  We present the notion of asymptotically non-terminating initial variable
values for linear loop programs. Those values are directly associated to
initial variable values for which the corresponding program does not terminate.
Our theoretical contributions provide us with powerful computational methods
for automatically generating sets of asymptotically non-terminating initial
variable values. Such sets are represented symbolically and exactly by a
semi-linear space, e.g., characterized by conjunctions and disjunctions of
linear equalities and inequalities. Moreover, by taking their complements, we
obtain a precise under-approximation of the set of inputs for which the program
does terminate. We can then reduce the termination problem of linear programs
to the emptiness check of a specific set of asymptotically non-terminating
initial variable values. Our static input data analysis is not restricted only
to programs where the variables are interpreted over the reals. We extend our
approach and provide new decidability results for the termination problem of
affine integer and rational programs.
