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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. 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.

**Authors:**

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

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