Skip to main content
Log in

Direct methods for linear systems with inexact input data

  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

We give a survey on direct methods for interval linear systems. We also consider various kinds of solution sets and show how the interval hull can be computed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Alefeld and J. Herzberger, Introduction to Interval Computations. Academic Press, New York, 1983.

    MATH  Google Scholar 

  2. G. Alefeld, V. Kreinovich and G. Mayer, On the shape of the symmetric, persymmetric and skew-symmetric solution set. SIAM J. Matrix Anal. Appl.,18 (1997), 693–705.

    Article  MATH  MathSciNet  Google Scholar 

  3. G. Alefeld, V. Kreinovich and G. Mayer, On the solution sets of particular classes of linear interval systems. J. Comp. Appl. Math.,152 (2003), 1–15.

    Article  MATH  MathSciNet  Google Scholar 

  4. G. Alefeld and G. Mayer, The Cholesky method for interval data. Linear Algebra Appl.,194 (1993), 161–182.

    Article  MATH  MathSciNet  Google Scholar 

  5. G. Alefeld and G. Mayer, On the symmetric and unsymmetric solution set of interval systems. SIAM J. Matrix Anal. Appl.,16 (1995), 1223–1240.

    Article  MATH  MathSciNet  Google Scholar 

  6. G. Alefeld and G. Mayer, Enclosing solutions of singular interval systems iteratively. Reliable Computing,11 (2005), 165–190.

    Article  MATH  MathSciNet  Google Scholar 

  7. G. Alefeld and G. Mayer, New criteria for the feasibility of the Cholesky method with interval data. SIAM J. Matrix Anal. Appl.,30 (2008), 1392–1405.

    Article  MATH  MathSciNet  Google Scholar 

  8. E.H. Bareiss, Numerical solution of linear equations with Toeplitz and vector Toeplitz matrices. Numer. Math.,13 (1969), 404–424.

    Article  MATH  MathSciNet  Google Scholar 

  9. W. Barth and E. Nuding, Optimale Lösung von Intervallgleichungssystemen, Computing,12 (1974), 117–125.

    Article  MATH  MathSciNet  Google Scholar 

  10. H. Beeck, Über Struktur und Abschätzungen der Lösungsmenge von linearen Gleichungssystemen mit Intervallkoeffizienten. Computing,10 (1972), 231–244.

    Article  MATH  MathSciNet  Google Scholar 

  11. H. Beeck, Zur scharfen Außenabschätzung der Lösungsmenge bei linearen Intervallgleichungssystemen. Z. Angew. Math. Mech.,54 (1974), T208-T209.

    MathSciNet  Google Scholar 

  12. C. Bliek, Computer methods for design automation. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1992.

    Google Scholar 

  13. A. Frommer, A feasibility result for interval Gaussian elimination relying on graph structure. Symbolic Algebraic Methods and Verification Methods, G. Alefeld, J. Rohn, S. Rump and T. Yamamoto (eds.), Springer, Wien, 2001, 79–86.

    Google Scholar 

  14. A. Frommer and G. Mayer, A new criterion to guarantee the feasibility of the interval Gaussian algorithm. SIAM J. Matrix Anal. Appl.,14 (1993), 408–419.

    Article  MATH  MathSciNet  Google Scholar 

  15. J. Garloff, Totally nonnegative interval matrices. Interval Mathematics 1980, K.L.E. Nickel (ed.), Academic Press, New York, 1980, 317–327.

    Google Scholar 

  16. J. Garloff, Solution of linear equations having a Toeplitz interval matrix as coefficient matrix. Opuscula Mathematica,2 (1986), 33–45.

    MathSciNet  Google Scholar 

  17. J. Garloff, Block methods for the solution of linear interval equations. SIAM J. Matrix Anal. Appl.,11 (1990), 89–106.

    Article  MATH  MathSciNet  Google Scholar 

  18. E.R. Hansen, Bounding the solution of interval linear equations. SIAM J. Numer. Anal.,29 (1992), 1493–1503.

    Article  MATH  MathSciNet  Google Scholar 

  19. M. Hebgen Eine scaling-invariante Pivotsuche für Intervallmatrizen. Computing,12 (1974), 99–106.

    Article  MATH  MathSciNet  Google Scholar 

  20. C. Jansson, Interval linear systems with symmetric matrices, skew-symmetric matrices and dependencies in the right hand side. Computing,46 (1991), 265–274.

