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We consider systems of recursively defined combinatorial structures. We givealgorithms checking that these systems are well founded, computing generatingseries and providing numerical values. Our framework is an articulation of theconstructible classes of Flajolet and Sedgewick with Joyal's species theory. Weextend the implicit species theorem to structures of size zero. A quadraticiterative Newton method is shown to solve well-founded systems combinatorially.From there, truncations of the corresponding generating series are obtained inquasi-optimal complexity. This iteration transfers to a numerical scheme thatconverges unconditionally to the values of the generating series inside theirdisk of convergence. These results provide important subroutines in randomgeneration. Finally, the approach is extended to combinatorial differentialsystems.

Algorithms for Combinatorial Systems: Well-Founded Systems and Newton Iterations