Research On Several Problems Of The Numerical Integration

Since Bernard Jacobson proposed the asymptotic property of integral medium points in 1982, it appealed to numerous scholars whose attainments cover many important properties of integral medium points and their applications. In the latest years, Liu Binqing and others further study asymptotic property and application of remainder term of numerical integration formula. In addition, numerous scholars at home and abroad study the integral formula, Gauss-Turán, with derived functions. Turán generalized classic Gauss formula thereby advancing Gauss-Turán whose highest algebraic accuracy reaches 2n(s + 1)-1. Other scholars proposed the explicit integral formula under the first kind Tchebyshev weight and the Gauss-Turán quadrature formula under Gori-Micchelli weight respectively.This thesis focuses on the asymptotic property and its application of remainder term of numerical integration formula. It proposes a more general outcome, on the basis of which correction formula of numerical integration is formed, whereby algebraic accuracy is promoted. With regard to Gauss-Turán quadrature formula, we adopt divided difference of multiple nodes to figure out the explicit expressions of numerical integration, and therefore the result is more concise.