## Iterative solution of fuzzy linear systems.(English)Zbl 1137.65336

One of the major applications using fuzzy number arithmetic is treating linear systems whose parameters are all or partially represented by fuzzy numbers. For solving a general fuzzy linear system, an useful idea proposed by Friedman et al. is the embedding method in which the original fuzzy system is transformed to a doubly-dimensional ordinary system and then a classical iteration such as Jacobi iteration is applied. Using this idea, in this paper, several well-known iterative methods (SOR, AOR, and so on) and their extrapolated form are extended for solving fuzzy linear systems. Convergence theorems are proved. Some numerical examples are presented to illustrate these algorithms.

### MSC:

 65F10 Iterative numerical methods for linear systems 15A06 Linear equations (linear algebraic aspects)
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### References:

 [1] Allahviranloo, T., Numerical methods for fuzzy system of linear equations, Appl. Math. Comput., 155, 493-502 (2004) · Zbl 1067.65040 [2] Allahviranloo, T., Successive overrelaxation iterative method for fuzzy system of linear equations, Appl. Math. Comput., 162, 189-196 (2005) · Zbl 1062.65037 [3] Avdelas, G.; Hadjidimos, A., Optimum accelerated overrelaxation method in a special case, Math. Comput., 36, 183-187 (1981) · Zbl 0463.65020 [4] Chang, S. L.; Zadeh, L. A., On fuzzy mapping and control, IEEE Trans., Syst. Man Cyb., 2, 30-34 (1972) · Zbl 0305.94001 [5] Datta, B. N., Numerical Linear Algebra and Applications (1995), Brooks/Cole Pub.: Brooks/Cole Pub. California [6] DeMarr, R., Nonnegative matrices with nonnegative inverses, Proc. Amer. Math. Soc., 307-308 (1972) · Zbl 0257.15002 [7] Evans, D. J., The extrapolated modified Aitken iteartion method for solving elliptic difference equations, Comput. J., 6, 193-201 (1963) · Zbl 0119.33404 [8] Friedman, M.; Ming, M.; Kandel, A., Fuzzy linear systems, FSS, 96, 201-209 (1998) · Zbl 0929.15004 [9] Hackbusch, W., Iterative Solution of Large Sparse Systems of Equations (1994), Springer-Verlag Inc.: Springer-Verlag Inc. New York [10] Hadjidimos, A., Accelerated overrelaxation method, Math. Comput., 32, 149-157 (1978) · Zbl 0382.65015 [11] Hageman, L. A.; Young, D. M., Applied Iterative Methods (1981), Academic Press: Academic Press New York · Zbl 0459.65014 [12] Kandel, A.; Friedamn, M.; Ming, M., Fuzzy linear systems and their solution, IEEE, 336-338 (1996) [13] Kincaid, D.; Cheney, W., Numerical Analysis Mathematics of Scientific Computing (1991), Brooks/Cole: Brooks/Cole California · Zbl 0745.65001 [14] Ming, M.; Kandel, A.; Friedman, M., A new approach for defuzzification, FSS, 111, 351-356 (2000) · Zbl 0968.93046 [15] Madalena Martins, M., On an accelerated overrelaxation iterative method for linear systems with strictly diagonally dominant matrix, Math. Comput., 152, 1269-1273 (1980) · Zbl 0463.65021 [16] Missirils, N. M.; Evans, D. J., On the convergence of some generalized preconditioned iterative methods, SIAM J. Numer. Anal., 18, 591-596 (1981) · Zbl 0464.65018 [17] Young, D. M., Iterative Solution of Large Linear Systems (1971), Academic Press: Academic Press New York · Zbl 0204.48102 [18] Young, D. M.; Gregory, R. T., A Survey of Numerical Mathematics, vol. 2 (1973), Dover Publications: Dover Publications New York [19] Zadeh, L. A., Fuzzy sets, Inform. Control, 8, 338-353 (1965) · Zbl 0139.24606 [21] Zimmermann, H.-J., Fuzzy Set Theory and its Applications (1996), Kluwer: Kluwer Dordrecht · Zbl 0845.04006
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