    Article  MATH  MathSciNet  Google Scholar 

  21. C. Jansson, Calculation of exact bounds for the solution set of linear interval systems. Linear Algebra Appl.,251 (1997), 321–340.

    Article  MATH  MathSciNet  Google Scholar 

  22. V. Kreinovich, A. Lakeyev, J. Rohn and P. Kahl, Computational Complexity and Feasibility of Data Processing and Interval Computations. Kluwer, Dordrecht, 1998.

    MATH  Google Scholar 

  23. G. Mayer, Old and new aspects of the interval Gaussian algorithm. Computer Arithmetic, Scientific Computation and Mathematical Modelling. E. Kaucher, S.M. Markov and G. Mayer (eds.), Baltzer, Basel, 1991, 329–349.

    Google Scholar 

  24. G. Mayer, Epsilon-inflation with contractive interval functions. Appl. Math.,43 (1998), 241–254.

    Article  MATH  MathSciNet  Google Scholar 

  25. G. Mayer, A contribution to the feasibility of the interval Gaussian algorithm. Reliable Computing,12 (2006), 79–98.

    Article  MATH  MathSciNet  Google Scholar 

  26. G. Mayer, On regular and singular interval systems, J. Comp. Appl. Math.,199 (2007), 220–228.

    Article  MATH  Google Scholar 

  27. G. Mayer, On the interval Gaussian algorithm. IEEE-Proceedings of SCAN 2006, IEEE Computer Society, Washington DC, 2007.

  28. G. Mayer and L. Pieper, A necessary and sufficient criterion to guarantee the feasibility of the interval Gaussian algorithm for a class of matrices. Appl. Math.,38 (1993), 205–220.

    MATH  MathSciNet  Google Scholar 

  29. G. Mayer and J. Rohn, On the applicability of the interval Gaussian algorithm. Reliable Computing,4 (1998), 205–222.

    Article  MATH  MathSciNet  Google Scholar 

  30. J. Mayer, An approach to overcome division by zero in the interval Gaussian algorithm. Reliable Computing,8 (2002), 229–237.

    Article  MATH  MathSciNet  Google Scholar 

  31. A. Neumaier, New techniques for the analysis of linear interval equations. Linear Algebra Appl.,58 (1984), 273–325.

    Article  MATH  MathSciNet  Google Scholar 

  32. A. Neumaier, Interval Methods for Systems of Equations. Cambridge University Press, Cambridge, 1990.

    MATH  Google Scholar 

  33. A. Neumaier, A simple derivation of the Hansen-Bliek-Rohn-Ning-Kearfott enclosure for linear interval equations. Reliable Computing5 (1999), 131–136.

    Article  MATH  MathSciNet  Google Scholar 

  34. M.K. Ng, Iterative Methods for Toeplitz Systems. Oxford Univ. Press, Oxford, 2004.

    MATH  Google Scholar 

  35. S. Ning and R.B. Kearfott, A comparison of some methods for solving linear interval equations. SIAM J. Numer. Anal.,34 (1997), 1289–1305.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  37. T. Ogita, S. Oishi and Y. Ushiro, Computation of sharp rigorous componentwise error bounds for the approximate solutions of systems of linear equations. Reliable Computing,9 (2003), 229–239.

    Article  MATH  MathSciNet  Google Scholar 

  38. S. Oishi and S.M. Rump, Fast verification of solutions of matrix equations. Numer. Math.,90 (2002), 755–773.

    Article  MATH  MathSciNet  Google Scholar 

  39. K. Reichmann, Ein hinreichendes Kriterium für die Durchführbarkeit des Intervall-Gauß-Algorithmus bei Intervall-Hessenberg-Matrizen ohne Pivotsuche. Z. Angew. Math. Mech.,59 (1979), 373–379.

    Article  MATH  MathSciNet  Google Scholar 

  40. K. Reichmann, Abbruch beim Intervall-Gauss-Algorithmus. Computing,22 (1979), 355–361.

    Article  MATH  MathSciNet  Google Scholar 

  41. J. Rohn, Inner solutions of linear interval systems. Interval Mathematics 1985, K. Nickel (ed.), Lecture Notes in Computer Science,212 Springer, Berlin, 1986, 157–158.

    Google Scholar 

  42. J. Rohn, Systems of linear interval equations. Linear Algebra Appl.,126 (1989), 39–78.

    Article  MATH  MathSciNet  Google Scholar 

  43. J. Rohn, Cheap and tight bounds: The recent result by E. Hansen can be made more efficient. Interval Computations,4 (1993), 13–21.

