Composite Gauss-Legendre Formulas for Solving Fuzzy Integration

where Ak ≥ 0, k = 0, 1, . . . , n, and xk ∈ [a, b], k = 0, 1, . . . , n, are called coefficients and nodes for mechanical quadrature, respectively.Once the coefficients andnodes are set down, the scheme (1) can be determined. Over years, some works have emerged about the asymptotic properties of numerical integration methods. However, their results are concise, but reasoning processes are very complicated [3–6].The topic of fuzzy integrationwas first discussed in [7]. In 2005, Allahviranloo [8]made a good attempt to use Newton Cot’s methods with positive coefficients for integration of fuzzy functions. For instance, he designed Trapezoidal integration rule and Simpson integration rule for fuzzy integral. Later, they applied the Gaussian quadrature method and Romberg method for approximation of fuzzy integral and fuzzy multiple integral, built a series of formulas for intricate fuzzy integral [8–11], and obtained some good results. But their methods did not have high convergence order. In this paper, we set up a class of high algebraic accuracy numerical integration methods which are proposed by compositing the two-point and three-point Gauss-Legendre formulas. We design these formulas to calculate integration of fuzzy functions. We also present the methods’ remainder terms and give corresponding convergence theorems. Compared with some approaches for approximating fuzzy integrations before, our methods are superior to those formulas on both amount of calculation and quadrature error. The structure of this paper is as follows. In Section 2, we recall some basic definitions and results on integration of fuzzy functions. In Section 3, we introduce the two-point and three-point Gauss-Legendre formulas and their composite method. Then we design them to solve fuzzy integration. We also put up methods’ reminder term representations and convergence theorems. The proposed algorithms are illustrated by solving two examples in Section 4 and the conclusion is drawn in Section 5.


Introduction
Numerical integration is one of the basic contents in numerical mathematics, and it always plays a vital role in engineering and science calculation.Numerical integration methods are introduced in detail [1].Numerical integration is always carried out by mechanical quadrature and its basic scheme [2] is as follows: where   ≥ 0,  = 0, 1, . . ., , and   ∈ [, ],  = 0, 1, . . ., , are called coefficients and nodes for mechanical quadrature, respectively.Once the coefficients and nodes are set down, the scheme (1) can be determined.Over years, some works have emerged about the asymptotic properties of numerical integration methods.However, their results are concise, but reasoning processes are very complicated [3][4][5][6].The topic of fuzzy integration was first discussed in [7].In 2005, Allahviranloo [8] made a good attempt to use Newton Cot's methods with positive coefficients for integration of fuzzy functions.For instance, he designed Trapezoidal integration rule and Simpson integration rule for fuzzy integral.Later, they applied the Gaussian quadrature method and Romberg method for approximation of fuzzy integral and fuzzy multiple integral, built a series of formulas for intricate fuzzy integral [8][9][10][11], and obtained some good results.But their methods did not have high convergence order.
In this paper, we set up a class of high algebraic accuracy numerical integration methods which are proposed by compositing the two-point and three-point Gauss-Legendre formulas.We design these formulas to calculate integration of fuzzy functions.We also present the methods' remainder terms and give corresponding convergence theorems.Compared with some approaches for approximating fuzzy integrations before, our methods are superior to those formulas on both amount of calculation and quadrature error.The structure of this paper is as follows.
In Section 2, we recall some basic definitions and results on integration of fuzzy functions.In Section 3, we introduce the two-point and three-point Gauss-Legendre formulas and their composite method.Then we design them to solve fuzzy integration.We also put up methods' reminder term representations and convergence theorems.The proposed algorithms are illustrated by solving two examples in Section 4 and the conclusion is drawn in Section 5.

Integration of Fuzzy Function.
Let  1 be the set of all real fuzzy numbers which are normal, upper semicontinuous, convex, and compactly supported fuzzy sets.
Definition 2 (see [14]).Assume  : [, ] →  1 .For each partition  = { 0 ,  1 , . . .,   } of [, ] and for arbitrary   : The definite integral of () over [, ] is provided that this limit exists in the metric .If the fuzzy function () is continuous in the metric , its definite integral exists.Furthermore, It should be noted that the fuzzy integral also can be defined using the Lebesgue-type approach [15,16].More details about properties of fuzzy integral are given in [17].

Gauss-Legendre Formulas.
Gauss quadrature formula is the highest algebraic accuracy of interpolation quadrature formula.By reasonably selecting quadrature nodes and quadrature coefficients of the form of we can obtain the interpolation quadrature formula with the highest algebraic accuracy; that is, 2 + 1.Using root nodes of  + 1 order Legendre orthogonal polynomial on special interval [−1, 1], we can propose Gauss-Legendre quadrature formula (7).

Convergence Order of Composite Method
Definition 4 (see [18]).Suppose  = ∫   (), and   is a composite numerical integration method.If ℎ → 0, it satisfies lim and we call   a  order convergent method.For instance, composite Trapezoid method and composite Simpson have two-order and four-order convergence property, respectively.
Proof.We first consider the remainder term of three-point Gauss-Legendre.By Lemma 3, ))
So the reminder terms of composite three-point Gauss-Legendre formulas (29) for fuzzy integration are (31).
it is clear that formula (37) holds.
We calculate numerically the above integral using Trapezoidal formula, Simpson formula, composite two-point Gauss-Legendre, and three-point Gauss-Legendre methods with ℎ = 1, ℎ = 1/2, and ℎ = 1/4.Some comparisons about the numerical solutions and the errors between the different methods are shown in Tables 1, 2, and 3.All data are denoted with eight-bit significant digits and errors are calculated by the distance between exact solution and numerical solution.
From the above tables' figures, we can clearly see that our methods have better approximation than the Trapezoidal formula and Simpson formula on the same fuzzy integration, in which the composite three-point Gauss-Legendre is really the case.

Conclusion
In this work, we applied composite Gauss-Legendre formulas to solve fuzzy integral over a finite interval [, ].Since this integration yields fuzzy number in parametric form, we use the parametric form of the methods.The integration of triangular fuzzy number is a triangular fuzzy number.Numerical examples showed that our methods are practical and efficient while computing fuzzy integral on a larger interval [, ].