QCD Evolution Equations: Numerical Algorithms from the Laguerre Expansion

A complete numerical implementation, in both singlet and non-singlet sectors, of a very elegant method to solve the QCD Evolution equations, due to Furmanski and Petronzio, is presented. The algorithm is directly implemented in x-space by a Laguerre expansion of the parton distributions. All the leading-twist distributions are evolved: longitudinally polarized, transversely polarized and unpolarized, to NLO accuracy. The expansion is optimal at finite x, up to reasonably small x-values ($x\approx 10^{-3}$), below which the convergence of the expansion slows down. The polarized evolution is smoother, due to the less singular structure of the anomalous dimensions at small-x. In the region of fast convergence, which covers most of the usual perturbative applications, high numerical accuracy is achieved by expanding over a set of approximately 30 polynomials, with a very modest running time.