Matrix approach to a numerical solution of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations

Abstract

A matrix-based approach to numerical integration of the DGLAP evolution equations is presented. The method arises naturally on discretisation of the Bjorken x variable, a necessary procedure for numerical integration. Owing to peculiar properties of the matrices involved, the resulting equations take on a particularly simple form and may be solved in closed analytical form in the variable t=ln(alpha_0/alpha). Such an approach affords parametrisation via data x bins, rather than fixed functional forms. Thus, with the aid of the full correlation matrix, appraisal of the behaviour in different x regions is rendered more transparent and free of pollution from unphysical cross-correlations inherent to functional parametrisations. Computationally, the entire programme results in greater speed and stability; the matrix representation developed is extremely compact. Moreover, since the parameter dependence is linear, fitting is very stable and may be performed analytically in a single pass over the data values.