Solving systems of linear fuzzy equations by parametric functions – an improved algorithm. (English) Zbl 1121.65026

Vroman, Annelies; Deschrijver, Glad; Kerre, Etienne E.

Fuzzy Sets Syst. 158, No. 14, 1515-1534 (2007).

Summary: J. J. Buckley and Y. Qu [ibid. 43, No. 1, 33–43 (1991; Zbl 0741.65023)] proposed a method to solve systems of linear fuzzy equations. Basically, in their method the solutions of all systems of linear crisp equations formed by the \(\alpha\)-levels are calculated. We propose a new method for solving systems of linear fuzzy equations based on a practical algorithm using parametric functions in which the variables are given by the fuzzy coefficients of the system. By observing the monotonicity of the parametric functions in each variable, i.e. each fuzzy coefficient in the system, we improve the algorithm by calculating less parametric functions and less evaluations of these parametric functions. We show that our algorithm is much more efficient than the method of Buckley and Qu [loc. cit.].

MSC:

65F05	Direct numerical methods for linear systems and matrix inversion
15A06	Linear equations (linear algebraic aspects)
15B33	Matrices over special rings (quaternions, finite fields, etc.)
08A72	Fuzzy algebraic structures

Keywords:

fuzzy number; system of fuzzy linear equations; numerical examples; algorithm

Citations:

Zbl 0741.65023

References:

[1]	Abramovich, F.; Wagenknecht, M.; Khurgin, Y. I., Solution of LR-type fuzzy systems of linear algebraic equations, Busefal, 35, 86-99 (1988) · Zbl 0659.15002
[2]	Alefeld, G.; Mayer, G., The Cholesky method for interval data, Linear Algebra Appl., 194, 161-182 (1993) · Zbl 0796.65032
[3]	Buckley, J. J.; Qu, Y., Solving systems of linear fuzzy equations, Fuzzy Sets and Systems, 43, 33-43 (1991) · Zbl 0741.65023
[4]	Fuller, R., On stability in possibilistic linear equality systems with Lipschitzian fuzzy numbers, Fuzzy Sets and Systems, 34, 347-353 (1990) · Zbl 0696.15003
[5]	Gay, D. M., Solving interval linear equations, SIAM J. Numer. Anal., 19, 14, 858-870 (1982) · Zbl 0497.65018
[6]	Golub, G. H.; Van Loan, C. F., Matrix Computations (1996), Johns Hopkins University Press: Johns Hopkins University Press Baltimore, MD · Zbl 0865.65009
[7]	Hansen, E., Interval arithmetic in matrix computations, part 1, SIAM J. Numer. Anal., 2, 308-320 (1965) · Zbl 0135.37303
[8]	Hanss, M.; Willner, K.; Guidati, S., On applying fuzzy arithmetic to finite element problems, (Proceedings of the 17th International Conference of the North American Fuzzy Information Processing Society—NAFIPS ’98. Proceedings of the 17th International Conference of the North American Fuzzy Information Processing Society—NAFIPS ’98, Pensacola Beach, FL, USA (1998)), 365-369
[9]	Jansson, C., Interval linear systems with symmetric matrices, skew-symmetric matrices and dependencies in the right hand side, Computing, 46, 265-274 (1991) · Zbl 0729.65016
[11]	Markov, S., Computation of algebraic solutions to interval systems via systems of coordinates, (Kraemer, W.; Wolff von Gudenberg, J., Scientific Computing, Validated Numerics, Interval Methods (2001), Kluwer: Kluwer Dordrecht), 103-114 · Zbl 1391.65131
[12]	Moens, D.; Vandepitte, D., Fuzzy finite element method for frequency response function analysis of uncertain structures, AIAA J., 40, 1, 126-136 (2002)
[13]	Moore, R., Interval Arithmetic (1996), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ, USA
[14]	Moore, R. E., Methods and Applications of Interval Analysis, SIAM Studies in Applied Mathematics (1979), SIAM: SIAM Philadelphia, PA
[16]	Neumaier, A., A simple derivation of the Hansen-Bliek-Rohn-Ning-Kearfott enclosure for interval equations, Reliable Comput., 5, 2, 131-136 (1999) · Zbl 0936.65055
[17]	Nguyen, H. T., A note on the extension principle for fuzzy sets, J. Math. Anal. Appl., 64, 369-380 (1978) · Zbl 0377.04004
[18]	Ning, S.; Kearfott, R. B., A comparison of some methods for solving linear interval equations, SIAM J. Numer. Anal., 34, 4, 1289-1305 (1997) · Zbl 0889.65022
[19]	Oettli, W., On the solution set of linear systems with inaccurate coefficients, J. Soc. Indust. and Appl. Math.: Ser. B Numer. Anal., 2, 1, 115-118 (1965) · Zbl 0146.13404
[20]	Ohta, T.; Ogita, T.; Rump, S. M.; Oishi, S., Numerical verification method for arbitrarily ill-conditioned linear systems, Trans. Japan Soc. Indust. and Appl. Math., 15, 3, 269-287 (2005)
[21]	Popova, E., Quality of the solution sets of parameter-dependent interval linear system, Z. Angew. Math. Mech., 82, 723-727 (2002) · Zbl 1013.65042
[22]	Rohn, J., Systems of linear interval equations, Linear Algebra Appl., 126, 39-78 (1989) · Zbl 0712.65029
[23]	Rump, S. M., Verified solution of large linear and nonlinear systems, (Bulgak, H.; Zenger, C., Error Control and Adaptivity in Scientific Computing (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 279-298 · Zbl 0943.65039
[24]	Rump, S. M.; Ogita, T., Super-fast validated solution of linear systems, J. Comput. Appl. Math., 199, 2, 199-206 (2007) · Zbl 1108.65020