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A fast algorithm for reversion of power series

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arxiv 1108.4772 v3 pith:ETN6UWMR submitted 2011-08-24 cs.SC

classification cs.SC
keywords algorithmmatrixmultiplicationpolynomialpowerreversionseriesachieve
verification ladder T0 review T1 audit T2 compute T3 formal

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abstract

We give an algorithm for reversion of formal power series, based on an efficient way to implement the Lagrange inversion formula. Our algorithm requires $O(n^{1/2}(M(n) + MM(n^{1/2})))$ operations where $M(n)$ and $MM(n)$ are the costs of polynomial and matrix multiplication respectively. This matches the asymptotic complexity of an algorithm of Brent and Kung, but we achieve a constant factor speedup whose magnitude depends on the polynomial and matrix multiplication algorithms used. Benchmarks confirm that the algorithm performs well in practice.

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  1. Corecursive Coding of High Computational Derivatives and Power Series

    cs.PL 2025-01 conditional novelty 4.0 of 10

    A Haskell tutorial for deriving infinite derivative chains and power series through lazy corecursion, including composition, reversion, and a Stirling-series example.

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