Solution of the fully fuzzy linear systems using iterative techniques

Abstract

Abstract This paper mainly intends to discuss the iterative solution of fully fuzzy linear systems which we call FFLS. We employ Dubois and Prade’s approximate arithmetic operators on LR fuzzy numbers for finding a positive fuzzy vector x ˜ which satisfies A ∼ x ˜ = b ∼ , where A ∼ and b ∼ are a fuzzy matrix and a fuzzy vector, respectively. Please note that the positivity assumption is not so restrictive in applied problems. We transform FFLS and propose iterative techniques such as Richardson, Jacobi, Jacobi overrelaxation (JOR), Gauss–Seidel, successive overrelaxation (SOR), accelerated overrelaxation (AOR), symmetric and unsymmetric SOR (SSOR and USSOR) and extrapolated modified Aitken (EMA) for solving FFLS. In addition, the methods of Newton, quasi-Newton and conjugate gradient are proposed from nonlinear programming for solving a fully fuzzy linear system. Various numerical examples are also given to show the efficiency of the proposed schemes.