Solution sets for systems of linear interval eguations

• Autorzy: Kulpa Z. , Rosłaniec K.
• Konferencja: Polish Conference on Computer methods in mechanics ; (14 ; 26-28.05.1999 ; Rzeszów, Poland
• Języki publikacji: EN

Abstrakty

EN

The paper discusses various classes of solution sets for linear interval systems of equations, and their properties. Interval methods constitute an important mathematical and computational tool for modelling real-world systems (especially mechanical) with (bounded) uncertainties of parameters, and for controlling rounding errors in computations. They are in principle much simpler than general probabilistic or fuzzy set formulation, while in the same time they conform very well with many practical situations. Linear interval systems constitute an important subclass of such interval models, still in the process of continuous development. Two important problems in this area are discussed in more detail - the classification of so-called united solution sets, and the problem of overestimation of interval enclosures (in the context of linear systems of equations called also a matrix coefficient dependence problem).

Słowa kluczowe

EN

solution set; linear interval equation

PL

rozwiązywanie zbioru; liniowe równanie całkowe

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Computer Assisted Mechanics and Engineering Sciences
• Rocznik: 2000
• Tom: Vol. 7, No. 4
• Strony: 625--639
• Opis fizyczny: Bibliogr. 26 poz., rys., wykr.

Twórcy

autor

Kulpa Z.

• Institute of Fundamental Technological Research, Polish Academy of Sciences, ul. Świętokrzyska 21, 00-049 Warsaw, Poland

autor

Rosłaniec K.

• Institute of Fundamental Technological Research, Polish Academy of Sciences, ul. Świętokrzyska 21, 00-049 Warsaw, Poland

Bibliografia

• [1] G. Alefeld, J. Herzberger. Introduction to Interval Computations. Academic Press, New York, 1983.
• [2] G. Alefeld, V. Kreinovich, G. Mayer. The shape of the symmetric solution set. In: R.B. Kearfott, V. Kreinovich, eds., Applications of Interval Computations. Kluwer Academic Publishers, Dordrecht 1996.
• [3] J. J. Buckley, Y. Qu. On using a-cuts to evaluate fuzzy equations. Fuzzy Sets and Systems, 38: 309-312, 1990.
• [4] E. Hansen. Global Optimization Using Interval Analysis. Marcel Dekker, New York, 1992.
• [5] D. J. Hartfiel, Concerning the solution set of Ax = b where P < A < Q and p < b < q. Numerische Mathematik, 1980
• [6] ] J. Kucwaj, Unstructured Grid Generation Package. Cracow University of Technology, Applied Mathematics Section, Report No 1/1997, September 1997.
• [7] H. U. Koyliioglu, A. S. Cakmak, S. R. K. Nielsen. Interval mapping in structural mechanics. In: Spanos, ed., Computational Stochastic Mechanics. 125-133, Balkema, Rotterdam 1995.
• [8] V. Kreinovich, A. Lakeyev, J. Rohn, P. Kahl. Computational Complexity and Feasibility of Data Processing and Interval Computations. Kluwer Academic Publishers, Dordrecht, 1998.
• [9] Z. Kulpa. Diagrammatic representation and reasoning. Machine GRAPHICS e.4 VISION, 3(1-2): 77-103, 1994.
• [10] Z. Kulpa. Diagrammatic representation for a space of intervals. Machine GRAPHICS E4 VISION, 6(1): 5-24, 1997.
• [11] Z. Kulpa. Diagrammatic representation of interval space in proving theorems about interval relations. Reliable Computing, 3(3): 209-217, 1997.
• [12] Z. Kulpa. Diagrammatic representation of interval space. Part I: Basics; Part II: Arithmetic. Internal Report DIAR-1/99 and DIAR-2/99, IFTR PAS, Warsaw, 1999.
• [13] Z. Kulpa. Diagrammatic representation for interval arithmetic. Linear Algebra Appl., 2001 (to appear).
• [14] Z. Kulpa, A. Pownuk, I. Skalna. Analysis of linear mechanical structures with uncertainties by means of interval methods. CA MES, 5(4): 443-477, 1998.
• [15] Z. Kulpa, A. Radomski, O. Gajl, M. Kleiber, I. Skalna. Hybrid expert system for qualitative and quantitative analysis of truss structures. Engin. Appl. Artif. Intell., 12: 229-240, 1999.
• [16] R. E. Moore. Interval Analysis. Prentice Hall, Englewood Cliffs, NJ, 1966.
• [17] A. Neumaier. Interval methods for systems of equations. Cambridge University Press, Cambridge, 1990.
• [18] W. Oettli, W. Prager. Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides. Numer. Math., 6: 405-409, 1964.
• [19] S. S. Rao, L. Berke. Analysis of uncertain structural systems using interval analysis. AIAA Journal, 35(4): 727-735, 1997.
• [20] J. Rohn. Systems of linear interval equations. Linear Algebra Appl., 126: 39-78, 1989.
• [21] J. Rohn. Cheap and tight bounds: The recent result by E. Hansen can be made more efficient. Interval Computations, 4: 13-21, 1993.
• [22] S. M. Rump. Verification methods for dense and sparse systems of equations. In: J. Herzberger, ed., Topics in Validated Computations. 63-135, Elsevier Science By., 1994.
• [23] S. P. Shary. Algebraic approach to the interval linear static identification, tolerance, and control problems, or one more application of Kaucher arithmetic. Reliable Computing, 2(1): 3-33, 1996.
• [24] Yu. I. Shokin. On interval problems, interval algorithms and their computational complexity. In: G. Alefeld, A. Frommer, B. Lang, eds., Scientific Computing and Validated Numerics. 314-328, Akademie-Verlag, Berlin 1996
• [25] J. Skrzypczyk. Fuzzy finite element methods—A new methodology. In: Computer Methods in Mechanics (Proc. XIII Polish Conference on Computer Methods in Mechanics, Poznań, Poland, May 5-8, 1997), 4: 1187-1194. Poznań University of Technology, Poznań 1997.
• [26] M. Warmus. Calculus of approximations. Bull. Acad. Pol. Sci., Cl. III, 4(5): 253-259, 1956.