On numerical improvement of Gauss-Legendre quadrature rules

Among all integration rules with n points, it is well-known that n-point Gauss-Legendre quadrature rule integral-(1)(1)f(x)dx similar or equal to Sigma(i=1)(n) w(j)f(x(j)) has the highest possible precision degree and is analytically exact for polynomials of degree at most 2n - 1, where nodes x(j) are zeros of Legendre polynomial P-j(x), and w(j)'s are corresponding weights. In this paper we are going to estimate numerical values of nodes x(j) and weights w(j) so that the absolute error of introduced quadrature rule is less than a preassigned tolerance epsilon(0), say epsilon(0) = 10(-8), for monomial functions f(x) = x(j), j = 0, 1, 2..., 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. (C) 2003 Elsevier Inc. All rights reserved.