The Shape of the Symmetric Solution Set

  • Götz Alefeld
  • Vladik Kreinovich
  • Günter Mayer
Part of the Applied Optimization book series (APOP, volume 3)


We give a new deduction of the set of inequalities which characterize the solution set S of real linear systems Ax = b with the n × n coefficient matrix A varying between a lower bound \(\underline A \) and an upper bound \(\overline A \), and with b similarly varying between \(\underline b \) and \(\overline b \). The idea of this deduction can also be used to construct a set of inequalities which describe the so-called symmetric solution set S sym, i.e., the solution set of Ax = b with A = A T varying between the bounds \(\underline A = {\underline A ^T}\) and \(\overline A = {\overline A ^T}.\) This is the main result of our paper. We show that in each orthant S sym is the intersection of S with sets of which the boundaries are quadrics.


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  1. [1]
    G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, N.Y., 1983.zbMATHGoogle Scholar
  2. [2]
    G. Alefeld and G. Mayer, “On the Symmetric and Unsymmetric Solution Set of Interval Systems”, SIAM J. Matrix Anal Appl, 1995 (to appear).Google Scholar
  3. [3]
    H. Beeck, “Über Struktur und Abschätzungen der Lösungsmenge von linearen Gleichungssystemen mit Intervallkoeffizienten”, Computing, 1972, Vol. 10, pp. 231–244.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    D. J. Hartfiel, “Concerning the Solution Set of Ax = b where P A Q and p b q ”Numer. Math., 1980, Vol. 35, p. 355–359.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990.zbMATHGoogle Scholar
  6. [6]
    W. Oettli and W. Prager, “Compatibility of Approximate Solution of Linear Equations with Given Error Bounds for Coefficients and Right-hand Sides”, Numer. Math., 1964, Vol. 6, pp. 405–409.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    J. Rohn, “Interval Linear Systems”, Freiburg er Intervall-Berichte, 1984, Vol. 84 /7, pp. 33–58.Google Scholar
  8. [8]
    A. Seidenberg, “A New Decision Method for Elementary Algebra”, Annals of Math., 1954, Vol. 60, pp. 365–374.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    A. Tarski, A Decision Method for Elementary Algebra and Geometry, 2nd ed., Berkeley and Los Angeles, 1951.zbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Götz Alefeld
    • 1
  • Vladik Kreinovich
    • 2
  • Günter Mayer
    • 3
  1. 1.Institut für Angewandte MathematikUniversität KarlsruheKarlsruheGermany
  2. 2.Department of Computer ScienceUniversity of Texas at El PasoEl PasoUSA
  3. 3.Fachbereich MathematikUniversität RostockRostockGermany

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