Numerical Algorithms

, Volume 78, Issue 2, pp 423–447 | Cite as

Enclosing the solution set of parametric interval matrix equation A(p)X = B(p)

  • Evgenija D. Popova
Original Paper


Consider the parametric matrix equation A(p)X = B(p), where the elements of the matrices A(p) and B(p) depend linearly on a number of uncertain parameters varying within given intervals. We prove that the united parametric solution sets of the matrix equation and that of the corresponding linear system with multiple right-hand sides, although different as sets, have the same interval hull. A generalization of the parametric Krawczyk iteration with low computational complexity for the matrix equation is presented. Some details improving the implementation and the application of this method are discussed. An interval method, designed by A. Neumaier and A. Pownuk for enclosing the united solution set of parametric linear systems with particular dependency structure, is generalized for arbitrary linear dependencies between the parameters and for systems with multiple right-hand sides. A new, more powerful, sufficient condition for regularity of a parametric interval matrix is proven. An important application of the linear systems with multiple right-hand sides is presented as a key methodology for feasibility in computing the interval hull of a class of united parametric solution sets that appear in practical problems.


Linear matrix equations Interval parameters Solution enclosure 

Mathematics subject classification (2010)

65G40 15A24 65F10 


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The author thanks the anonymous reviewers for their comments which helped improving the manuscript.


  1. 1.
    Dehghani-Madiseh, M., Dehghan, M.: Parametric AE-solution sets to the parametric linear systems with multiple right-hand sides and parametric matrix equation A(p)X = B(p). Numerical Algorithms 73, 245–279 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Hashemi, B., Dehghan, M.: Results concerning interval linear systems with multiple right-hand sides and the interval matrix equation A x = B. J. Comput. Appl. Math 235, 2969–2978 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Hladík, M.: Enclosures for the solution set of parametric interval linear systems. Int. J. Appl. Math. Comput. Sci. 22(3), 561–574 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Jansson, C.: Interval linear systems with symmetric matrices, skew-symmetric matrices, and dependencies in the right hand side. Computing 46, 265–274 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kolev, L.: A method for outer interval solution of parametric systems. Reliab. Comput. 10, 227–239 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kolev, L.: Componentwise determination of the interval hull solution for linear interval parameter systems. Reliab. Comput. 20, 1–24 (2014)MathSciNetGoogle Scholar
  7. 7.
    Muhanna, R.L.: Benchmarks for interval finite element computations web site (2004)
  8. 8.
    Muhanna, R.L., Mullen, R.L.: Uncertainty in mechanics problems interval-based approach. J. Eng. Mech. 127(6), 557–566 (2001)CrossRefGoogle Scholar
  9. 9.
    Neumaier, A.: Interval methods for systems of equations. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  10. 10.
    Neumaier, A., Pownuk, A.: Linear systems with large uncertainties, with applications to truss structures. Reliab. Comput. 13, 149–172 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Popova, E.D.: Strong regularity of parametric interval matrices. In: Dimovski, I.I. et al. (eds.) Mathematics and Education in Mathematics, Proceedings of the 33rd Spring Conference of the Union of Bulgarian Mathematicians, Borovets, Bulgaria, BAS, pp. 446–451 (2004)Google Scholar
  12. 12.
    Popova, E.D.: Generalizing the parametric fixed-point iteration. Proc. Appl. Math. Mech. 4, 680–681 (2004)CrossRefzbMATHGoogle Scholar
  13. 13.
    Popova, E.D.: Computer-assisted proofs in solving linear parametric problems, Post-proceedings of 12Th GAMM–IMACS Int. Symp. on Scientific Computing, Computer Arithmetic and Valiyeard Numerics. IEEE Computer Society Press, Duisburg, Germany (2006)Google Scholar
  14. 14.
    Popova, E.D.: Visualizing parametric solution sets. BIT Numer. Math. 48(1), 95–115 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Popova, E.D.: Explicit description of 2D parametric solution sets. BIT Numer. Math. 52, 179–200 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Popova, E.D.: On overestimation-free computational version of interval analysis. Int. J. Comput. Methods Eng. Sci. Mech. 14(6), 491–494 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Popova, E.D.: Improved enclosure for some parametric solution sets with linear shape. Computers and Mathematics with Applications 68(9), 994–1005 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Popova, E.D., Krämer, W.: Inner and outer bounds for the solution set of parametric linear systems. J. Comput. Appl. Math. 199, 310–316 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rama Rao, M.V., Muhanna, R.L., Mullen, R.L.: Interval finite element analysis of thin plates. In: Freitag, S., Muhanna, R.L., Mullen, R.L. (eds.) Proceedings of the NSF Workshop on Reliable Engineering Computing, Ruhr Univ. Bochum, Germany, pp. 111–130 (2016)Google Scholar
  20. 20.
    Rohn, J., Kreinovich, V.: Computing exact componentwise bounds on solutions of linear systems with interval data is NP-hard. SIAM J. Matrix Anal. Appl. 16, 415–420 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rump, S.M.: Verification methods for dense and sparse systems of equations. In: Herzberger, J. (ed.) Topics in Valiyeard Computations, pp. 63–135. Elsevier Science B. V (1994)Google Scholar
  22. 22.
    Rump, S.M.: A note on epsilon-inflation. Reliab. Comput. 4, 371–375 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rump, S.M.: Verification methods: rigorous results using floating-point arithmetic. Acta Numer 19, 287–449 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Skalna, I.: A method for outer interval solution of systems of linear equations depending linearly on interval parameters. Reliab. Comput. 12(2), 107–120 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Skalna, I.: A comparison of methods for solving parametric interval linear systems with general dependencies. Numerical Methods and Applications, Lecture Notes in Computer Science 6046, 494–501 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Skalna, I.: Strong regularity of parametric interval matrices. Linear and Multilinear Algebra. doi: 10.1080/03081087.2016.1277687 (2017)

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria

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