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On computing reciprocals of power series

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Abstract

It is shown that root-finding iterations can be used in the field of power series. As a consequence, we obtain a class of new algorithms for computing reciprocals of power series. In particular, we show that the recent sieveking algorithm for computing reciprocals is just Newton iteration. Moreover, ifL n is the number of non scalar multiplications needed to compute the firstn+1 terms of the reciprocal of a power series, we show that

$$n + 1 \leqq L_n \leqq 4n - \log _2 n$$

and conjecture that

$$L_n = 4n - lowerorderterms.$$

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Authors and Affiliations

  1. Dept. of Computer Science, Carnegie-Mellon University, 15213, Pittsburgh, Pa., USA

    H. T. Kung

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  1. H. T. Kung
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Additional information

This research was supported in part by the National Science Foundation under Grant GJ 32111 and the Office of Naval Research under Contract N00014-67-A-0314-0010, NR 044-422.

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Kung, H.T. On computing reciprocals of power series. Numer. Math. 22, 341–348 (1974). https://doi.org/10.1007/BF01436917

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  • DOI: https://doi.org/10.1007/BF01436917

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