We present a new 1D algorithm for computing the global one-dimensional unstable manifold of a saddle point of a map. Our method can be generalized to compute two-dimensional unstable manifolds of maps with three-dimensional state spaces. This is shown here with a quasi-2D (Q2D) algorithm for the special case of a quasiperiodically forced map, which allows for a substantial simplification of the general case described in Krauskopf and Osinga (1998). The key idea is to "grow" the manifold in steps, which consist of finding a new point on the manifold at a prescribed distance from the last point. The speed of growth is determined only by the curvature of the manifold, and not by the dynamics. The performance of the 1D algorithm is demonstrated with a constructed test example, and it is then used to compute one-dimensional manifolds of a map modeling mixing in a stirring tank. With the Q2D algorithm we compute two-dimensional unstable manifolds in the quasiperiodically forced Hénon map. © 1998 Academic Press.