A linearized finite-difference scheme is used to transform the initial/boundary-value problem associated with the nonlinear Schrödinger equation into a linear algebraic system. This method is developed by replacing the time and the space partial derivatives by parametric finite-difference replacements and the nonlinear term by an appropriate parametric linearized scheme based on Taylor's expansion. The resulting finite-difference method is analysed for stability and convergence. The results of a number of numerical experiments for the single-soliton wave are given.