Worst Cases and Lattice Reduction

Damien Stehlé 1 Vincent Lefèvre 1 Paul Zimmermann 1
1 SPACES - Solving problems through algebraic computation and efficient software
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : We propose a new algorithm to find worst cases for correct rounding of an analytic function. We first reduce this problem to the real small value problem --- i.e. for polynomials with real coefficients. Then we show that this second problem can be solved efficiently, by extending Coppersmith's work on the integer small value problem --- for polynomials with integer coefficients --- using lattice reduction [4,5,6]. For floating-point numbers with a mantissa less than N, and a polynomial approximation of degree d, our algorithm finds all worst cases at distance < N [power](-d2/2d-1) from a machine number in time [OMICRON](N[power]((d-1)/(2d-1)+e)). For d=2, this improves on the [OMICRON](N- [power]((2/3)+e)) complexity from Lefèvre's algorithm [12,13] to [OMICRON](N[p- ower]((3/5)+e)). We exhibit some new worst cases found using our algorithm, for double-extended and quadruple precision. For larger d, our algorithm can be used to check that there exist no worst cases at distance < N[power]-k in time [OMICRON](N[power](1/2+[OMICRON](1/k))).
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Submitted on : Tuesday, May 23, 2006 - 7:30:10 PM
Last modification on : Thursday, January 11, 2018 - 6:20:00 AM
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Damien Stehlé, Vincent Lefèvre, Paul Zimmermann. Worst Cases and Lattice Reduction. [Research Report] RR-4586, INRIA. 2002. ⟨inria-00071999⟩

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