Finding optimal step of fuzzy Newton-Cotes integration rules by using the CESTAC method

The aim of this work, is to evaluate the value of a fuzzy integral by applying the Newton-Cotes integration rules via a reliable scheme. In order to perform the numerical examples, the CADNA (Control of Accuracy and Debugging for Numerical Applications) library and the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method are applied based on the stochastic arithmetic. By using this method, the optimal number of points in the fuzzy numerical integration rules and the optimal approximate solution are obtained. Also, the accuracy of the fuzzy quadrature rules are discussed. An algorithm is given to illustrate the implementation of the method. In this case, the termination criterion is considered as the Hausdorff distance between two sequential results to be an informatical zero. Two sample fuzzy integrals are evaluated based on the proposed algorithm to show the importance and advantage of using the stochastic arithmetic in place of the floating-point arithmetic.


Introduction
One of the important subjects in computational mathematics is numerical integration which is always carried out by mechanical quadrature with following basic scheme [18]: where f i = f (a + ih), h = b−a n and λ i are crisp coefficients.A family of quadrature methods is the Newton-Cotes integration rules [27].In recent years, many authors presented and improved different applicable algorithms about Newton-Cotes integration rules.Simos et al. in [42,52,53,54,55], Burg in [19] and Sandu in [50] used this method for solving many applicable problems and Dehghan et al. in [28,29], Hashemiparast et al. in [38,39] and Eslahchi et al. in [31] improved this method.The results of these works are obtained in the common computer arithmetic and the termination criterion usually depends on a positive and small real number like ε.Hence, the validation of the computed results is important.In this case, because of the round-off error propagation, the computer is not able to improve the accuracy of the computed solution.
For remedy of this shortages, the CESTAC method has been used to validate the numerical algorithms in different topics such as interpolation polynomials [1], ill-condition functions [2], numerical integration [3,4,34], linear algebra [43,44] and the others [20,21,22,23,24,25,56,58]. In the floating-point arithmetic, the results may be false without awareness of the user.But using the stochastic arithmetic, the validation of the results is determined and if the results are not reliable, a warning message is flagged for the user.When the number of the significant digits corresponding to a result is zero, the result is not valid.In this case, the result is called an informatical zero which is denoted by @.0.The floating-point arithmetic is not able to detect an informatical zero.In other word, the evaluation of a fuzzy definite integral is one of the important topics in fuzzy mathematics [7,8,9,10,11,12].Let Ĩ be a the exact value of a fuzzy integral and Ĩn be an approximate value of Ĩ by using a numerical integration rule as follows: where fi = f (a + ih), h = b−a n and λ i are crisp coefficeints.The topic of fuzzy integration was first discussed in [61].Particularly, it has been studied by Kaleva [41], Goetschel and Voxman [37], Nanda [47], Ralescu and Adams [49], Wang [59], Bede and Gal [16] and others [32,33].The fuzzy Riemann integral and its numerical integration was investigated by Wu in [60].Allahviranloo in [9] applied the Newton-Cotes methods with positive coefficients for integration of fuzzy functions.For instance, the Trapezoidal and Simpson integration rules for fuzzy definite integrals were considered.In [1], the authors applied the stochastic arithmetic in order to validate the results of evaluation a given definite integral by the Newton-Cotes integration rules.They simulated the CESTAC method to implement the discrete stochastic arithmetic on the Fortran codes in Windows operating system.In this work, this idea is developed on fuzzy definite integrals by applying the fuzzy Newton-Cotes integration rules.In this case, the CADNA library is used to automatic implementation the fuzzy CESTAC method on the C++ codes in Linux machine.The aim of this paper, is to implement and validate the fuzzy Newton-Cotes integration rules such as Simpson, Trapezoidal and Midpoint rules and to find the optimal number of iteration and calculate the optimal approximate of fuzzy functions by using the stochastic arithmetic and the CESTAC method.For this purpose, the CADNA library should be used.This paper is organized as follows: In section 2, some basic definitions in fuzzy sets theory are reminded.Also, the fuzzy Newton-Cotes integration rules such as Simpson, Trapezoidal and Midpoint integration rules are introduced.The fuzzy CESTAC method and CADNA library are introduced too.Section 3 is the main idea.In this section, the accuracy of the proposed method is discussed based on the concept of the common significant digits.In section 4, an applicable algorithm is presented and two sample fuzzy integrals are evaluated via the proposed algorithm.Finally, section 5 contains some conclusion remarks.

