Quadrature Formulas Based On Orthogonal Rational Functions

This thesis is mainly devoted to the rational Gauss-type quadrature formulas based on orthogonal rational functions. It can be divided into four chapters:In chapter one,we start by pointing out that when the integrand has poles, rational Gauss-type quadrature formulas have some advantages over classical Gauss-type quadrature formulas. So a study on rational Gauss-type quadrature formulas is appropriate. Then a survey of the recent researches on rational quadrature formulas is given. A summary of our study is also given.In chapter two,we are mainly concerned with rational Gauss-Lobatto quadrature formulas which include both endpoints as nodes. These quadrature formulas are all interpolation quadrature formulas based on orthogonal rational functions, and their coefficients are nonnegative.Hence they are numerical stable. For general rational Lobatto quadrature formulas, it's difficult to compute their nodes and coefficients. In the third section of this chapter, explicit expressions of orthogonal rational functions with respect to some special weight functions are given. Thus we provide an efficient way to compute the nodes and coefficients in corresponding quadrature formulas. The above results can be used to compute the quadrature formulas in [8] [16]. This is one of the main purposes of our study. In the last section,we discuss the rational Gauss-Radau quadrature formulas briefly.These results are similar to the results of rational Gauss-Lobatto quadrature formulas.In chapter three,we proof the numerical stabilities of rational Gauss quadrature formulas by constructing rational Hermite interpolations. The Lebesgue...