A geometrically invariant concept of singularly perturbed systems of ordinary differential equations (singularly perturbed vector fields) is proposed in this paper. Singularly perturbed vector fields can be represented locally as singularly perturbed systems (for corresponding coordinate system choice. The paper focuses on possible ways of fast and slow directions/manifolds evaluations. A special algorithm for the evaluation is proposed. The algorithm is called as a global quasi-linearization procedure. A practical application of the proposed algorithm for numerical simulations is the main issue of the paper.