zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Solving systems of linear fuzzy equations by parametric functions -- an improved algorithm. (English) Zbl 1121.65026
Summary: {\it J. J. Buckley} and {\it 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:
65F05Direct methods for linear systems and matrix inversion (numerical linear algebra)
15A06Linear equations (linear algebra)
15B33Matrices over special rings (quaternions, finite fields, etc.)
08A72Fuzzy algebraic structures
WorldCat.org
Full Text: DOI
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, No. 14, 858-870 (1982) · Zbl 0497.65018
[6] Golub, G. H.; Van Loan, C. F.: Matrix computations. (1996) · 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, 365-369 (1998)
[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
[10] E.E. Kerre, Fuzzy Sets and Approximate Reasoning, Xian Jiaotong University Press, Xian, People’s Republic of China, 1999.
[11] Markov, S.: Computation of algebraic solutions to interval systems via systems of coordinates. Scientific computing, validated numerics, interval methods, 103-114 (2001)
[12] Moens, D.; Vandepitte, D.: Fuzzy finite element method for frequency response function analysis of uncertain structures. Aiaa j. 40, No. 1, 126-136 (2002)
[13] Moore, R.: Interval arithmetic. (1996)
[14] Moore, R. E.: Methods and applications of interval analysis, SIAM studies in applied mathematics. (1979)
[15] A. Neumaier, Interval Methods for Systems of Equations, Encyclopedia of Mathematics and its Application, 1990. · Zbl 0715.65030
[16] Neumaier, A.: A simple derivation of the hansen -- bliek -- rohn -- ning -- kearfott enclosure for interval equations. Reliable comput. 5, No. 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, No. 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, No. 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, No. 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. Error control and adaptivity in scientific computing, 279-298 (1999) · Zbl 0943.65039
[24] Rump, S. M.; Ogita, T.: Super-fast validated solution of linear systems. J. comput. Appl. math. 199, No. 2, 199-206 (2007) · Zbl 1108.65020
[25] A. Vroman, G. Deschrijver, E.E. Kerre, Solving systems of linear fuzzy equations by parametric functions, IEEE Trans. Fuzzy Systems, forthcoming. · Zbl 1077.03033