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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,

$S=\left[\begin{array}{cc}P& Q\\ Q& P\end{array}\right]$

with ${p}_{i,j}=max\left(0,{a}_{i,j}\right)$, ${q}_{i,j}=-min\left(0,{a}_{i,j}\right)$, $i,j=1,\cdots ,n$, 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.

##### MSC:
 65F30 Other matrix algorithms 15A06 Linear equations (linear algebra) 08A72 Fuzzy algebraic structures 15A33 Matrices over special rings 65F10 Iterative methods for linear systems