  A Comprehensive Grobner system for a parametric ideal I in K(A)[X] represents
the collection of all Grobner bases of the ideals I' in K[X] obtained as the
values of the parameters A vary in K.
  The recent algorithms for computing them comprehensive Grobner systems
consider the corresponding ideal J in K[A,X], and are based on stability of
Grobner bases of ideals under specializations of the parameters. Starting from
a Grobner basis of J, the computation splits recursively depending on the
vanishing of the evaluation of some ``coefficients'' in K[A].
  In this paper, taking inspiration from the algorithm described by Nabeshima,
we create a new iterative algorithm to compute comprehensive Grobner systems.
We show how we keep track of the sub-cases to be considered, and how we avoid
some redundant computation branches using ``comparatively-cheap''
ideal-membership tests, instead of radical-membership tests.
