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Year 2021, Volume: 5 Issue: 4, 568 - 579, 30.12.2021
https://doi.org/10.31197/atnaa.890281

Abstract

References

  • [1] R.P. Agarwal, B. Xu and W. Zhang, Stability of functional equations in single variable, J. Math. Anal. Appl., 288 (2003), 852-869.
  • [2] J.A. Baker, The stability of certain functional equations, Proc. Amer. Math. Soc., 112 (1991), 729-732.
  • [3] 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.
  • [4] 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.
  • [5] 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.
  • [6] C.G. Broyden, A class of methods for solving nonlinear simultaneous equations, Math. Comput., 19 (1965), 577-593.
  • [7] 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.
  • [8] 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.
  • [9] 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.
  • [10] 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.
  • [11] 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.
  • [12] 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.
  • [13] 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.
  • [14] C.T. Kelley, Solving nonlinear equations with Newtons method, SIAM, Philadelphia (2003).
  • [15] J.E. Dennis and R.B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, SIAM, Philadelphia (1993).
  • [16] W. Sun and Y.X. Yuan, Optimization theory and methods; Nonlinear programming, Springer, New York (2006).
  • [17] 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.
  • [18] 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.
  • [19] 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.
  • [20] 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.
  • [21] 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.
  • [22] 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.
  • [23] 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.
  • [24] 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.
  • [25] 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.
  • [26] 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.
  • [27] 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.
  • [28] 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.
  • [29] 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.
  • [30] 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.
  • [31] Y.H. Dai and L.Z. Lio, New conjugacy conditions and related nonlinear conjugate gradient methods, Appl. Math. Optim., 43 (2001), 87-101.
  • [32] W.W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods, Pac. J. Optim., 2 (2006), 35-58.
  • [33] 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.
  • [34] 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.
  • [35] L. Li and D. Wu, The convergence of Ishikawa iteration for generalized Φ-contractive mappings, Results in Nonlinear Analysis, 4(1) (2021), 47-56.
  • [36] 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.
  • [37] K. Meintjes, and A.P. Morgan, A methodology for solving chemical equilibrium systems, Appl. Math. Comput., 22 (1987), 333-361.
  • [38] 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.
  • [39] 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.
  • [40] 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.
  • [41] J.E. Dennis and R.B Schnabel, Numerical method for unconstrained optimization and non-linear equations, practice Hall, Englewood cliffs, NJ, USA, 1983.
  • [42] D.W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Indust. Appl. Math., 11 (1963), 431-441.
  • [43] 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.
  • [44] 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.
  • [45] 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.
  • [46] 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.

On efficient matrix-free method via quasi-Newton approach for solving system of nonlinear equations

Year 2021, Volume: 5 Issue: 4, 568 - 579, 30.12.2021
https://doi.org/10.31197/atnaa.890281

Abstract

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

  • [1] R.P. Agarwal, B. Xu and W. Zhang, Stability of functional equations in single variable, J. Math. Anal. Appl., 288 (2003), 852-869.
  • [2] J.A. Baker, The stability of certain functional equations, Proc. Amer. Math. Soc., 112 (1991), 729-732.
  • [3] 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.
  • [4] 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.
  • [5] 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.
  • [6] C.G. Broyden, A class of methods for solving nonlinear simultaneous equations, Math. Comput., 19 (1965), 577-593.
  • [7] 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.
  • [8] 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.
  • [9] 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.
  • [10] 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.
  • [11] 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.
  • [12] 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.
  • [13] 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.
  • [14] C.T. Kelley, Solving nonlinear equations with Newtons method, SIAM, Philadelphia (2003).
  • [15] J.E. Dennis and R.B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, SIAM, Philadelphia (1993).
  • [16] W. Sun and Y.X. Yuan, Optimization theory and methods; Nonlinear programming, Springer, New York (2006).
  • [17] 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.
  • [18] 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.
  • [19] 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.
  • [20] 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.
  • [21] 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.
  • [22] 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.
  • [23] 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.
  • [24] 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.
  • [25] 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.
  • [26] 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.
  • [27] 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.
  • [28] 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.
  • [29] 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.
  • [30] 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.
  • [31] Y.H. Dai and L.Z. Lio, New conjugacy conditions and related nonlinear conjugate gradient methods, Appl. Math. Optim., 43 (2001), 87-101.
  • [32] W.W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods, Pac. J. Optim., 2 (2006), 35-58.
  • [33] 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.
  • [34] 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.
  • [35] L. Li and D. Wu, The convergence of Ishikawa iteration for generalized Φ-contractive mappings, Results in Nonlinear Analysis, 4(1) (2021), 47-56.
  • [36] 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.
  • [37] K. Meintjes, and A.P. Morgan, A methodology for solving chemical equilibrium systems, Appl. Math. Comput., 22 (1987), 333-361.
  • [38] 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.
  • [39] 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.
  • [40] 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.
  • [41] J.E. Dennis and R.B Schnabel, Numerical method for unconstrained optimization and non-linear equations, practice Hall, Englewood cliffs, NJ, USA, 1983.
  • [42] D.W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Indust. Appl. Math., 11 (1963), 431-441.
  • [43] 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.
  • [44] 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.
  • [45] 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.
  • [46] 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.
There are 46 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Muhammad Abdullahı This is me 0000-0002-3307-4482

Abubakar Halilu This is me 0000-0002-9680-266X

Aliyu Awwal This is me 0000-0002-1040-3626

Nuttapol Pakkaranang 0000-0002-0224-4661

Publication Date December 30, 2021
Published in Issue Year 2021 Volume: 5 Issue: 4

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