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Solving systems of two-sided (max, min)-linear equations. (English) Zbl 1195.65037

Summary: A finite iteration method for solving systems of (max, min)-linear equations is presented. The systems have variables on both sides of the equations. The algorithm has polynomial complexity and may be extended to wider classes of equations with a similar structure.

MSC:

65F10 Iterative numerical methods for linear systems
08A72 Fuzzy algebraic structures
65Y20 Complexity and performance of numerical algorithms
15A80 Max-plus and related algebras
15B15 Fuzzy matrices
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References:

[1] Baccelli, F. L., Cohen, G., Olsder, G. J., Quadrat, J. P.: Synchronization and Linearity. An Algebra for Discrete Event Systems. Wiley, Chichester, 1992. · Zbl 0824.93003
[2] Butkovič, P., Hegedüs, G.: An elimination method for finding all solutions of the system of linear equations over an extremal algebra. Ekonomicko-matematický obzor 20 (1984), 203-215. · Zbl 0545.90101
[3] Butkovič, P., Zimmermann, K.: A strongly polynomial algorithm for solving two-sided linear systems in max-algebra. Discrete Applied Mathematics 154 (2006), 437-446. · Zbl 1090.68119 · doi:10.1016/j.dam.2005.09.008
[4] Cechlárová, K.: Efficient computation of the greatest eigenvector in fuzzy algebra. Tatra Mt. Math. Publications 12 (1997), 73-79. · Zbl 0963.65041
[5] Cechlárová, K.: Eigenvectors of interval matrices over max-plus algebra. Discrete Applied Mathematics 150 (2005), Nos. 1-3, 2-15. · Zbl 1086.15009 · doi:10.1016/j.dam.2005.02.016
[6] Cuninghame-Green, R. A.: Minimax Algebra. (Lecture Notes in Economics and Mathematical Systems 166.) Springer-Verlag, Berlin 1979. · Zbl 0739.90073 · doi:10.1016/0165-0114(91)90130-I
[7] Cuninghame-Green, R. A., Butkovič, P.: The equation \(A \otimes x = B \otimes y\) over (max,+). Theoretical Computer Science 293 (2003), 3-12. · Zbl 1021.65022 · doi:10.1016/S0304-3975(02)00228-1
[8] Cuninghame-Green, R. A., Zimmermann, K.: Equation with residual functions. Comment. Math. Univ. Carolinae 42 (2001), 729-740. · Zbl 1068.93039
[9] Sanchez, E.: Resolution of eigen fuzzy sets equations. Fuzzy Sets and Systems 1 (1978), 69-74. · Zbl 0366.04001 · doi:10.1016/0165-0114(78)90033-7
[10] Sanchez, E.: Inverses of fuzzy relations. Applications to possibility distributions and medical diagnosis. Fuzzy Sets and Systems 1 (1978), 75-86. · Zbl 0399.03040 · doi:10.1016/0165-0114(79)90017-4
[11] Vorobjov, N. N.: Extremal algebra of positive matrices (in Russian). Datenverarbeitung und Kybernetik 3 (1967), 39-71.
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