Modular static analyzers use procedure abstractions, a.k.a. summarizations, to ensure that their execution time increases linearly with the size of analyzed programs A similar abstraction mechanism is also used within a procedure to perform a bottom-up analysis. For instance, a sequence of instructions is abstracted by combining the abstractions of its components, or a loop is abstracted using the abstraction of its loop body: fixed point iterations for a loop can be replaced by a direct computation of the transitive closure of the loop body abstraction.

More specifically, our abstraction mechanism uses affine constraints, i.e. polyhedra, to specify pre- and post-conditions as well as state transformers. We present an algorithm to compute the transitive closure of such a state transformer  and we illustrate its performance on various examples. Our algorithm is simple, based on discrete differentiation and integration: it is very different from the usual abstract interpretation fixed point computation based on widening.  Experiments are carried out using previously published examples. We obtain the same results directly, without using any heuristic.