Pairwise summation

In numerical analysis, pairwise summation, also called cascade summation, is a technique to sum a sequence of finite-precision floating-point numbers that substantially reduces the accumulated round-off error compared to naively accumulating the sum in sequence. Although there are other techniques such as Kahan summation that typically have even smaller round-off errors, pairwise summation is nearly as good (differing only by a logarithmic factor) while having much lower computational cost—it can be implemented so as to have nearly the same cost (and exactly the same number of arithmetic operations) as naive summation.In pseudocode, the pairwise summation algorithm for an array x of length n > 0 can be written:Suppose that one is summing n values xi, for i = 1, ..., n. The exact sum is:In the C Sharp language, HPCsharp nuget package implements pairwise summation, as well as Kahan summation (Neumaier variation): serial, data parallel using SIMD processor instructions, and parallel multi-core.