In this paper, first we give a new characterization for the upper bound epsilon* of the parasitic parameter epsilon in singularly perturbed systems, which ensures stability of such systems if 0 <epsilon <epsilon*. It will be shown that this upper bound is just the minimum positive eigenvalue of a matrix pair, which can be explicitly constructed from the system matrix. Secondly, based on the new characterization for the stability upper bound, an algorithm for computing this upper bound epsilon* is established. Thirdly, in order to improve the upper bound epsilon* via state feedback, an algorithm is developed. Finally, several examples are presented to illustrate the algorithms proposed in this paper.