FLINT3.7.0-dev
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_d_vec_dot_thrice

Synopsis#include <flint/d_vec.h>
double _d_vec_dot_thrice(const double * vec1, const double * vec2, slong len2, double * err)source

Returns the dot product computed as if in three times the working precision and then rounded to double, using error-free transformations (the algorithm Dot3 of Ogita, Rump and Oishi [ORO2005]). The result is accurate to nearly full double precision unless the dot product is extremely ill-conditioned. If err is not NULL, sets *err to the error bound of [ORO2005] in terms of the returned value rr, evaluated in double precision: (ε+2γ2)∣r∣+γ3∥x∥2∥y∥2(\varepsilon + 2\gamma^2) |r| + \gamma^3 \|x\|_2 \|y\|_2 with ε=2−52\varepsilon = 2^{-52} and γ=(4n−2)ε/(1−(4n−2)ε)\gamma = (4n-2)\varepsilon / (1 - (4n-2)\varepsilon).

Each product xiyix_i y_i is split into hi+eih_i + e_i exactly by Dekker's product with Veltkamp splitting (the code does not use FMA), the sum of the hih_i is accumulated with Knuth's TwoSum, keeping the n−1n - 1 rounding errors, and the resulting 2n2n terms are summed again with a cascade of TwoSums followed by an ordinary sum of the errors; see Error-free transformations for the error-free transformations. The products must not overflow, and products below about 2−9692^{-969} lose the exactness of the splitting. A temporary vector of 2n2n doubles is allocated.

References#

  • [ORO2005] T. Ogita, S. M. Rump and S. Oishi. Accurate sum and dot product. SIAM Journal on Scientific Computing 26:6 (2005), 1955–1988. https://doi.org/10.1137/030601818
Cite this page

The FLINT team, “_d_vec_dot_thrice (d_vec.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/d_vec/_d_vec_dot_thrice.html (generated 2026-11-02).