This paper addresses the problem of computing an exact and effective representation of the set of reachable configurations of a linear hybrid automaton. Our solution is based on accelerating the state-space exploration by computing symbolically the repeated effect of cycles in the automaton control graph. The computed sets of configurations are represented by Real Vector Automata (RVA), the expressive power of which is beyond that of the first-order additive theory of reals and integers. This approach makes it possible to compute in finite time sets of configurations that cannot be expressed as finite unions of convex sets. The main technical contributions of the paper consist in a powerful sufficient criterion for checking whether a hybrid transformation (i.e., with both discrete and continuous features) can be accelerated, as well as an algorithm for applying such an accelerated transformation on RVA. Our results have been implemented and successfully applied to several case studies, including the well-known leaking gas burner, and a simple communication protocol with timers.