From implicit to recursive equations
Joris van der Hoeven · Applicable Algebra in Engineering Communication and Computing · 2018
The technique of relaxed power series expansion provides an efficient way to solve so called recursive equations of the form $$F = \varPhi (F)$$ , where the unknown F is a vector of power series, and where the solution can be obtained as the limit of the sequence $$0, \varPhi (0), \varPhi (\varPhi (0)), \ldots $$ . 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 $$\varPhi $$ . In this paper, we consider more general implicit equations of the form $$\varPhi (F) = 0$$ . Under mild assumptions on such an equation, we will show that it can be rewritten as a recursive equation.