Accurate sum and dot product

Takeshi Ogita, Siegfried M. Rump, Shin'ichi Oishi

published in
SIAM Journal on Scientific Computing, 26:6 (2005), pp. 1955-1988.

Algorithms for summation and dot product of floating point numbers are presented which are fast in terms of measured computing time. We show that the computed results are as accurate as if computed in twice or K-fold working precision, K >= 3. For twice the working precision our algorithms for summation and dot product are some 40 % faster than the corresponding XBLAS routines while sharing similar error estimates. Our algorithms are widely applicable because they require only addition, subtraction and multiplication of floating point numbers in the same working precision as the given data. Higher precision is unnecessary, algorithms are straight loops without branch, and no access to mantissa or exponent is necessary.