USING FUZZY LOGIC TO APPROXIMATE THE ACCURACY RATE IN SOLVING A SYSTEM OF LINEAR EQUATIONS

This paper proposes a numerical procedure for determining rate of accuracy of the numerical method used for solving a system of linear equations Ax=b. The idea of the method depends on the fuzzy value of the condition number and singularity rate instead of the crisp values. The procedure will be implemented on the mathematical code MATLAB and it’s Simulink features. Finally the applicability and efficiency of the procedure is illustrated by a numerical example.


INTRODUCTION
The term fuzzy logic was introduced with the 1965 proposal of fuzzy set theory by Lotfi Zadeh. [13] Fuzzy logic had, however, been studied since the 1920s, as infinite-valued logicnotably by Łukasiewicz and Tarski. [8,16] Fuzzy logic provided mathematicians with an appropriate tool for modelling vagueness phenomenon and shed new light into the control theory for engineers. [4,6,7,8,12] Later, in 1974 it was applied and implemented practically for the first 2 AWNI M. ABU-SAMAN time by Mamdani [9] in steam engine control theory. Fuzzy sets and logic were made practically useful in the 1990s [16]. Several models based on the fuzzy sets, the fuzzy transform has been proposed and widely used as a numerical tools for solving differential equations [1,2,3]. Fuzzy logic allows us to model vogue human language notions, especially, so called linguistic expressions (small, very big, more or less … etc.). Fuzzy logic controllers have the many advantages over the conventional controllers [6]: they are cheaper to develop, they cover a wider range of operating conditions, and they are more readily customizable in natural language terms. [8,14].
One way to measure the magnification factor is by means of the quantity A with respect to its inverse. The condition number determines the loss of precision due to round-off error in Gaussian elimination and can be used to estimate the accuracy of results obtained from matrix inversion and linear system solution. As a rule of thump, if the condition number to the beginning of the next interval will give different number of loss of precision although they are almost equal. The fuzzy logic with the membership functions gives more reasonable values for the loss precisions. Fuzzy logic allows us also to model a non-specific mathematical and scientific language notions, especially, so called vogue expressions (small rate, very small determinant, large condition number, nearly singular matrix … etc.). It prevents different actions for almost same behaviours. [14] The main aim of this paper is to overview the main concepts of fuzzy logic and its applications.
Moreover, we will explain a construction of approximation technique with membership functions to use Fuzzy logic, condition number and degree of singularity of the matrix to approximate the decimal digits of accuracy in solving a System of linear equations. An example will be given with condition number and degree of singularity as input parameters, and the accuracy as output parameter.
Fuzzy logic is considered to be a basic control system that relies on the degrees of state of the input and the output depends on the state of the input and rate of change of this state. [5] In other words, a fuzzy logic system works on the principle of assigning a particular output (action) 4 AWNI M. ABU-SAMAN depending on the probability of the state of the input. An interesting property is that the behaviour of a fuzzy system is not described using algorithms and formulas, but rather as a set of rules that may be expressed in natural language.

NUMERICAL COMPUTATIONS
According to figure one, we used MATLAB Simulink [10] features to describe and design a controller to determine the rate of a curacy of a system of linear equations Ax=b solver as an output variable. Assume the input variables are the condition number ϰ(A) and rate of singularity of the coefficient matrix. Four descriptor were used for the condition number, three descriptor for singularity rate input variables and five descriptor used for the output variable. The centroid defuzzification option and non-uniform triangular memberships in MATLAB [10] are used.

USING FUZZY LOGIC TO APPROXIMATE THE ACCURACY RATE
The first input variable is considered to be the condition number of the coefficient matrix.

Crisp Condition
No. Rate (CR) The fuzzy descriptor for the condition number worse rate variable is defined as: The second input variable is considered to be the singularity rate of the coefficient matrix. Assume it's in % according to its worseness and defined as it's in the below table. Table 2 The singularity rate approximate value.   1 The non-uniform triangular membership functions used to define the condition number input. Fig. 2 The uniform triangular membership functions used to define the singularity input. Fig. 3 The non-uniform triangular membership functions used to define the accuracy output.   Table 3 and fig. 4 show the accuracy action which should be taken according to the condition number and singularity rates. They indicate clearly that for large condition number rate with large singularity rate, a large accuracy action should be taken. For example, if the condition number and singularity rates 80% and 75% respectively, it's necessary to increase calculation digits by approximately 69%, while for the rates 80% and 20% , the action which should be taken is 53.3%, which is clearly less than the previous value due the lower rate of singularity. It's also clear that, for condition number rates 34.8 % and 35.2% with same singularity rate, approximately the same accuracy action should be taken although they belong to two separate intervals. While in crisp logic two completely different actions should be taken for almost the two close values.

Example
In this example the rate of accuracy will be evaluated for 60% as condition number worse rate and 30% as singularity rate. The fuzzy rules are considered as in table 4. Table 4 The applied fuzzy rules.  Which correspond to rule 4, Condition number is large and singularity is medium and has maximum strength 3 5 . To find out the final defuzzified value, we now take average of ( ), So the mean deffuzzified value for the accuracy rate for the condition number rate 60% and singularity rate 30% is: 47+63 2 = 55, which is the same results obtained by MATLAB in fig. 4.