We propose a numerical technique for computing the H-x norm of input-output maps arising from nonlinear state-space systems. We show that computing the H-x norm of a nonlinear system involves a sequence of steps, one of which entails the solution of a first-order nonlinear partial differential inequality (PDI). The proposed numerical method relies on the recently developed viscosity solution framework for understanding this PDI, and is based on the finite difference numerical scheme. Results of numerical experiments carried out on Sun 4 and Connection Machine computers are provided.