Worst cases and lattice reduction

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 (D. Coppersmith, 1996; 2001). 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/sup -d2//(2d+1) from a machine number in time O(N/sup ((d+1)/(2d+1))+/spl epsiv//). For d=2, this improves on the O(N/sup 2/(3+/spl epsiv/)/) complexity from Lefevre's algorithm (V. Lefevre, 2000; V. Lefevre et al., 2001) to O(N/sup 3/(5+/spl epsiv/)/). 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/sup -k/ in time O(N/sup (1/2)+O(1/k)/).