Numerical validation for systems of absolute value equations

  • 248 Accesses

  • 2 Citations


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 to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.


  1. 1.

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

  2. 2.

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

  3. 3.

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

  4. 4.

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

  5. 5.

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

  6. 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)

  7. 7.

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

  8. 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)

  9. 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)

  10. 10.

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

  11. 11.

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

  12. 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. 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. 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)

  15. 15.

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

  16. 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)

  17. 17.

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

  18. 18.

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

Download references


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

Author information

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

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).

Download citation


  • Systems of absolute value equations
  • Poincaré–Miranda theorem
  • Error bounds

Mathematics Subject Classification

  • 65K05
  • 90C30