Preliminaries
2.1 Fuzzy arithmetic Definition 2.1.[41].A fuzzy number is a fuzzy set with membership function p : R → [0, 1] which satisfies There are real numbers a, b : c ≤ a ≤ b ≤ d for which • p(x) is monotonic decreasing on [b, d], • p Definition 2.2.[46].The parametric form of fuzzy number p is presented by (p(r), p(r)) which is satisfies in the following requirements: International Scientific Publications and Consulting Services • p(r) is a bounded left-continuous non-decreasing function over [0, 1], • p(r) is a bounded right-continuous non-increasing function over [0, 1], • p(r) < p(r), for all 0 ≤ r ≤ 1.
If p = (p(r), p(r)), q = (q(r), q(r)) are the arbitrary fuzzy numbers and λ is a real number then the definitions of addition and the scalar multiplication of the fuzzy numbers p and q are presented as follows • p = q if and only if p(r) = q(r) and p(r) = q(r), • p ⊕ q = (p(r) + q(r), p(r) + q(r)), Definition 2.3.The Hausdorff distance between p = (p, p) and q = (q, q) is defined as Definition 2.4.[13].Let p, q : [a, b] → R F , be fuzzy real number valued functions.The uniform distance between p, q is defined by Goetschel and Voxman in [37] proved that if the fuzzy function ϕ (x) is continuous in the metric D, its definite integral exists and also, where ϕ (x) = (ϕ (x; r), ϕ (x; r)).
It should be noted that the fuzzy integral can be also defined using the Lebesgue-type approach [47].However, if ϕ (x) be continuous, both approaches yield the same value.Moreover, the representation of the fuzzy integral is more convenient for numerical calculations.
The details of fuzzy sets theory and fuzzy calculus can be found in [13,16,37,61].

Fuzzy Newton-Cotes integration rules
Let f be a fuzzy function.For any natural number n, the Newton-Cotes formulas (1.1) provide approximate values for ∫ b a f (x)dx.The parametric form of (1.1) is as follows [9]: The weights λ i , i = 1, ..., n, are rational numbers with the property ∑ n i=1 λ i = n.This follows (2.2) when applied to f (x; r) = f (x; r) = 1.It can be shown that the approximate error may be expressed as follows: (2. 3) The values of p and K depend only on n but not on the integrand f .For large n, some of the values λ i become negative and the corresponding formulas are unsuitable for numerical purposes, as cancelations tend to occur in computing the sum (2.3).Let (2.4) Thus from (2.2) we get (2.5) The following theorem proves that

Simpson integration rule
From (2.2) we have: International Scientific Publications and Consulting Services where (2.9) The same results can be found for the 3  8 -Simpson integration rule.

