Speeding up the inversion of power series

Guillaume Hanrot 1 Paul Zimmermann 1
1 POLKA - Polynomials, Combinatorics, Arithmetic
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : We present a new algorithm to compute the $n$ middle coefficients of a $2n \times n$ product in $K(n)$ ring operations, where $K(n)$ is the number of operations needed by Karatsuba's algorithm for a full $n \times n$ product. Used in Newton iteration, and together with previous work of Mulders, Karp and Markstein, this algorithm enables one to compute an inverse in $\sim 0.904 \, K(n)$ operations, a quotient in $\sim 1.173 \, K(n)$ operations, and a square root in $\sim 0.891 \, K(n)$ operations. These results apply both to power series and polynomials.
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https://hal.inria.fr/inria-00099294
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Submitted on : Tuesday, September 26, 2006 - 8:52:33 AM
Last modification on : Thursday, January 11, 2018 - 6:19:48 AM

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  • HAL Id : inria-00099294, version 1

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Guillaume Hanrot, Paul Zimmermann. Speeding up the inversion of power series. [Intern report] A00-R-067 || hanrot00a, 2000, 8 p. ⟨inria-00099294⟩

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