TITLE: A parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps: numerical algorithms. AUTHORS: Alex Haro (1) and Rafael de la Llave (2) (1) Departament de Matematica Aplicada i Analisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona (Spain) E-mail: haro@mat.ub.es (2) Department of Mathematics, University of Texas at Austin, Austin, TX 78712 (USA). E-mail: llave@math.utexas.edu ABSTRACT: We develop numerical algorithms for the computation of invariant manifolds in quasi-periodically forced systems. We show how to compute invariant tori and invariant manifolds associated to them. In particular, the stable and unstable manifolds of invariant tori, but also non-resonant invariant manifolds associated to spaces invariant under the linearization. These non-resonant manifolds include the slow manifolds which dominate the asymptotic behavior. The algorithms are based on the parameterization method. Rigorous results about this method are proved in a companion paper. In this paper, we concentrate on numerical issues of algorithm. Examples of implementations of the algorithms appear in another companion paper.