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Abstract:

For a computer algebra system, it is crucial to optimize the arithmetical operations on basic objects|numbers, polynomials, series,... In fact, two classes of objects can be distinguished: integers and polynomials, which require exact operations; oating-point numbers and series, for which only the most signicant part of the exact result is needed. The best algorithms currently known for multiplication, division, and square root on integers and oating-point numbers are mostly recent. We present and analyse them using complexity models based on three dierent multiplication algorithms (naive, Karatsuba, and FFT). The MPFR library developed by Guillaume Hanrot and Paul Zimmermann is a C library for multiprecision oating-point computations with exact rounding [6]. Its main purpose is to achieve eciency with a well-dened semantics. Beside the elementary operations +, , , and /, it provides routines for square root (with remainder in the integer case, without remainder in the oating-point case), logarithm and exponential. The longer-term goal is to integrate routines for the numerical evaluation of other elementary and special functions as well. Paul Zimmermann's algorithm for square roots [8] originates in this work. It is reported on here, as well as other recent fast algorithms for multiplications, divisions, and square roots. They all base

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