration parameter for Solving systems of nonlinear equations”,Open Journal of Science and Technology, 2 (3) (2019) 1-4.
[29] Musa, Y. B., Waziri, M.Y., and Halilu, A.S., “On computing the regularization Parameter for the Levenberg-Marquardt method via the spectral radius approach to solving systems of nonlinear equations”,Journal of Numerical Mathematics and Stochastics, 9 (2017) 80 · Zbl 1438.65110
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