Advertisement

Calcolo

, Volume 54, Issue 3, pp 669–683 | Cite as

Numerical validation for systems of absolute value equations

  • H. J. WangEmail author
  • D. X. Cao
  • H. LiuEmail author
  • L. Qiu
Article

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.

Keywords

Systems of absolute value equations Poincaré–Miranda theorem Error bounds 

Mathematics Subject Classification

65K05 90C30 

Notes

Acknowledgments

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

References

  1. 1.
    Rohn, J.: A theorem of the alternatives for the equation \(Ax + B|x| = b\). Linear Multilinear Algebra 52(6), 421–426 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Mangasarian, O.L., Meyer, R.R.: Absolute value equations. Linear Algebra Appl. 419(2), 359–367 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Hu, S.L., Huang, Z.H.: A note on absolute value equations. Optim. Lett. 4(3), 417–424 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Mangasarian, O.L.: Absolute value equation solution via concaveminimization. Optim. Lett. 1, 3–8 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Prokopyev, O.: On equivalent reformulations for absolute value equations. Comput. Optim. Appl. 44(3), 363–372 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  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)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Rohn, J.: An algorithm for computing all solutions of an absolute value equation. Optim. Lett. 6(5), 851–856 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  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)CrossRefzbMATHMathSciNetGoogle Scholar
  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)CrossRefzbMATHMathSciNetGoogle Scholar
  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)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3, 101–108 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  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)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Miranda, C.: Un’osservazione su un teorema di Brouwer. Bollettino dell’Unione Matematica Italiana 3, 5–7 (1940)zbMATHMathSciNetGoogle Scholar
  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)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Marco, S.: Steigungen höherer Ordnung zur verifizierten globalen Optimierung. Universitätsverlag Karlsruhe, c/o Universitätsbibliothek, Karlsruhe (2007)Google Scholar
  18. 18.
    Alefeld, G.E., Mayer, G.: Interval analysis: theory and applications. J. Comput. Appl. Math. 121, 421–464 (2000)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Italia 2016

Authors and Affiliations

  1. 1.College of ScienceChina University of Mining and TechnologyXuzhouPeople’s Republic of China
  2. 2.College of SciencesNanjing Tech UniversityNanjingPeople’s Republic of China

Personalised recommendations