Fuzzy CESTAC method-CADNA library
The main idea of the CESTAC method was designated by La Porte and Vignes [58].This method is able to perform the discrete stochastic arithmetic for estimating the round-off error and determining the accuracy of the result.The CESTAC method is based on an approach which replaces the floating-point arithmetic by a new arithmetic [1,2,3,4,21,23,25,58] which is called the stochastic arithmetic.By using this method n runs of the computer program take place in parallel.The details about this arithmetic and its properties can be found in [1,2,3,4,22,24,26].Let a fuzzy value z(r) = (z(r), z(r)) is represented in the form of Z(r) = (Z(r), Z(r)) in the computer.In the binary floating-point arithmetic with P mantissa bits, the rounding error stems from assignment operator is (2.12) where, ε 1 and ε 2 are the signs of z(r) and z(r) respectively and 2 −P γ and 2 −P γ are the lost part of the mantissa due to round-off error and E 1 and E 2 are the binary exponents of the results.For a personal computer, in single precision case, P = 24 and in double precision case, P = 53.According to (2.12), in order to perturb the last mantissa bit of the value z = (z, z) then, it is sufficient that the values γ and γ are considered as random variables uniformly distributed on [−1, 1].Thus Z = (Z, Z), the calculated result, is a random variable and its precision depends on its mean (µ, µ) and its standard deviation (σ , σ ).If a computer program is performed n times, the distribution of the results (Z i , Z i ), i = 1, ..., n, is Gaussian which their mean is equal to the exact real value (z, z), that is (E(Z), E(Z)) = (z, z).This n samples are used for estimating the values (µ, µ) and (σ , σ ).The value n can be chosen any natural number like 2, 3, 5, 7, but in order to avoid the execution time, usually n = 3.In International Scientific Publications and Consulting Services this case, the number of decimal significant digits common to (Z, Z) and to the exact value (z, z) can be estimated by, where (σ , σ ) is the standard deviation of the samples (Z, Z).
Definition 2.5.A computed result (Z, Z) using the CESTAC method is an "informatical zero", denoted by @.0, if and only if In the CESTAC method, during the run of the algorithm, if (Z, Z) = @.0, the informatical result (Z, Z) is insignificant and it means a numerical instability exists in its related line.
CADNA is a library which was designated by Chesneaux is able to implement the CESTAC method automatically on any code written by C++ or Fortran [20,21,22,23,25,58,43,44,56].This library allows, during the execution of a code, the estimation of the accuracy of each result and the detection of numerical instabilities at each line of the code.These cases are advantages of stochastic arithmetic in comparison with floating-point arithmetic.Furthermore, stochastic arithmetic is able to find the optimal step and also eliminate the unnecessary iterations of the iterative methods which the floating-point arithmetic is not able to distinguish them.

Main idea
In this section, the accuracy of the fuzzy Newton-Cotes integration rule is discussed to estimate the fuzzy definite integral (I, based on the concept of the common significant digits which is important in the implementation of the CESTAC method.Then, it is concluded, one can find the optimal number of the iterations in these rules, which minimizes the global error.At first a theorem and a proposition are presented which can be proved similar to the proof in crisp case discussed in [17][18]. by using the Newton-Cotes integration rule with p + 1 points, where x j = x 0 + jh, j = 0, 1, ..., p and h = there exists a point like (ξ , ξ ) in (x 0 , x p ) such that and ) , be the approximation of (I, I) by using the composite Newton-Cotes integration rule with p + 1 points, where x j = a + jh, j = 0, 1, ..., n and h = b−a n , n = pl, l ≥ 1.Then, there exists a point like Now, the following lemma is proved as a development of the results mentioned in [1] in fuzzy case.
where K p is a constant which depends only on p, f (m) (b; Proof.Let p be even, then m = p + 1.According to (3.14), where Ap = (3.17) The values α and β are evaluated so that this formula be exact for the polynomials of degree at most p + Therefore, the following system is obtained: After solving the system,

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One can observe, this formula is also exact if f (x) = x p+3 .For this purpose, it is sufficient to prove, Since, x i = −h + i 2h p , a p−i = a i , i = 0, 1, ..., p and a p 2 = 0, hence ( Besides, So, the relation can be concluded for the interval [x 0 , x n ].If p is odd then, the similar proof is presented. In Eq. (3.15), since h = b−a n hence ) , where . The value K p,m is small in comparison with n, as n → ∞.Now, we need the following definition [7]: Definition 3.1.Let I and J be two real numbers, the number of significant digits that are common to I and J, denoted by C I,J , can be defined by (1) for (2) for all real numbers I, C I,I = +∞.
One can use this definition in order to find the accuracy of the fuzzy Newton-Cotes integration rules.The following theorem is proved for these rules in closed form.The same result can be proved in open form.) , where m = p + 1 if p is even and m = p, if p is odd.According to (3.19), ) , and ) , hence ) ( ) , ) ( ) .
Eq. (3.21) shows that for n enough large the number of the common siginificant digits between two sequential results I P n and I P n+1 are almost equal to the common significant digits of I P n and the exact value I and also the number of the common siginificant digits between two sequential results I ) << 1.

