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Computer Arithmetic, IEEE Symposium on (2003)
Santiago de Compostela, Spain
June 15, 2003 to June 18, 2003
ISSN: 1063-6889
ISBN: 0-7695-1894-X
pp: 142
Damien Stehlé , ENS Paris
Vincent Lefèvre , Technopôle de Nancy-Brabois
Paul Zimmermann , Technopôle de Nancy-Brabois
<p>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 coef.cients. Then we show that this second problem can be solved ef.ciently, by extending Coppersmith?s work on the integer small value problem — for polynomials with integer coefficients — using lattice reduction [4, 5, 6].</p> <p>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\frac{{ - d^2 }}{{2d + 1}} from a machine number in time 0(N^{\frac{{d + 1}}{{2d + 1}} + \varepsilon } ). For d = 2, this improves on the 0(N^{{2 \mathord{\left/ {\vphantom {2 {3 + \varepsilon }}} \right. \kern-\nulldelimiterspace} {3 + \varepsilon }}} ) complexity from Lefèvre?s algorithm [15, 16] to 0(N^{{3 \mathord{\left/ {\vphantom {3 {5 + \varepsilon }}} \right. \kern-\nulldelimiterspace} {5 + \varepsilon }}} ). 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^{ - k} in time 0(N^{\frac{1}{2} + 0(\frac{1}{k})} ).</p>

V. Lefèvre, D. Stehlé and P. Zimmermann, "Worst Cases and Lattice Reduction," Computer Arithmetic, IEEE Symposium on(ARITH), Santiago de Compostela, Spain, 2003, pp. 142.
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