Constructions And Studies Of Several Kinds Of Numerical Integration Formulaue Of High Degree

Numerical integration is a numerical approximation to an integration. It is used wildly both in mathematics itself and in applied fields, such as physics, engineering and machinery. One major work of numerical integration is to study how to construct the numerical inte-gration formulae, which are also called quadrature formulae for one dimension or cubature formulae for higher dimensions, such that they satisfy some criterions. In the light of the performance of the formulae acting on polynomials, two of the classical criterions are the algebraic degree and the trigonometric degree. Although the research has been studied for hundreds of years, there are still many unsolved problems. In this paper, it is the con-structions of cubature formulae of algebraic degree for higher dimensions and of quadrature formulae of trigonometric degree for one dimension that are discussed. The details are as follows.It is always important and difficult to construct the cubature formulae for higher di-mensions. In applications, the product cubature formulae over regions of product forms are used very widely. In order to be of algebraic degree as high as possible, such cubature formulae are constructed by the product form of some one-dimensional Gaussian quadra-ture formulae. However, the product cubature formulae contains a number of nodes, which increases exponentially with dimensions. The rapid increase makes such formulae inconve-nient when used in applications, especially for higher dimensions. In order to reduce the number of nodes, the Smolyak formulae rise. The advantages of Smolyak formulae are not only that they are constructed in some product forms, but also that they contain apparently fewer nodes than the Gaussian product cubature formulae. In this paper, the cubature for-mulae of degree4over general product regions are studied. In this case for dimension n, the Gaussian product formulae use3n nodes; the Smolyak formulae use at most about2n2nodes, while the formulae constructed in this paper use only about n2nodes, which is the minimal number of nodes to the best knowledge of the author. Moreover, the scheme is very convenient in applications. It transforms the problem of higher dimensions into a sequence of one-dimensional moment problems, which not only reduces the amount of computation in a great deal, but also ensures the success of construction. Besides, all the cubature formulae in this paper have explicit expressions, while the Smolyak formulae does not. In applications, it is very important to estimate the errors of cubature formulae. The popular method is to use the difference between two or more cubature formulae to estimate the error of the one of lower degree. In order to reduce the amount of computation as much as possible, the nodes of these formulae are often supposed to be embedded. Such formulae are also called embedded cubature formulae. Most of the known embedded cubature formu-lae are constructed by adding or deleting nodes. However, almost all the known methods deal with the higher-dimensional problems directly, which are not effective in applications. In this paper, a new strategy to construct the embedded cubature formulae in planar re-gions is studied. In the process, this strategy transforms the problem for given degrees into some lower-degree problem such that the amount of computation can be reduced to some degree. Moreover, for some special cases, this strategy only deals with some one-dimensional problems and the amount of computation will be reduced further. Also, a recursive algo-rithm is given to construct such formulae under some conditions. This strategy is easy and applicable.The quadrature formulae of trigonometric degree are very useful when the integrals are periodic functions. As it is well known, the maximum trigonometric degree of n-point quadrature formula is n-1. However, the maximum trigonometric degree is not given uniformly for any n when there are m prescribed nodes. The positive answers are given to the cases of m=1and m=2. One part of this paper deals with the problem that how to construct the quadrature formulae of maximum trigonometric degrees by adding some new nodes to the prescribed ones. First, the maximum degree for any given integer m and the vanishing property of the new nodes are given. Then, a constructive algorithm is given such that all the new nodes are simple, that is, they are all real, pairwise different and inside the integration region.