Skip to main content
SpringerLink
Log in

Momentum acceleration-based matrix splitting method for solving generalized absolute value equation

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, a momentum acceleration-based matrix splitting iteration method is presented for solving generalized absolute value equation. The convergence of the accelerated iteration method is studied in detail. And the optimal iteration parameters are studied. In particular, we present the approximate optimal iteration parameters which are independent of the number of iterations. Numerical experiments show that the proposed method with suitable parameters is efficient and accelerate the convergence performance with less CPU time and the number of iteration steps than some existing iteration methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Data Availability

Data sharing is not applicable to this article as no datasets were analyzed during the current study. In particular, the data studied were generated randomly and we explained how they were explicitly generated.

References

  • Axelsson O (1996) Iterative solution methods. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Bai Z-Z (2010) Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer Linear Algebra Appl 17(6):917–933

    MathSciNet  MATH  Google Scholar 

  • Bai Z-Z, Yang X (2009) On HSS-based iteration methods for weakly nonlinear systems. Appl Numer Math 59:2923–2936

    MathSciNet  MATH  Google Scholar 

  • Bai Z-Z, Zhang L-L (2013) Modulus-based synchronous multisplitting iteration methods for linear complementarity problems. Numer Linear Algebra Appl 20:425–439

    MathSciNet  MATH  Google Scholar 

  • Berman A, Plemmons RJ (1979) Nonnegative matrices in the mathematical sciences. Academic, New York

    MATH  Google Scholar 

  • Bhaya A, Kaszkurewicz E (2004) Steepest descent with momentum for quadratic functions is a version of the conjugate gradient method. Neural Netw 17:65–71

    MATH  Google Scholar 

  • Chung S-J (1989) NP-completeness of the linear complementarity problem. J Optim Theory Appl 60:393–399

    MathSciNet  MATH  Google Scholar 

  • Cottle RW, Pang J-S, Stone RE (1992) The Linear Complementarity Problem. Academic, San Diego

    MATH  Google Scholar 

  • Dehghan M, Hajarian M (2011) Two class of synchronous matrix multisplitting schemes for solving linear complementarity problems. J Comput Appl Math 235:4325–4336

    MathSciNet  MATH  Google Scholar 

  • Dehghan M, Shirilord A (2020) Matrix multisplitting Picard-iterative method for solving generalized absolute value matrix equation. Appl Numer Math 158:425–438

    MathSciNet  MATH  Google Scholar 

  • Dehghan M, Shirilord A (2021) On the Hermitian and skew-Hermitian splitting-like iteration approach for solving complex continuous-time algebraic Riccati matrix equation. Appl Numer Math 170:109–127

    MathSciNet  MATH  Google Scholar 

  • Dehghan M, Shirilord A (2022) The use of homotopy analysis method for solving generalized Sylvester matrix equation with applications. Eng Comput 38(301):2699–2716

    Google Scholar 

  • Dehghan M, Shirilord A (2022) A new approximation algorithm for solving generalized Lyapunov matrix equations. J Comput Appl Math 404:113898

    MathSciNet  MATH  Google Scholar 

  • Dehghan M, Shirilord A (2022) Approximating optimal parameters for generalized preconditioned Hermitian and skew-Hermitian splitting (GPHSS) method. Comput Appl Math 41:72

    MathSciNet  MATH  Google Scholar 

  • Guo P, Wu S-L, Li C-X (2019) On the SOR-like iteration method for solving absolute value equations. Appl Math Lett 97:107–113

    MathSciNet  MATH  Google Scholar 

  • Hladík M (2018) Bounds for the solutions of absolute value equations. Comput Optim Appl 69(1):243–266

    MathSciNet  MATH  Google Scholar 

  • Ke Y-F (2020) The new iteration algorithm for absolute value equation. Appl Math Lett 99:105990

    MathSciNet  MATH  Google Scholar 

  • Ke Y-F, Ma C-F (2017) SOR-like iteration method for solving absolute value equations. Appl Math Comput 311:195–202

    MathSciNet  MATH  Google Scholar 

  • Li C-X (2016) A modified generalized Newton method for absolute value equations. J Optim Theory Appl 170:1055–1059

    MathSciNet  MATH  Google Scholar 

  • Li C-X (2017) A preconditioned AOR iterative method for the absolute value equations. Int J Comput Methods 14:1–12

    MathSciNet  Google Scholar 

  • Li X, Li X-Y, Dou Y (2023) Shift-splitting fixed point iteration method for solving generalized absolute value equations. Numer Algorithm 93:695–710

    MathSciNet  MATH  Google Scholar 

  • Liao L-D, Zhang G-F, Zhu M-Z (2019) A block product preconditioner for saddle point problems. J Comput Appl Math 352:426–436

    MathSciNet  MATH  Google Scholar 

  • Mangasarian OL (2009) A generalized Newton method for absolute value equations. Optim Lett 3:101–108

    MathSciNet  MATH  Google Scholar 

  • Mangasarian OL, Meyer RR (2006) Absolute value equations. Linear Algebra Appl 419:359–367

    MathSciNet  MATH  Google Scholar 

  • Mansoori A, Erfanian M (2018) A dynamic model to solve the absolute value equations. J Comput Appl Math 333:28–35

    MathSciNet  MATH  Google Scholar 

  • Mansoori A, Eshaghnezhad M, Effati S (2017) An efficient neural network model for solving the absolute value equations. IEEE Trans Circuits Syst II Express Briefs 65(3):391–395

