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SOLVING FUZZY LINEAR SYSTEMS IN THE STOCHASTIC
ARITHMETIC BY APPLYING CADNA LIBRARY
Mohammad Ali Fariborzi Araghi
1
and Hassan Fattahi
2
1
Department of Mathematics, Faculty of Science, Islamic Azad University, Central Tehran Branch
P.O.Box 13185.768, Tehran, Iran
2
Department of Mathematics, Islamic Azad University, Kermanshah Branch, Kermanshah, Iran
K
eywords:
Iterative methods, Fuzzy Linear Systems (FLS), Stochastic arithmetic, CESTAC method, CADNA library,
Hausdorff distance, Fuzzy number.
Abstract:
In this paper, a fuzzy linear system with crisp coefficient matrix is considered in order to solve in the stochastic
arithmetic. The fuzzy CESTAC method is applied in order to validate the computed results. The Gauss-Seidel
and Jacobi iterative methods are used for solving a given fuzzy linear system. In order to implement the
proposed algorithm, the CADNA library is applied to find the optimal number of iterations. Finally, two
numerical examples are solved based on the given algorithm in the stochastic arithmetic.
1 INTRODUCTION
A general model for solving a FLS whose coefficient
matrix is crisp and the right-hand side column is an
arbitrary fuzzy number was first proposed by Fried-
man, Ming and Kandel, (1998). They used the em-
bedding method and replaced the original fuzzy lin-
ear system by a crisp linear system with a nonnegative
coefficients matrix. In the sequel, solving this kind of
fuzzy linear system based on the numerical and itera-
tive methods were proposed by others. Some of these
works were presented by Abbasbandy et al. (2006),
Allahviranloo (2004)(2005), Dehghan and Hashemi
(2006). Since, the results of the iterative methods
are obtained in the floating-point arithmetic, the ter-
mination criterion depends on a positive number like
ε. So, the final results may not be accurate or the
number of iterations may increase without increas-
ing the accuracy of the results. Therefore, the vali-
dation of the computed results is important. In this
case, because of the round-off error propagation, the
computer may not able to improve the accuracy of the
computed solution. By using the stochastic arithmetic
in place of the traditional floating-point arithmetic,
one can rely the results and estimate the accuracy of
them (Abbasbandy and Fariborzi Araghi, 2004; Ches-
neaux, 1992; Fariborzi Araghi, 2008; Vignes, 1993).
CESTAC (Controle et Estimation Stochastique des
Arrondis de Calculs) method is an efficient method
in order to estimate the accuracy of the results and
find the optimal number of iterations (Vignes,1993).
CADNA (Control of Accuracy and Debugging for
Numerical Applications) library is a tool to imple-
ment the stochastic arithmetic automatically. The first
goal of this software is the estimation of the accuracy
of each computed result. CADNA detects numerical
instabilities (informatical zero) during the run of the
program. CADNA works on Fortran or C++ codes.
When a result is a stochastic zero (i.e. is insignif-
icant), the symbol @.0 is printed. CADNA detects
numerical instabilities during the run of the program
(Jezequel and Chesneaux, 2008). For more details
about this library we refer the reader to ”http://www-
pequan.lip6.fr/cadna”.
By using the CESTAC method, N runs of the com-
puter program take place in parallel. In this way,
one runs every arithmetical operation N times syn-
chronously before running the next operation. In this
method, by running the program N times, for each re-
sult of any floating-point arithmetic operation, a set
of N computed results X
i
; i = 1, 2, ...N, is obtained.
N can be chosen any natural number like 2,3, 5, 7, but
in order to decrease operations cost, usually N = 3 is
considered. This method is able to estimate the round-
off error on each result and determine the accuracy of
it.
Let F be the set of all values represented on the
computer. Thus, any real value x is represented in the
form of X ∈F on the computer. It has been mentioned
in (Vignes, 1993) that in a binary floating-point arith-
metic with P mantissa bits, the rounding error stems
from assignment operator is
446
Fariborzi Araghi M. and Fattahi H..
SOLVING FUZZY LINEAR SYSTEMS IN THE STOCHASTIC ARITHMETIC BY APPLYING CADNA LIBRARY.
DOI: 10.5220/0003675804460450
In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (FCTA-2011), pages 446-450
ISBN: 978-989-8425-83-6
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)