Numerical methods for fuzzy system of linear equations. (English) Zbl 1067.65040

The paper deals with systems of linear equations \(Ax = b\) with fuzzy right-hand side (\(b\) and \(x\) are vectors of fuzzy numbers). It is a continuation of papers by M. Friedman [Fuzzy Sets Syst. 96, 201–209 (1998; Zbl 0929.15004)] and M. Friedman, A. Kandel and M. Ma [Fuzzy Sets Syst. 109, 55–58 (2000; Zbl 0945.15002)], where the associated matrix \(S\) was introduced, \[ S = \left[\begin{matrix} P & Q \\ Q & P \end{matrix}\right] \] with \(p_{i,j} =\max(0,a_{i,j})\), \(q_{i,j} = -\min(0,a_{i,j})\), \(i,j =1,\dots,n\), where the system \(Sy=c\) is crisp (real matrix and vectors). If the matrix \(A\) is diagonally dominant, then \(S\) has also this property (Theorem 3.2). Therefore, approximate solutions can be obtained by Jacobi iterations or Gauss-Seidel iterations.


65F30 Other matrix algorithms (MSC2010)
15A06 Linear equations (linear algebraic aspects)
08A72 Fuzzy algebraic structures
15B33 Matrices over special rings (quaternions, finite fields, etc.)
65F10 Iterative numerical methods for linear systems
Full Text: DOI


[1] Buckley, J. J., Fuzzy eigenvalues and input-output analysis, FSS, 34, 187-195 (1990) · Zbl 0687.90021
[2] Zadeh, L. A., The concept of a linguistic variable and its application to approximate reasoning, Inform. Sci., 8, 199-249 (1975) · Zbl 0397.68071
[3] Friedman, M.; Ming, M.; Kandel, A., Fuzzy linear systems, FSS, 96, 201-209 (1998) · Zbl 0929.15004
[4] Diamond, P., Fuzzy least squares, Inform. Sci., 46, 144-157 (1988) · Zbl 0663.65150
[5] Ortega, J. M., Numerical Analysis a Second Course (1990), Siam · Zbl 0701.65002
[6] Chang, S. L.; Zadeh, L. A., On fuzzy mapping and control, IEEE Trans., Syst. Man Cyb., 2, 30-34 (1972) · Zbl 0305.94001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.