Outer Estimation of Generalized Solution Sets to Interval Linear Systems
Tóm tắt
The work advances a numerical technique for computing enclosures of generalized AE-solution sets to interval linear systems of equations. We develop an approach (called algebraic) in which the outer estimation problem reduces to a problem of computing algebraic solutions of an auxiliary interval equation in Kaucher complete interval arithmetic.
Tài liệu tham khảo
Alefeld, G. and Herzberger, J.: Introduction to Interval Computations, Academic Press, New York, 1983.
Collatz, L.: Funktionalanalysis und Numerische Mathematik, Springer-Verlag, Berlin—Höttingen—Heidelberg, 1964.
Gardeñes, E. and Trepat, A.: Fundamentals of SIGLA, an Interval Computing System over the Completed Set of Intervals, Computing 24 (1980), pp. 161–179.
Gardeñes, E., Trepat, A., and Mielgo, H.: Present Perspective of the SIGLA Interval System, Freiburger Intervall-Berichte (82/9) (1982), pp. 1–65.
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.
Kelling, B.: Geometrische Untersuchungen zur eigenschränkte Lösungsmenge Intervallgleichungssysteme, ZAMM 74 (1994), pp. 625–628.
Kleene, S. C.: Mathematical Logic, John Wiley, New York, 1967.
Lakeyev, A. V.: Computational Complexity of Estimation of Generalized Solution Sets to Interval Linear Systems, in: Proceedings of XI International Conference “Optimization Methods and Their Applications”, Baikal, July 5–12, 1998 (section 4), Irkutsk, 1998, pp. 115–118.
Neumaier, A.: Tolerance Analysis with Interval Arithmetic, Freiburger Intervall-Berichte (86/9) (1986), pp. 5–19.
Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990.
Ortega, J. M. and Rheinboldt, W. C.: Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
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 in the “Outer Problem” for Interval Linear Equations, Reliable Computing 3(1) (1997), pp. 103–135.
Shary, S. P.: Algebraic Approach to the Analysis of Linear Static Systems under Interval Uncertainty, Izvestiya Akademii Nauk. Control Theory and Systems (3) (1997), pp. 51–61 (in Russian).
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, Akademie Verlag, Berlin, 1996, pp. 224–233.
Shary, S. P.: Interval Gauss-Seidel Method for Generalized Solution Sets to Interval Linear Systems, in: MISC'99—Workshop on Applications of Interval Analysis to Systems and Control, Girona, Spain, February 24–26, 1999, Universitat de Girona, 1999, pp. 51–65.
Shary, S. P.: Solving the Linear Interval Tolerance Problem, Mathematics and Computers in Simulation 39 (1995), pp. 53–85.
Vatolin, A. A.: On Linear Programming Problems with Interval Coefficients, J. Comp. Mathem. and Math. Phys. 24 (1984), pp. 1629–1637 (in Russian).
Walter, E. and Pronzato, L.; Identification of Parametric Models from Experimental Data, Springer, Berlin-Heidelberg, 1997.