On numerical improvement of Gauss–Legendre quadrature rules

摘要

Among all integration rules with n points, it is well-known that n-point Gauss–Legendre quadrature rule∫−11f(x)dx≃∑i=1nwif(xi)has the highest possible precision degree and is analytically exact for polynomials of degree at most 2n−1, where nodes xi are zeros of Legendre polynomial Pn(x), and wi's are corresponding weights.In this paper we are going to estimate numerical values of nodes xi and weights wi so that the absolute error of introduced quadrature rule is less than a preassigned tolerance ϵ0, say ϵ0=10−8, for monomial functionsf(x)=xj,j=0,1,…,2n+1.(Two monomials more than precision degree of Gauss–Legendre quadrature rules.) We also consider some conditions under which the new rules act, numerically, more accurate than the corresponding Gauss–Legendre rules. Some examples are given to show the numerical superiority of presented rules.