Worst Cases and Lattice Reduction
Résumé
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))).