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.