 As shown in the paper, the method was able to compute the nodes at a problem size of one billion in 11 seconds.Since the nodes near the endpoints of formula_4 become very close to each other at this problem size, this method of computing the nodes is sufficient for essentially any practical application in double-precision floating point.Gauss-Legendre quadrature is optimal in a very narrow sense for computing integrals of a function "f" over , since no other quadrature rule integrates all degree polynomials exactly when using "n" sample points.