Numerical illustrations
In this section, two fuzzy integrals are evaluated and estimated by the CESTAC method.When the floating-point arithmetic is applied to find the approximate value of fuzzy numerical integration, since it depends on a tolerance as accuracy, the following termination criterion can be applied: where, ε is an arbitrary positive value.This criterion may not be acceptable.If we choose very large ε, the iterations are stopped before getting access to a suitable approximation.If we choose very small ε then, unnecessary iterations are done without improving the accuracy of the results.In this case, Ĩn has finite significant digits and the termination criterion is meaningless.In the stochastic arithmetic, in place of above criterion, the following statement is used: Now, the following algorithm is presented to show how the CADNA library can be applied to find the optimal step and optimal approximation of fuzzy Newton-Cotes integration rules.Also, in this algorithm the Hausdorff distance mentioned in Definition 2.3 and termination criterion (4.23) are applied.Also, d as the increment of do loop is a small positive value like 0.1.
Step  where k = (r − 1, 1 − r).By using the CESTAC method the fuzzy Newton-Cotes integration rules are applied to approximate the fuzzy integral (4.24).The numerical results are presented in Tables 1, 2, 3, 4 based on the proposed algorithm and in Table 5, the optimal step of iterations are demonstrated.
International Scientific Publications and Consulting Services   where k = (r, 2 − r) is illustrated and by using presented method, the fuzzy integral (4.25) is estimated.The numerical results are shown in Tables 6,7,8,9.The optimal step of iteration in fuzzy Newton-Cotes integration rules are compared in Table 10.
Table 6: Numerical results of Example 4.2 by using 3 8 -Simpson integration rule.

Conclusion
In this paper, the fuzzy Newton-Cotes integration rules was validated by using the CESTAC method and the CADNA library to find the approximate solution of fuzzy integrals.The CESTAC method is based on the stochastic arithmetic.The advantages of this arithmetic in comparison with the floating-point arithmetic were discussed.The presented theorem show the efficiency and accuracy of the CESTAC method.This method was applied to find the optimal iteration of fuzzy Newton-Cotes integration rules and the optimal approximation of the examples.

Theorem 3 . 1 .
Let (I p , I p ) = ( p ∑ j=0 a j f (x j ; r), p ∑ j=0 a j f (x j ; r)), be the approximation of (I, I) = ( x p −x 0 p .Then,International Scientific Publications and Consulting Services

Theorem 3 . 2 .
Let (I p n , I p n ) be the approximate value of (I, I) = ( ∫ b a f (x)dx, ∫ b a f (x)dx) computed using the closed International Scientific Publications and Consulting Services Newton-Cotes integration rule with p + 1 points over [a, b] and step size h = b−a n .If f ∈ C m+3 [a, b], m 1, then,

Proof.
By using Eq.(3.20) we obtain ( m (b − a) m+1 , K p,m (b − a) m+1 ).Also, from (3.22), equal to the common significant digits of I P n and the exact value I in companion with the term log 10 2 m+1 2 m+1 −1 which is a small value and O (

Table 2 :
Numerical results of Example 4.1 by using 3 8 -Simpson integration rule.
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Table 4 :
Numerical results of Example 4.1 by using Midpoint integration rule.

Table 5 :
Optimal step of Example 4.1 for presented integration rules.

Table 9 :
Numerical results of Example 4.2 by using Midpoint integration rule.

Table 10 :
Optimal step of Example 4.2 for presented integration rules.
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