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

Allahviranloo, Tofigh

Appl. Math. Comput. 155, No. 2, 493-502 (2004).

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.

Reviewer: Józef Drewniak (Rzeszów)

MSC:

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

Keywords:

diagonal dominance; fuzzy number; fuzzy solution; Jacobi iteration; Gauss-Seidel iteration; fuzzy system of linear equations; iterative methods

Citations:

Zbl 0929.15004; Zbl 0945.15002

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