Speeding up the Division and Square Root of Power Series

Guillaume Hanrot (1) , Michel Quercia , Paul Zimmermann (1)

1 POLKA - Polynomials, Combinatorics, Arithmetic

We present new algorithms for the inverse, quotient, or square root of power series. The key trick is a new algorithm -- RecursiveMiddleProduct or RMP -- computing the $n$ middle coefficients of a $2n x n$ product in essentially the same number of operations -- $K(n)$ -- than a full $n x n$ product with Karatsuba's method. This improves previous work of Mulders, Karp and Markstein, Burnikel and Ziegler. These results apply both to series, polynomials, and multiple precision floating-point numbers.

Mots clés: division; square root; Newton method; Karatsuba's algorithm

Domaines: Autre [cs.OH]

Guillaume Hanrot, Michel Quercia, Paul Zimmermann. Speeding up the Division and Square Root of Power Series. [Research Report] RR-3973, INRIA. 2000, pp.20. ⟨inria-00072675⟩