Some Investigations On Numerical Integration

In this paper, the following problems in numerical integration are studied, best quadrature, best quadrature formulas with weight functions and numerical evaluation of integral defined by Hadamard finite part.For any natural number r and any positive number K, let KW^r[a, b] be the rth Sobolev class consisting of every function, defined on the interval [a, b], whose (r-1)st derivative f^r-1 is absolutely continuous and its rth derivative f^r satisfies Now, suppose that function f∈KW^r[a, b], we only know the values of f and its derivatives at a set of nodes x := (x1, x2,..., xn)∈Rⁿ, not expression of f. These known values of f are defined by we name it as Hermite information. Suppose for the given data Y, we have We hope to obtain the best quadrature formula and its error bound for integral∫ab f(t)dt based on the given information Y. That is to say, we need to find a quadrature formula Q* for the integral∫ab f(t)dt, whose function f belongs to the Sobolev class KW^r[a, b] and satisfies Hxr(f) = Y, minimizing the biggest quadrature error bound among all the functionals Q: Hxr(KW^r[a, b])→R, i.e., Then the quadrature formula Q*(Y) is said to be the best quadrature formula based on the given information Y. Correspondingly, the error bound R(Y) is called the radius of the Hermite information Y.For extremal problem of quadrature formula, best quadrature formulas in the sense of Sard and Kolmogorov-Nikolskii-Sehoenberg are considered. There are some differences between the best quadrature formulas above and the best quadrature formula based on the given information. Firstly, the former are linear functionals instead of all functionals Q. Secondly, they searches the upper bound of the error for all the functions in KW^r[a, b] instead of using the given information Y. As we know, the given information are always obtained expensively, thus, any disuse of them is not judicious. From some point of view, they are not the best quadrature formulas. In this paper, we will discuss the relations among the three best quadrature formulas detailedly, and provide a method to obtain the other two quadrature formulas from the best quadrature formula based on the given information.Wang and Mi [92] first proposed a definition for the best quadrature based on the given information, and obtained results for r=2. In [71], the results for r=1 are also given. Now, we obtain the explicit results for r=3, 4 based on the given information making use of some algebraic techniques. Farther development of this problem can see [76, 77, 79, 94].Furthermore, using the explicit quadrature formulas and error bounds, we produce the extension and generalizations of Iyengar inequality to the class KW^r[a,b] (r=3,4). Since 1938, considerable efforts have contributed to the extensions and generalization of the Iyengar inequality. Results are obtained under additional conditions. In this paper, we give a real generalization of the Iyengar inequality. In this paper, we also obtain the best quadrature formulas based on the given information and error bounds with weight functions on rth Sobolev class KW^r[a, b]. The weight functions are in the following form, p(t) = (1－t²)^m－1/2, (t－x_i)^m－1/2(x_i＋1－t)^m－1/2(i=1,2...,n－1), sin mt, cosmt, where m is nonnega tire integers. We compare our quadrature formula with Gauss-Tu(?)an quadrature formula with Chebyshev function of the first kind.The rest part in this paper focuses on the numerical evaluation of Hadamard finite-part integrals. The concept of finite-part integral seems to have been first introduced by Hadamard in 1923. He discard the divergent terms in Cauchy principal value integral, therefore, we define the following whereζ∈(a, b) and p∈N₀ := {0, 1...}. In general, the second integral on the right-hand side of (1) can properly be computed. Therefore, we emphasize the numerical evaluation for the integral of the first term.First of all, we can rewrite the first integral in the following divided difference form To the approximate evaluation of integral (2) in spite of using an ordinary Gaussian rule or interpolatory quadrature formula, there are all concerned with the problem of calculating the divided difference f[x_k,(?), where x_k is one of the quadrature nodes. When the distance between the node x_k andζis very small, the divided difference f[x_k,(?) of function f is very difficult to compute. In this paper, we use the divided difference of the Lagrange interpolation of f to approximate the divided difference of f. The Lagrange interpolating polynomial based on another suitable nodes a₀,a₁,..., a_n. Although, the divided difference is computed based on a₀, a₁,... , a_n which are assumed that the distances between the nodes are large enough such that the computation of the divided difference can be carried out smoothly.Then, the quadrature formula is expressed explicitly by means of cycle indicator polynomials of symmetric group and some numerical examples are also given.