Fast composition of numeric power series
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Abstract

Let and be two convergent power series in or , whose first terms are given numerically with a -bit precision for a fixed constant . Assuming that , we will show in this paper that the first coefficients of can be computed with a -bit precision in time . Using Newton iteration, a similar complexity bound holds for power series reversion of . Our method relies on fast multi-point evaluation, which will be recalled and further detailed for numeric polynomials. We also discuss relaxed variants of our algorithm.

Keywords: power series, composition, FFT, multi-point evaluation, algorithm

A.M.S. subject classification: 40-04, 42-04, 68W40

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Remark. The anonymous referees pointed me to [1], which contains many of the results presented in this paper. Nevertheless, some details in the presentation may still be of interest, such as the results in section 3.2. Also, we discuss relaxed variants of the main composition algorithm.

[1] P Ritzmann. A fast numerical algorithm for the composition of power series with complex coefficients. TCS, 44(1):1–16, 1986.