From implicit to recursive equations

Joris van der Hoeven (1)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

The technique of relaxed power series expansion provides an efficient way to solve so called recursive equations of the form F=Phi(F), where the unknown F is a vector of power series, and where the solution can be obtained as the limit of the sequence 0, Phi(0), Phi(Phi(0)), ... With respect to other techniques, such as Newton's method, two major advantages are its generality and the fact that it takes advantage of possible sparseness of Phi. In this paper, we consider more general implicit equations of the form Phi(F). Under mild assumptions on such an equation, we will show that it can be rewritten as a recursive equation. If we are actually computing with analytic functions, then recursive equations also provide a systematic device for the computation of verified error bounds. We will show how to apply our results in this context.

Mots clés: Implicit equation; relaxed power series; algorithm

Domaines: Logiciel mathématique [cs.MS]

Joris van der Hoeven. From implicit to recursive equations. Applicable Algebra in Engineering, Communication and Computing, 2019, 30 (3), pp.243-262. ⟨10.1007/s00200-018-0370-2⟩. ⟨hal-00583125v2⟩