    Google Scholar 

  • Miao X-H, Yang J-T, Hu S, Chen J-S (2015) A generalized Newton method for absolute value equations associated with circular cones. Appl Math Comput 269:155–168

    MathSciNet  MATH  Google Scholar 

  • Miao X-H, Yang J-T, Saheya B, Chen J-S (2017) A smoothing Newton method for absolute value equation associated with second-order cone. Appl Numer Math 120:82–96

    MathSciNet  MATH  Google Scholar 

  • Nguyen CT, Saheya B, Chang Y-L, Chen J-S (2019) Unified smoothing functions for absolute value equation associated with second-order cone. Appl Numer Math 135:206–227

    MathSciNet  MATH  Google Scholar 

  • Ortega JM, Rheinboldt WC (1970) Iterative solution of nonlinear equations in several variables. Academic Press, New York

    MATH  Google Scholar 

  • Qian N (1999) On the momentum term in gradient descent learning algorithms. Neural Netw 12:145–151

    Google Scholar 

  • Rohn J (2004) A theorem of the alternatives for the equation \(Ax+B|x|=b\). Linear Multilinear Algebra 52:421–426

    MathSciNet  MATH  Google Scholar 

  • Rohn J (2009) On unique solvability of the absolute value equation. Optim Lett 3:603–606

    MathSciNet  MATH  Google Scholar 

  • Rohn J (2009) An algorithm for solving the absolute value equations. Electron J Linear Algebra 18:589–599

    MathSciNet  MATH  Google Scholar 

  • Rohn J (2009) Description of all solutions of a linear complementarity problem. Electron J Linear Algebra 18:246–252

    MathSciNet  MATH  Google Scholar 

  • Rohn J, Hooshyarbakhsh V, Farhadsefat R (2014) An iterative method for solving absolute value equations and sufficient conditions for unique solvability. Optim Lett 8:35–44

    MathSciNet  MATH  Google Scholar 

  • Rumelhart DE, Hinton GE, Williams RJ (1986) Learning representations by back-propagating errors. Nature 323:533–536

    MATH  Google Scholar 

  • Saad Y (1997) Iterative methods for sparse linear systems. PWS Press, New York

    Google Scholar 

  • Salkuyeh DK (2014) The Picard-HSS iteration method for absolute value equations. Optim Lett 8:2191–2202

    MathSciNet  MATH  Google Scholar 

  • Shen S-Q, Huang T-Z (2006) Convergence and comparison theorems for double splittings of matrices. Comput Math Appl 51:1751–1760

    MathSciNet  MATH  Google Scholar 

  • Shi X-J, Yang L, Huang Z-H (2016) A fixed point method for the linear complementarity problem arising from American option pricing. Acta Math Appl Sin Engl Ser 32:921–932

    MathSciNet  MATH  Google Scholar 

  • Wang A, Cao Y, Chen J-X (2019) Modified Newton-type iteration methods for generalized absolute value equations. J Optim Theory Appl 181:216–230

    MathSciNet  MATH  Google Scholar 

  • Wu S-L, Guo P (2018) Modulus-based matrix splitting algorithms for the quasi-complementarity problems. Appl Numer Math 132:127–137

    MathSciNet  MATH  Google Scholar 

  • Wu S-L, Li C-X (2020) A note on unique solvability of the absolute value equation. Optim Lett 14:1957–1960

    MathSciNet  MATH  Google Scholar 

  • Wu S-L, Shen S-Q (2021) On the unique solution of the generalized absolute value equation. Optim Lett 15:2017–2024

    MathSciNet  MATH  Google Scholar 

  • Wu A-L, Sun H-J, Zhang Y (2018) An SOR implicit iterative algorithm for coupled Lyapunov equations. Automatica 97:38–47

    MathSciNet  MATH  Google Scholar 

  • Zhang N-M (2013) Semi-stability of steepest descent with momentum for quadratic functions. Neural Comput 25(5):1277–1301

    MathSciNet  MATH  Google Scholar 

  • Zhang J-L, Zhang G-F, Liang Z-Z (2022) A modified generalized SOR-like method for solving an absolute value equation. Linear Multilinear Algebra 71(9):1578–1595

    MathSciNet  MATH  Google Scholar 

  • Zhou H-Y, Wu S-L, Li C-X (2021) Newton-based matrix splitting method for generalized absolute value equation. J Comput Appl Math 394:113578

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Economics and Management, Lanzhou University of Technology, Lanzhou, 730050, People’s Republic of China

    Jia-Lin Zhang

  2. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Jia-Lin Zhang, Guo-Feng Zhang & Zhao-Zheng Liang

  3. Department of Mathematics, Nanchang University, Nanchang, 330031, People’s Republic of China

    Li-Dan Liao

Authors
  1. Jia-Lin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guo-Feng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhao-Zheng Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Li-Dan Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guo-Feng Zhang.

Ethics declarations

Conflict of interest

No potential conflict of interest was reported by the author(s).

Additional information

Communicated by Jinyun Yuan.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the National Natural Science Foundation of China (nos. 11771193 and 11801242), the Jiangxi Provincial Natural Science Fund (nos. 20202BAB211002 and 20232BAB201018).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, JL., Zhang, GF., Liang, ZZ. et al. Momentum acceleration-based matrix splitting method for solving generalized absolute value equation. Comp. Appl. Math. 42, 300 (2023). https://doi.org/10.1007/s40314-023-02436-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-023-02436-1

Keywords

Mathematics Subject Classification

Search

Navigation