Soft Computing

, Volume 22, Issue 9, pp 2881–2890 | Cite as

An estimation of algebraic solution for a complex interval linear system

  • Mojtaba Ghanbari
Methodologies and Application


In this paper, we introduce an algorithm for presentation of an inner estimation of the solution set of a complex interval linear system, where the coefficient matrix is a crisp complex-valued matrix and the right-hand-side vector is an interval complex-valued vector. Also, we show that under some certain conditions, the obtained inner estimation is, in fact, an algebraic solution.


Complex interval number Complex interval linear system Complex limiting factor Solution set Algebraic solution 



The author would like to thank referees for their helpful comments.

Compliance with ethical standards

Conflict of interest

The author declares that there is no conflict of interests regarding the publication of this paper.

Ethical approval

This article does not contain any studies with human participants or animals performed by the author.


  1. Alefeld G (1968) Intervallrechnung uber den komplexen Zahlen und einige Anwendungen. PhD thesis, Universität Karlsruhe, KarlsruheGoogle Scholar
  2. Alefeld G, Mayer G (1995) On the symmetric and unsymmetric solution set of interval systems. SIAM J Matrix Anal Appl 16:1223–1240MathSciNetCrossRefzbMATHGoogle Scholar
  3. Alefeld G, Mayer G (2000) Interval analysis: theory and applications. J Comput Appl Math 121:421–464MathSciNetCrossRefzbMATHGoogle Scholar
  4. Barmish BR (1994) New tools for robustness of linear systems. MacMillan, New YorkzbMATHGoogle Scholar
  5. Boche R (1966) Complex interval arithmetic with some applications. Technical report LMSC4-22-66-1, Lockheed Missiles & Space Company, SunnyvaleGoogle Scholar
  6. Candau Y, Raissi T, Ramdani N, Ibos L (2006) Complex interval arithmetic using polar form. Reliab Comput 12:1–20MathSciNetCrossRefzbMATHGoogle Scholar
  7. Djanybekov BS (2006) Interval householder method for complex linear systems. Reliab Comput 12:35–43MathSciNetCrossRefzbMATHGoogle Scholar
  8. Dreyer A (2005) Interval analysis of analog circuits with component tolerances. Shaker Verlag, Aachen. Doctoral thesis, TU KaiserslauternGoogle Scholar
  9. Friedman M, Ming M, Kandel A (1998) Fuzzy linear systems. Fuzzy Sets Syst 96:201–209MathSciNetCrossRefzbMATHGoogle Scholar
  10. Garajová E, Mečiar M (2016) Solving and visualizing nonlinear set inversion problems. Reliab Comput 22:104–115MathSciNetGoogle Scholar
  11. Gargantini I, Henrici P (1971) Circular arithmetic and the determination of polynomial zeros. Numer Math 18(4):305–320MathSciNetCrossRefzbMATHGoogle Scholar
  12. Garloff J (2009) Interval Gaussian elimination with pivot tightening. SIAM J Matrix Anal Appl 30:1761–1772MathSciNetCrossRefzbMATHGoogle Scholar
  13. Ghanbari M, Allahviranloo T, Haghi E (2012) Estimation of algebraic solution by limiting the solution set of an interval linear system. Soft Comput 16:2135–2142CrossRefzbMATHGoogle Scholar
  14. Henrici P (1971) Circular arithmetic and the determination of polynomial zeros. In: Conference on applications of numerical analysis, volume 228 of Lecture Notes in Mathematics, Dundee, pp 86–92Google Scholar
  15. Hladik M (2010) Solution sets of complex linear interval systems of equations. Reliab Comput 14:78–87MathSciNetGoogle Scholar
  16. Hladik M (2014) Strong solvability of linear interval systems of inequalities with simple dependencies. Int J Fuzzy Comput Modell 1:3–14CrossRefGoogle Scholar
  17. Kolev LV (1993) Interval methods for circuit analysis. World Scientific, SingaporeCrossRefzbMATHGoogle Scholar
  18. Kolev LV, Vladov SS (1989) Linear circuit tolerance analysis via systems of linear interval equations. In: ISYNT89 6th international symposium on networks, systems and signal processing, June 28 July 1, Zagreb, pp 57–60Google Scholar
  19. Mayer G (2006) A contribution to the feasibility of the interval Gaussian algorithm. Reliab Comput 12:79–98MathSciNetCrossRefzbMATHGoogle Scholar
  20. Popova E, Kolev L, Kramer W (2010) A Solver for complex-valued parametric linear systems. Serdica J Comput 4:123–132zbMATHGoogle Scholar
  21. Rump SM (1999) INTLAB-INTerval LABoratory. In: Csendes T (ed) Developments in reliable computing. Kluwer Academic Publishers, Dordrecht, pp 77–104CrossRefGoogle Scholar
  22. Sevastjanov P, Dymova L (2009) A new method for solving interval and fuzzy equations: linear case. Inf Sci 179:925–937MathSciNetCrossRefzbMATHGoogle Scholar
  23. Shary SP (2002) A new technique in systems analysis under interval uncertainty and ambiguity. Reliab Comput 8(5):321–418MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Aliabad Katoul BranchIslamic Azad UniversityAliabad KatoulIran

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