Article information
2016 , Volume 21, ¹ 5, p.111-118
Shary S.P.
New characterizations of the solution set for interval systems of linear equations
This note presents new analytical characterizations for the solution set for interval linear equation systems, which are alternatives to the well-known Oettli - Pager inequality. The new characterizations have the form of vector inequalities involving interval magnitude function. Based on the new characterization, we introduce so-called recognizing functionals of the solution set that determine, for a given point, an aggregated quantitative measure on how the point is compatible (consistent) with the interval data of the system. The recognizing functionals prove to be useful in investigation of whether the solution set is empty or not, as well as in finding the points that possess some optimality properties with respect to the interval linear equation system, e. g., in data fitting problems under interval uncertainty.
[full text] Keywords: interval linear equation, solution set, characterization, recognizing functional
Author(s): Shary Sergey Petrovich Dr. , Senior Scientist Position: Leading research officer Office: Federal Research Center for Information and Computational Technologies Address: 630090, Russia, Novosibirsk, Ac. Lavrentiev ave, 6
Phone Office: (3832) 30 86 56 E-mail: shary@ict.nsc.ru SPIN-code: 9938-9344 References: [1] Kearfott, R.B., Nakao, M.T., Neumaier A., Rump S.M ., Shary S.P., van Hentenryck P. Standardized notation in interval analysis. Computational Technologies. 2010; 15(1):713. [2] Shary, S.P. A new technique in systems analysis under interval uncertainty and ambiguity. Reliable Computing. 2002; 8(5):321418. Available at: http://www.nsc.ru/interval/shary/Papers/ANewTech.pdf [3] Oettli, W., Prager, W. Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides. Numerische Mathematik. 1964; 6(1):405409. [4] Beeck, H. Uber die Struktur und Absch¨atzungen der Losungsmenge von linearen Gleichungssystemen mit Intervallkoeffizienten. Computing. 1972; 10(3):231244. [5] Shary, S.P. Konechnomernyy interval'nyy analiz [Finite-dimensional Interval Analysis]. Novosibirsk: Institute of Computational Technologies; 2016: 617. Available at: http://www.nsc.ru/interval/index.php? j=Library/InteBooks/index ( In Russ.) [6] Neumaier, A. Interval methods for systems of equations. Cambridge: Cambridge University Press; 1990: 255. [7] Shary S.P. New characterizations for the solution set to interval linear systems of equations. Applied Mathematics and Computation. 2015; (265):570573. [8] Alefeld, G., Herzberger, J. Introduction to Interval Computations. New York: Academic Press; 1983: 345. [9] Kalmykov, S.A., Shokin, Yu.I., Yuldashev, Z.Kh. Metody interval'nogo analiza [Methods of interval analysis]. Novosibirk: Nauka; 1986: 222. ( In Russ.) [10] Moore, R.E., Kearfott, R.B., Cloud, M.J. Introduction to interval analysis. Philadelphia: SIAM; 2009: 223. [11] Shary, S.P. Solvability of interval linear equations and data analysis under uncertainty. Automation and Remote Control. 2012; 73(2):310322. DOI: 10.1134/S0005117912020099 [12] Shary, S.P., Sharaya, I.A. Recognizing solvability of interval equations and its application to data analysis. Computational Technologies. 2013; 18(3):80109. (In Russ.) [13] Shary, S.P. Maximum consistency method for data fitting under interval uncertainty. Journal of Global Optimization. 2016; 66(1):111126. DOI: 10.1007/s10898-015-0340-1 [14] Shary, S.P., Sharaya, I.A. On solvability recognition for interval linear systems of equations. Optimization Letters. 2016; 10(2):247260. DOI: 10.1007/s11590-015-0891-6 [15] Fiedler, M., Nedoma, J., Ramik, J., Rohn, J., Zimmermann, K. Linear optimization problems with inexact data. New York: Springer Science+Business Media; 2006: 214. [16] Kreinovich, V., Lakeyev, A., Rohn, J., Kahl, P. Computational complexity and feasibility of data processing and interval computations. Dordrecht: Kluwer; 1998: 459.
Bibliography link: Shary S.P. New characterizations of the solution set for interval systems of linear equations // Computational technologies. 2016. V. 21. ¹ 5. P. 111-118
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