Skip to main content
SpringerLink
Log in

Numerical validation for systems of absolute value equations

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

This paper establishes a numerical validation test for solutions of systems of absolute value equations based on the Poincaré–Miranda theorem. In this paper, the Moore–Kioustelidis theorem is generalized for nondifferential systems of absolute value equations. Numerical results are reported to show the efficiency of the new test method.

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

Similar content being viewed by others

References

  1. Rohn, J.: A theorem of the alternatives for the equation \(Ax + B|x| = b\). Linear Multilinear Algebra 52(6), 421–426 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Mangasarian, O.L., Meyer, R.R.: Absolute value equations. Linear Algebra Appl. 419(2), 359–367 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Hu, S.L., Huang, Z.H.: A note on absolute value equations. Optim. Lett. 4(3), 417–424 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Mangasarian, O.L.: Absolute value equation solution via concaveminimization. Optim. Lett. 1, 3–8 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Prokopyev, O.: On equivalent reformulations for absolute value equations. Comput. Optim. Appl. 44(3), 363–372 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Noor, M.A., Iqbal, K.J., Noor, I., Al-Said, E.: On an iterative method for solving absolute value equations. Optim. Lett. 6(5), 1027–1033 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Rohn, J.: An algorithm for computing all solutions of an absolute value equation. Optim. Lett. 6(5), 851–856 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Rohn, J., Hooshyarbakhsh, V., Farhadsefat, R.: An iterative method for solving absolute value equations and sufficient conditions for unique solvability. Optim. Lett. 8(1), 35–44 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  9. Caccetta, L., Qu, B., Zhou, G.: A globally and quadratically convergent method for absolute value equations. Comput. Optim. Appl. 48(1), 45–58 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Wang, H., Liu, H., Cao, S.Y.: A verification method for enclosing solutions of absolute value equations. Collect. Math. 64, 17–38 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3, 101–108 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Zhu, M.Z., Zhang, G.F., Liang, Z.Z.: The nonlinear Hss-like iteration method for absolute value equations (2014). arXiv:1403.7013v2

  13. Farhad, K.H.: On generalized Traubs method for absolute value equations. J. Optim. Theory Appl. (2015). doi:10.1007/s10957-015-0712-1

  14. Iqbal, J., Iqbal, A., Arif, M.: Levenberg–Marquardt method for solving systems of absolute value equations. J. Comput. Appl. Math. 282, 134–138 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  15. Miranda, C.: Un’osservazione su un teorema di Brouwer. Bollettino dell’Unione Matematica Italiana 3, 5–7 (1940)

    MATH  MathSciNet  Google Scholar 

  16. Moore, R.E., Kioustelidis, J.B.: A simple test for accuracy of approximate solutions to nonlinear systems. SIAM J. Numer. Anal. 17(4), 521–529 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  17. Marco, S.: Steigungen höherer Ordnung zur verifizierten globalen Optimierung. Universitätsverlag Karlsruhe, c/o Universitätsbibliothek, Karlsruhe (2007)

  18. Alefeld, G.E., Mayer, G.: Interval analysis: theory and applications. J. Comput. Appl. Math. 121, 421–464 (2000)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors thank the referee for his/her valuable comments and suggestions which improved the original manuscript of this paper.

Author information

Authors and Affiliations

  1. College of Science, China University of Mining and Technology, Xuzhou, 221008, People’s Republic of China

    H. J. Wang, D. X. Cao & L. Qiu

  2. College of Sciences, Nanjing Tech University, Nanjing, 211816, People’s Republic of China

    H. Liu

Authors
  1. H. J. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. X. Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. L. Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to H. J. Wang or H. Liu.

Additional information

This work is supported by the Jiangsu Province Natural Science Foundation of China (BK20151139) and the Fundamental Research Funds for the Central Universities (2012LWA10).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, H.J., Cao, D.X., Liu, H. et al. Numerical validation for systems of absolute value equations. Calcolo 54, 669–683 (2017). https://doi.org/10.1007/s10092-016-0204-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10092-016-0204-1

Keywords

Mathematics Subject Classification

Search

Navigation