Interval Gauss-Seidel Method for Generalized Solution Sets to Interval Linear Systems
Tóm tắt
In the paper, we advance a numerical technique for enclosing generalized AE-solution sets to interval linear systems. The main result of the paper is an extension of the well-known interval Gauss-Seidel method to the problems of outer estimation of these generalized solution sets. We give a theoretical study of the new method, prove an optimality property for the generalized interval Gauss-Seidel iteration applied to the systems with the interval M-matrices.
Tài liệu tham khảo
Alefeld, G. and Herzberger, J.: Introduction to Interval Computations, Academic Press, New York, 1983.
Barth, W. and Nuding, E.: Optimale Lösung von Intervallgleichungssystemen, Computing 12 (1974), pp. 117-125.
Berman, A. and Plemmons, R. J.: Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
Gardeñes, E. and Trepat, A.: Fundamentals of SIGLA, an Interval Computing System over the Completed Set of Intervals, Computing 24 (1980), pp. 161-179.
Kaucher, E.: Interval Analysis in the Extended Interval Space ℝ, Computing Suppl. 2 (1980), pp. 33-49.
Kearfott, R. B.: Rigorous Global Search: Continuous Problems, Kluwer Academic Publishers, Dordrecht, 1996.
Kreinovich, V., Lakeyev, A., Rohn, J., and Kahl, P.: Computational Complexity and Feasibility of Data Processing and Interval Computations, Kluwer Academic Publishers, Dordrecht, 1997.
Lakeyev, A. V.: Computational Complexity of Estimation of the Generalized Solution Sets to Interval Linear Systems, in: Proceedings of XI International Conference “Optimization Methods and Their Applications”, Irkutsk, Baikal, July 5–12, 1998, section 4, Irkutsk, 1998, pp. 115-118 (in Russian).
Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990.
Shary, S. P.: A New Approach to the Analysis of Static Systems under Interval Uncertainty, in: Alefeld, G., Frommer, A., and Lang, B. (eds), Scientific Computing and Validated Numerics, Akademie Verlag, Berlin, 1996, pp. 118-132.
Shary, S. P.: Algebraic Approach to the Interval Linear Static Identification, Tolerance and Control Problems, or One More Application of Kaucher Arithmetic, Reliable Computing 2 (1) (1996), pp. 3-33.
Shary, S. P.: Algebraic Solutions to Interval Linear Equations and Their Applications, in: Alefeld, G. and Herzberger, J. (eds), Numerical Methods and Error Bounds, Berlin, Akademie Verlag, 1996, pp. 224-233.
Shary, S. P.: Linear Static Systems under Interval Uncertainty: Algorithms to Solve Control and Stabilization Problems, in: Reliable Computing. Supplement. Extended Abstracts of APIC'95, International Workshop on Applications of Interval Computations, El Paso, TX, 1995, pp. 181-184.
Shary, S. P.: Outer Estimation of Generalized Solution Sets to Interval Linear Systems, in: SCAN-98, IMACS/GAMM International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, Budapest, September 22–25, 1998, Volume of extended abstracts, Budapest, 1998, pp. 159-160.
Shary, S. P.: Outer Estimation of Generalized Solution Sets to Interval Linear Systems, Reliable Computing 5 (3) (1999), pp. 323-336. Also published in: Csendes, T. (ed.), Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999.