Abstract
This is a contribution to solvability of linear interval equations and inequalities. In interval analysis we usually suppose that values from different intervals are mutually independent. This assumption can be sometimes too restrictive. In this article we derive extensions of Oettli-Prager theorem and Gerlach theorem for the case where there is a simple dependence structure between coefficients of an interval system. The dependence is given by equality of two submatrices of the constraint matrix.
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Alefeld, G., Kreinovich, V., and Mayer, G.: On Symmetric Solution Sets, in: Herzberger, J. (ed.), Inclusion Methods for Nonlinear Problems. Proceedings of the International GAMM-Workshop, Munich and Oberschleissheim, 2000, Wien, Springer, Comput. Suppl. 16 (2003), pp. 1–22.
Alefeld G., Kreinovich V., Mayer G. (1998) The Shape of the Solution Set for Systems of Interval Linear Equations with Dependent Coefficients. Math. Nachr. 192: 23–36
Fiedler M.,Nedoma J., \(\hbox{Ram\'ik}\) J.,Rohn J., Zimmermann K. (2006). LinearOptimization Problems with Inexact Data. Springer-Verlag, New York
Gerlach W. (1981) Zur Lösung linearer Ungleichungssysteme bei Störung der rechten Seite und der Koeffizientenmatrix, Math. Operationsforsch. Stat., Ser. Optimization 12: 41–43
Kolev L.V. (1993). Interval Methods for Circuit Analysis. Word Scientific, Singapore
Kolev L.V. (2004) Solving Linear Systems Whose Elements Are Nonlinear Functions of Interval Parameters. Numerical Algorithms 37: 199–212
Kolev, L. V. and Vladov, S. S.: Linear Circuit Tolerance Analysis via Systems of Linear Interval Equations, ISYNT’89 6th International Symposium on Networks, Systems and Signal Processing, June 28–July 1, Zagreb, Yugoslavia, 1989, pp. 57–69.
Oettli W., Prager W. (1964) Compatibility ofApproximate Solution of Linear Equations withGiven Error Bounds for Coefficients and Right-Hand Sides. Numer. Math. 6: 405–409
Popova E. (2001). On the Solution of Parameterised Linear Systems. In: Kraemer W., Wolff von Gudenberg J. (eds) Scientific Computing,Validated Numerics, Interval Methods. Kluwer Academic Publishers, Boston, pp. 127–138
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Hladík, M. Solution Set Characterization of Linear Interval Systems with a Specific Dependence Structure. Reliable Comput 13, 361–374 (2007). https://doi.org/10.1007/s11155-007-9033-x
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DOI: https://doi.org/10.1007/s11155-007-9033-x