From implicit to recursive equations

The technique of relaxed power series expansion provides an efficient way to solve so called recursive equations of the form F= Φ(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 , Φ(0) , Φ(Φ(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 Φ. In this paper, we consider more general implicit equations of the form Φ(F) = 0. Under mild assumptions on such an equation, we will show that it can be rewritten as a recursive equation.