On efficient matrix-free method via quasi-Newton approach for solving system of nonlinear equations
Keywords:
Matrix-free, Descent Direction, Global Convergent, Acceleration parameterAbstract
In this paper. a matrix-free method for solving large-scale system of nonlinear equations is presented. The method is derived via quasi-Newton approach, where the approximation to the Broyden's update is sufficiently done by constructing diagonal matrix using acceleration parameter. A fascinating feature of the method is that it is a matrix-free, so is suitable for solving large-scale problems. Furthermore, the convergence analysis and preliminary numerical results that is reported using a benchmark test problems, shows that the method is promising.
References
R.P. Agarwal, B. Xu and W. Zhang, Stability of functional equations in single variable, J. Math. Anal. Appl., 288 (2003), 852-869.
J.A. Baker, The stability of certain functional equations, Proc. Amer. Math. Soc., 112 (1991), 729-732.
A.S. Halilu and M.Y. Waziri, An improved derivative-free method via double direction approach for solving systems of nonlinear equations, J. of the Ramanujan Math. Soc., 33 (2018), 75-89.
J.E. Dennis and J.J. More, A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Comp., 28 (1974), 549-560.
M. Mamat, K. Muhammad and M.Y. Waziri, Trapezoidal Broyden's method for solving systems of nonlinear equations, Appl. Math. Sci., 6 (2014), 251-260.
C.G. Broyden, A class of methods for solving nonlinear simultaneous equations, Math. Comput., 19 (1965), 577-593.
M.Y. Waziri, Y.M. Kufena, and A.S. Halilu, Derivative-free three-term spectral conjugate gradient method for symmetric nonlinear equations, Thai J. Math., 18 (2020), 1417-1431.
M. Ziani and H.F. Guyomarch, An autoadaptative limited memory Broyden's method to solve systems of nonlinear equa- tions, Appl. Math. Comput., 205 (2008), 205-215.
D. Li and M. Fukushima, A global and superlinear convergent Gauss-Newton based BFGS method for symmetric nonlinear equation, SIAM J. Numer. Anal., 37 (1999), 152-172.
M.Y. Waziri and L.Y. Uba, Three-step derivative-Free diagonal updating method for solving large-scale systems of nonlinear equations, J. Numer. Math. Stoch., 6 (2014), 73-83.
W. Leong, M.A. Hassan and M.Y. Waziri, A matrix-free quasi-Newton method for solving large-scale nonlinear systems, Comput. Math. Appl., 62 (2011), 2354-2363.
A. Ramli, M.L. Abdullah and M. Mamat, Broyden's method for solving fuzzy nonlinear equations, Adv. Fuzzy Syst., (2010), Art. ID 763270, 6 pages.
A.S. Halilu and M.Y. Waziri, A transformed double step length method for solving large-scale systems of nonlinear equa- tions, J. Numer. Math. Stoch., 9 (2017), 20-32.
C.T. Kelley, Solving nonlinear equations with Newtons method, SIAM, Philadelphia (2003).
J.E. Dennis and R.B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, SIAM, Philadelphia (1993).
W. Sun and Y.X. Yuan, Optimization theory and methods; Nonlinear programming, Springer, New York (2006).
A.S. Halilu and M.Y Waziri, En enhanced matrix-free method via double step lengthapproach for solving systems of nonlinear equations, International Journal of Applied Mathematical Research, 6(4) (2017), 147-156.
A.S. Halilu,A. Majumder, M.Y. Waziri, H. Abdullahi, Double direction and step length method for solving system of nonlinear equations, Euro. J. Mol. Clinic. Med., 7(7) (2020), 3899-3913.
S. Aji, P. Kumam, A.M. Awwal, M.M. Yahaya and K. Sitthithakerngkiet, An eficient DY-type spectral conjugate gradient method for system of nonlinear monotone equations with application in signal recovery, AIMS Math., 6 (2021): 8078-8106.
A.M. Awwal, P. Kumam and H. Mohammad, Iterative algorithm with structured diagonal Hessian approximation for solving nonlinear least squares problems, J. Nonlinear Convex Anal., 22(6) (2021), 1173-1188.
A.M. Awwal, P. Kumam, K. Sitthithakerngkiet, A.M. Bakoji, A.S. Halilu and I.M. Sulaiman, Derivative-free method based on DFP updating formula for solving convex constrained nonlinear monotone equations and application, AIMS Math., 6(8) (2021), 8792-8814.
S. Aji, P. Kumam, A.M. Awwal, M.M. Yahaya and W. Kumam, Two hybrid spectral methods with inertial effect for solving system of nonlinear monotone equations with application in robotics, IEEE Access, 9 (2021), 30918-30928.
