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Practical Need for Algebraic (Equality-Type) Solutions of Interval Equations and for Extended-Zero Solutions

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Parallel Processing and Applied Mathematics (PPAM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10778))

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Abstract

One of the main problems in interval computations is solving systems of equations under interval uncertainty. Usually, interval computation packages consider united, tolerance, and control solutions. In this paper, we explain the practical need for algebraic (equality-type) solutions, when we look for solutions for which both sides are equal. In situations when such a solution is not possible, we provide a justification for extended-zero solutions, in which we ignore intervals of the type \([-a,a]\).

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References

  1. Chakraverty, S., Hladík, M., Behera, D.: Formal solution of an interval system of linear equations with an application in static responses of structures with interval forces. Appl. Math. Model. 50, 105–117 (2017)

    Article  MathSciNet  Google Scholar 

  2. Chakraverty, S., Hladík, M., Mahato, N.R.: A sign function approach to solve algebraically interval system of linear equations for nonnegative solutions. Fundam. Inform. 152, 13–31 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Jaulin, L., Kiefer, M., Dicrit, O., Walter, E.: Applied Interval Analysis. Springer, London (2001). https://doi.org/10.1007/978-1-4471-0249-6

    Book  Google Scholar 

  4. Kreinovich, V., Starks, S.A., Mayer, G.: On a theoretical justification of the choice of epsilon-inflation in PASCAL-XSC. Reliab. Comput. 3(4), 437–452 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lakeyev, A.: On the computational complexity of the solution of linear systems with moduli. Reliab. Comput. 2(2), 125–131 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)

    Book  MATH  Google Scholar 

  7. Nickel, K.: Die Auflösbarkeit linearer Kreisscheineb- und Intervall-Gleichingssyteme. Linear Algebra Appl. 44, 19–40 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. Rabinovich, S.G.: Measurement Errors and Uncertainty: Theory and Practice. Springer, Berlin (2005). https://doi.org/10.1007/0-387-29143-1

    MATH  Google Scholar 

  9. Ratschek, K., Sauer, W.: Linear interval equations. Computing 25, 105–115 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Sevastjanov, P., Dymova, L.: Fuzzy solution of interval linear equations. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Wasniewski, J. (eds.) PPAM 2007. LNCS, vol. 4967, pp. 1392–1399. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-68111-3_147

    Chapter  Google Scholar 

  11. Sevastjanov, P., Dymova, L.: A new method for solving interval and fuzzy equations: linear case. Inf. Sci. 17, 925–937 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Shary, S.P.: Algebraic approach to the interval linear static identification, tolerance, and control problems, or one more application of Kaucher arithmetic. Reliab. Comput. 2(1), 3–33 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Shary, S.P.: Algebraic approach in the ‘outer problem’ for interval linear equations. Reliab. Comput. 3(2), 103–135 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Shary, S.P.: A new technique in systems analysis under interval uncertainty and ambiguity. Reliab. Comput. 8, 321–418 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sheskin, D.J.: Handbook of Parametric and Nonparametric Statistical Procedures. Chapman and Hall/CRC, Boca Raton (2011)

    MATH  Google Scholar 

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Acknowledgments

This work was supported in part by the grant DEC-2013/11/B/ST6/00960 from the National Science Center (Poland), by the US National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721, and by an award “UTEP and Prudential Actuarial Science Academy and Pipeline Initiative” from Prudential Foundation.

The authors are thankful to the anonymous referees for valuable suggestions.

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Authors and Affiliations

  1. Institute of Computer and Information Science, Czestochowa University of Technology, Dabrowskiego 73, 42-200, Czestochowa, Poland

    Ludmila Dymova & Pavel Sevastjanov

  2. Computational Science Program, University of Texas at El Paso, El Paso, TX, 79968, USA

    Andrzej Pownuk & Vladik Kreinovich

Authors
  1. Ludmila Dymova
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  2. Pavel Sevastjanov
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  3. Andrzej Pownuk
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  4. Vladik Kreinovich
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Corresponding author

Correspondence to Andrzej Pownuk .

Editor information

Editors and Affiliations

  1. Czestochowa University of Technology, Czestochowa, Poland

    Roman Wyrzykowski

  2. University of Tennessee, Knoxville, Tennessee, USA

    Jack Dongarra

  3. University of Southern California, Marina Del Rey, California, USA

    Ewa Deelman

  4. Czestochowa University of Technology, Czestochowa, Poland

    Konrad Karczewski

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Dymova, L., Sevastjanov, P., Pownuk, A., Kreinovich, V. (2018). Practical Need for Algebraic (Equality-Type) Solutions of Interval Equations and for Extended-Zero Solutions. In: Wyrzykowski, R., Dongarra, J., Deelman, E., Karczewski, K. (eds) Parallel Processing and Applied Mathematics. PPAM 2017. Lecture Notes in Computer Science(), vol 10778. Springer, Cham. https://doi.org/10.1007/978-3-319-78054-2_39

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  • DOI: https://doi.org/10.1007/978-3-319-78054-2_39

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-78053-5

  • Online ISBN: 978-3-319-78054-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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