Dynamical Control of Computations Using the Iterative Methods to Solve Fully Fuzzy Linear Systems

A linear system with fuzzy coefficients matrix, unknown and right hand side fuzzy vectors is called a fully fuzzy linear system (FFLS). Solving these kinds of systems via iterative methods to find the optimal number of iterations and optimal solution is important computationally. In this study, a FFLS is solved in the stochastic arithmetic to find this optimal solution. To this end, the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Application) library are considered to evaluate the round-off error effect on computed results. The Gauss-Seidel, Jacobi, Richardson and SOR iterative methods are used to solve FFLS. Also, an efficient algorithm is presented based on the proposed approach to compute the optimal results. Finally, two numerical examples are solved to validate the results and show the importance of using the stochastic arithmetic in comparison with the common floating-point arithmetic.

[1]  Abraham Kandel,et al.  Fuzzy linear systems , 1998, Fuzzy Sets Syst..

[2]  J. Ortega.  Numerical Analysis: A Second Course , 1974 .

[3]  Hasan Barzegar Kelishami,et al.  Dynamical control of accuracy in the fuzzy Runge-Kutta methods to estimate the solution of a fuzzy differential equation , 2016 .

[4]  S. Abbasbandy,et al.  A stochastic scheme for solving definite integrals , 2005 .

[5]  Tofigh Allahviranloo,et al.  Bounded and symmetric solutions of fully fuzzy linear systems in dual form , 2011, WCIT.

[6]  Mohammad Ali Fariborzi Araghi,et al.  Solving Fuzzy Linear Systems in the Stochastic Arithmetic by Applying CADNA Library , 2011, IJCCI.

[7]  Didier Dubois,et al.  Fuzzy sets and systems ' . Theory and applications , 2007 .

[8]  Davod Khojasteh Salkuyeh,et al.  Numerical accuracy of a certain class of iterative methods for solving linear system , 2006, Appl. Math. Comput..

[9]  Krzysztof Siminski,et al.  Interval type-2 neuro-fuzzy system with implication-based inference mechanism , 2017, Expert Syst. Appl..

[10]  Saeid Abbasbandy,et al.  Conjugate gradient method for fuzzy symmetric positive definite system of linear equations , 2005, Appl. Math. Comput..

[11]  S. Abbasbandy,et al.  The use of the stochastic arithmetic to estimate the value of interpolation polynomial with optimal degree , 2004 .

[12]  M. Otadi,et al.  Iterative method for approximate solution of fuzzy integro-differential equations , 2016 .

[13]  Jean Vignes,et al.  A stochastic arithmetic for reliable scientific computation , 1993 .

[14]  Mehdi Dehghan,et al.  Solution of the fully fuzzy linear systems using iterative techniques , 2007 .

[15]  J. Buckley,et al.  Solving systems of linear fuzzy equations , 1991 .

[16]  S. A. Edalatpanah,et al.  A nonlinear model for fully fuzzy linear programming with fully unrestricted variables and parameters , 2016 .

[17]  Reza Ghanbari,et al.  Solutions of fuzzy LR algebraic linear systems using linear programs , 2015 .

[18]  N. Amiri,et al.  NEW MODELS AND ALGORITHMS FOR SOLUTIONS OF SINGLE-SIGNED FULLY FUZZY LR LINEAR SYSTEMS , 2012 .