Given floating-point arithmetic with t-digit base-β significands in which all arithmetic operations are performed as if calculated to infinite precision and rounded to a nearest representable value, we prove that the product of complex values z 0 and z 1 can be computed with maximum absolute error |z 0 z 1 | 1 2 β 1−t √ 5. In particular, this provides relative error bounds of 2 −24 √ 5 and 2 −53 √ 5 for IEEE 754 single and double precision arithmetic respectively, provided that overflow, underflow, and denormals do not occur. We also provide the numerical worst cases for IEEE 754 single and double precision arithmetic.
Abstract. We describe a method for searching for ideal symmetric solutions to the Prouhet-Tarry-Escott Problem. We report results of extensive searches for solutions of sizes up to 12. We found two solutions of size 10 that are smaller by two orders of magnitude than the solution found by A. Letac in the 1940s, which was the smallest size 10 solution known before our search.
Abstract. We extend the work of Richard Crandall et al. to demonstrate how the Discrete Weighted Transform (DWT) can be applied to speed up multiplication modulo any number of the form a ± b where p|ab p is small. In particular this allows rapid computation modulo numbers of the form k ·2 n ±1.In addition, we prove tight bounds on the rounding errors which naturally occur in floating-point implementations of FFT and DWT multiplications. This makes it possible for FFT multiplications to be used in situations where correctness is essential, for example in computer algebra packages.
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