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The technique of relaxed power series expansion provides an efficient
way to solve so called recursive equations of the form , where the unknown is a vector of power series, and where the solution
can be obtained as the limit of the sequence . 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 .
In this paper, we consider more general implicit equations of the form
. Under mild assumptions on
such an equation, we will show that it can be rewritten as a recursive
equation. *Preprint version only:* 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.

**Author:**

**Keywords: **implicit equation, relaxed power series,
algorithm

**A.M.S. subject classification: **68W25, 42-04, 68W30,
65G20, 30B10