A.S. Halilu and M.Y. Waziri, Solving systems of nonlinear equations using improved double direction method, J. Nigerian Math. Soc., 39(2) (2020), 287-301.
A.M. Awwal, P. Kumam, L. Wang, S. Huang and W. Kumam, Inertial-based derivative-free method for system of monotone nonlinear equations and application, IEEE Access, 8 (2020) 226921-226930.
A.M. Awwal, L. Wang, P. Kumam, H. Mohammad and W. Watthayu, A projection Hestenes-Stiefel method with spectral parameter for nonlinear monotone equations and signal processing, Math. Comput. Appl., 25(2) (2020), Art. ID 27-28, 29 pages.
Y.B. Zhao and D. Li, Monotonicity of fixed point and normal mapping associated with variational inequality and its application, SIAM J. Optim., 11 (2001), 962-997.
M. Li, H. Liu and Z. Liu, A new family of conjugate gradient methods for unconstrained optimization, J. Appl. Math. Comput., 58 (2018) 219-234.
A.S. Halilu, and M.Y. Waziri, Inexact double step length method for solving systems of nonlinear equations, Stat. Optim. Inf. Comput., 8(1) (2020), 165-174.
H.Abdullahi, A.S. Halilu, and M.Y. Waziri, 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.
Y.H. Dai and C.X. Kou, A nonlinear conjugate gradient algorithm with an optimal property and an improved wolfe line search, SIAM J. Optim., 23 (2013), 296-320.
Y.H. Dai and L.Z. Lio, New conjugacy conditions and related nonlinear conjugate gradient methods, Appl. Math. Optim., 43 (2001), 87-101.
W.W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods, Pac. J. Optim., 2 (2006), 35-58.
A.S. Halilu, M.Y. Waziri and I. Yusuf, E?cient matrix-free direction method with line search for solving large-scale system of nonlinear equations, Yugosl. J. Oper. Res., 30(4) (2020), 399-412.
M.Y. Waziri, K. Ahmad, and A.S. Halilu, Enhanced Dai-Liao conjugate gradient methods for systems of monotone nonlinear equations, SeMA J., 78(1) (2020), 15-51.
L. Li and D. Wu, The convergence of Ishikawa iteration for generalized Φ-contractive mappings, Results in Nonlinear Analysis, 4(1) (2021), 47-56.
P. Lo'lo' and M. Shabibi, Common best proximity points theorems for H-contractive non-self mappings, Advances in the Theory of Nonlinear Analysis and its Application, 5(2) (2021), 173-179.
K. Meintjes, and A.P. Morgan, A methodology for solving chemical equilibrium systems, Appl. Math. Comput., 22 (1987), 333-361.
M. Sun, M.Y. Tian, and Y.J. Wang, Multi-step discrete-time Zhang neural networks with application to time-varying nonlinear optimization, Discrete Dyn. Nat. Soc., Art. ID 4745759, (2019) 1-14.
A.M. Awwal, L. Wang, P. Kumam, and H. Mohammad, A two-step spectral gradient projection method for system of nonlinear monotone equations and image deblurring problems, Symmetry, 12(6) (2020), Art. ID 874, 20 pages.
N. Wairojjana, N. Pakkaranang, I. Uddin, P. Kumam and A.M. Awwal, Modi?ed proximal point algorithms involving convex combination technique for solving minimization problems with convergence analysis, Optimization, 69(7-8) (2020), 1655-1680.
J.E. Dennis and R.B Schnabel, Numerical method for unconstrained optimization and non-linear equations, practice Hall, Englewood cliffs, NJ, USA, 1983.
D.W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Indust. Appl. Math., 11 (1963), 431-441.
Y.B. Musa, M.Y. Waziri, and A.S. Halilu, On computing the regularization Parameter for the Levenberg-Marquardt method via the spectral radius approach to solving systems of nonlinear equations, J. Numer. Math. Stoch., 9(1) (2017), 80-94.
A.S. Halilu, M.K. Dauda, M.Y. Waziri, and M. Mamat, A derivative-free decent method via acceleration parameter for solving systems of nonlinear equations, Open J. sci. tech., 2(3) (2019) 1-4.
A.S. Halilu, A. Majumder, M.Y. Waziri, K. Ahmed, Signal recovery with convex constrained nonlinear monotone equations through conjugate gradient hybrid approach, Math. Comput. Simulation, 187 (2021), 520-539.
M.Y. Waziri, H.U. Muhammad, A.S. Halilu, and K. Ahmad, Modified matrix-free methods for solving system of nonlinear equations, (2020) DOI: 10.1080/02331934.2020.1778689.