    MathSciNet  Google Scholar 

  44. J. Rohn, Interval matrices: Singularity and real eigenvalues. SIAM J. Matrix Anal. Appl.,14 (1993), 82–91.

    Article  MATH  MathSciNet  Google Scholar 

  45. J. Rohn, On overestimation produced by the interval Gaussian algorithm. Reliable Computing,3 (1997), 363–368.

    Article  MATH  MathSciNet  Google Scholar 

  46. J. Rohn, A Handbook of Results on Interval Linear Problems. April 7, 2005, http://www.cs.cas.cz/~rohn/handbook/.

  47. J. Rohn and V. Kreinovich, Computing exact componentwise bounds on solutions of linear systems with interval data is NP-hard. SIAM J. Matrix Anal. Appl.,16 (1995), 415–420.

    Article  MATH  MathSciNet  Google Scholar 

  48. S.M. Rump, Solving algebraic problems with high accuracy. A New Approach to Scientific Computation, U.W. Kulisch and W.L. Miranker (eds.), Academic Press, New York, 1983, 51–120.

    Google Scholar 

  49. S.M. Rump, Rigorous sensitivity analysis for systems of linear and nonlinear equations. Math. Comp.,54 (1990), 721–736.

    Article  MATH  MathSciNet  Google Scholar 

  50. S.M. Rump, On the solution of interval linear systems. Computing,47 (1992), 337–353.

    Article  MATH  MathSciNet  Google Scholar 

  51. S.M. Rump, Verification methods for dense and sparse systems of equations. Topics in Validated Computations, J. Herzberger (ed.), Elsevier, Amsterdam, 1994, 63–135.

    Google Scholar 

  52. U. Schäfer, The feasibility of the interval Gaussian algorithm for arrowhead matrices. Reliable Computing,7 (2001), 59–62.

    Article  MATH  MathSciNet  Google Scholar 

  53. U. Schäfer, Two ways to extend the Cholesky decomposition to block matrices with interval entries. Reliable Computing,8 (2002), 1–20.

    Article  MATH  MathSciNet  Google Scholar 

  54. F. Schätzle, Überschätzung beim Gauss-Algorithmus für lineare Intervallgleichungssysteme. Freiburger Intervall-Berichte,84/3 (1984), 1–122.

    Google Scholar 

  55. H. Schwandt, An interval arithmetic approach for the construction of an almost globally convergent method for the solution of nonlinear Poisson equation on the unit square. SIAM J. Sci. Statist. Comput.,5 (1984), 427–452.

    Article  MATH  MathSciNet  Google Scholar 

  56. H. Schwandt, Cyclic reduction for tridiagonal systems of equations with interval coefficients on vector computers. SIAM J. Numer. Anal.,26 (1989), 661–680.

    Article  MATH  MathSciNet  Google Scholar 

  57. S.P. Shary, Optimal solution of interval linear algebraic systems, I. Interval Computations,2 (1991), 7–30.

    MathSciNet  Google Scholar 

  58. S.P. Shary, A new technique in systems analysis under interval uncertainty and ambiguity. Reliable Computing,8 (2002), 321–418.

    Article  MATH  MathSciNet  Google Scholar 

  59. G.W. Stewart, Matrix Algorithms, Vol. I: Basic Decompositions. SIAM, Philadelphia, 1998.

    Google Scholar 

  60. W.F. Trench, An algorithm for the inversion of finite Toeplitz matrices. J. Soc. Indust. Appl. Math.,12 (1964), 515–522.

    Article  MATH  MathSciNet  Google Scholar 

  61. P. Wongwises, Experimentelle Untersuchungen zur numerischen Auflösung von linearen Gleichungssystemen mit Fehlererfassung. Interval Mathematics, G. Goos and J. Hartmanis (eds.), Lecture Notes in Computer Science,29, Springer, Berlin, 1975, 316–325.

    Google Scholar 

  62. S. Zohar, The solution of a Toeplitz set of linear equations. J. Assoc. Comput. Mach.,21 (1974), 272–276.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Günter Mayer.

About this article

Cite this article

Mayer, G. Direct methods for linear systems with inexact input data. Japan J. Indust. Appl. Math. 26, 279–296 (2009). https://doi.org/10.1007/BF03186535

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03186535

Key words

Navigation