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Interval Linear Systems

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Encyclopedia of Optimization

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Article Outline

Keywords

An Iterative Interval Method

Optimal Bounds

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References

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References

  1. Alefeld G (1994) Inclusion methods for systems of nonlinear equations. In: Herzberger J (ed) Topics in Validated Computations: Studies in Computational Mathematics. North-Holland, Amsterdam, pp 7–26

    Google Scholar 

  2. Alefeld G, Herzberger J (1983) Introduction to interval computations Acad. Press, New York

    MATH  Google Scholar 

  3. Arithmos (1986) Benutzerhandbuch, Siemens, Bibl.-Nr. U 2900-I-Z87-1.

    Google Scholar 

  4. Hammer R, Hocks M, Kulisch U, Ratz D (1993) PASCAL-XSC: Basic numerical problems Springer, Berlin

    MATH  Google Scholar 

  5. Hansen ER (1992) Bounding the solution set of interval linear systems. SIAM J Numer Anal 29:1493–1503

    Article  MathSciNet  MATH  Google Scholar 

  6. Hansen E, Sengupta S (1981) Bounding solutions of systems of equations using interval analysis. BIT 21:203–211

    Article  MathSciNet  MATH  Google Scholar 

  7. Hansen E, Smith R (1967) Interval arithmetic in matrix computations. SIAM Numer Anal 2(4):1–9

    Article  MathSciNet  Google Scholar 

  8. IBM (1986) High-accuracy arithmetic subroutine library (ACRITH). Program Description and User's Guide SC 33-6164-02

    Google Scholar 

  9. Jansson C (1997) Calculation of exact bounds for the solution set of linear interval equations. Linear Alg & Its Appl 251:321–340

    Article  MathSciNet  MATH  Google Scholar 

  10. Jansson C, Rohn J (1999) An algorithm for checking regularity of interval matrices. SIAM J Matrix Anal Appl 20(3):756–776

    Article  MathSciNet  MATH  Google Scholar 

  11. Kearfott RB (1990) Preconditioners for the interval-Gauss-Seidel method. SIAM J Numer Anal 27(3):804–822

    Article  MathSciNet  MATH  Google Scholar 

  12. Kearfott RB (1996) Rigorous global search: continuous problems. Kluwer, Dordrecht

    MATH  Google Scholar 

  13. Knüppel O (1994) PROFIL/BIAS: A fast interval library. Computing 53:277–287

    Article  MathSciNet  MATH  Google Scholar 

  14. Krawczyk R (1969) Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing 4:187–201

    Article  MathSciNet  MATH  Google Scholar 

  15. Moore RE (1966) Interval analysis. Prentice-Hall, Englewood Cliffs, NJ

    MATH  Google Scholar 

  16. Moore RE (1979) Methods and applications of interval analysis. SIAM, Philadelphia

    MATH  Google Scholar 

  17. Neumaier A (1984) New techniques for the analysis of linear interval equations. Linear Alg & Its Appl 58:273–325

    Article  MathSciNet  MATH  Google Scholar 

  18. Neumaier A (1990) Interval methods for systems of equations. Encycl Math Appl. Cambridge Univ. Press, Cambridge

    MATH  Google Scholar 

  19. Oettli W (1965) On the solution set of a linear system with inaccurate coefficients. SIAM J Numer Anal 2:115–118

    MathSciNet  Google Scholar 

  20. Oettli W, Prager W (1964) Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides. Numer Math 6:405–409

    Article  MathSciNet  MATH  Google Scholar 

  21. Papadimitriou CH, Steiglitz K (1982) Combinatorial optimization: Algorithms and complexity. Prentice-Hall, Englewood Cliffs, NJ

    MATH  Google Scholar 

  22. Ris FN (1972) Interval analysis and applications to linear algebra. PhD Thesis Oxford Univ.

    Google Scholar 

  23. Rohn J (1986) A note on the sign-accord algorithm. Freiburger Intervall-Ber 86(4):39–43

    Google Scholar 

  24. Rohn J (1989) Systems of linear interval equations. Linear Alg & Its Appl 126:39–78

    Article  MathSciNet  MATH  Google Scholar 

  25. Rohn J (1991) Linear interval equations: computing enclosures with bounded relative or absolute overestimation is NP-hard. In: Kearfott RB, Kreinovich V (eds) Applications of Interval Computations. Kluwer, Dordrecht, pp 81–89

    Google Scholar 

  26. Rohn J (1992) Cheap and tight bounds: the result of E. Hansen can be made more efficient. Interval Comput 4:13–21

    Google Scholar 

  27. Rohn J (1994) NP-hardness results for linear algebraic problems with interval data. In: Herzberger J (ed) Topics in Validated Computations: Studies in Computational Mathematics. Elsevier, Amsterdam, 463–472

    Google Scholar 

  28. Rump SM (1983) Solving algebraic problems with high accuracy. Habilitationsschrift. In: Kulisch UW, Miranker WL (eds) A New Approach to Scientific Computation. Acad. Press, New York, pp 51–120

    Google Scholar 

  29. Rump SM (1993) Validated solution of large linear systems. In: Albrecht R, Alefeld G, Stetter HJ (eds) Computing Supplementum 9, Validation Numerics. Springer, Berlin, pp 191–212

    Google Scholar 

  30. Rump SM (1994) Verification methods for dense and sparse systems of equations. In: Herzberger J (ed) Topics in Validated Computations: Studies in Computational Mathematics. Elsevier, Amsterdam, pp 63 –136

    Google Scholar 

  31. Shary SP (1991) Optimal solutions of interval linear algebraic systems. Interval Comput 2:7–30

    MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Informatik III, Techn. Universität Hamburg-Harburg, Hamburg, Germany

    C. Jansson

Authors
  1. C. Jansson
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Editor information

Editors and Affiliations

  1. Department of Chemical Engineering, Princeton University, Princeton, NJ, 08544-5263, USA

    Christodoulos A. Floudas

  2. Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 32611-6595, USA

    Panos M. Pardalos

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© 2008 Springer-Verlag

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Jansson, C. (2008). Interval Linear Systems . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_307

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