In numerical analysis, pairwise summation, also called cascade summation, is a summation algorithm, i.e. 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 particular, pairwise summation of a sequence of n numbers xn works by recursively breaking the sequence into two halves, summing each half, and adding the two sums: a divide and conquer algorithm. Its worst-case roundoff errors grow asymptotically as at most O(ε log n), where ε is the machine precision (assuming a fixed condition number, as discussed below). In comparison, the naive technique of accumulating the sum in sequence (adding each xi one at a time for i = 1, ..., n) has roundoff errors that grow at worst as O(εn). Kahan summation has a worst-case error of roughly O(ε), independent of n, but requires several times more arithmetic operations. If the roundoff errors are random, and in particular have random signs, then they form a random walk and the error growth is reduced to an average of O ( ε log n ) {\displaystyle O(\varepsilon {\sqrt {\log n}})} for pairwise summation. A very similar recursive structure of DFT decomposition is found in many fast Fourier transform (FFT) algorithms, and is responsible for the same slow roundoff accumulation of those FFTs when implemented with a breadth-first technique due to poor memory locality.
The algorithm In pseudocode, the pairwise summation algorithm for an array x of length n ≥ 0 can be written:
s = pairwise(x[1...n]) if n ≤ N base case: naive summation for a sufficiently small array s = 0 for i = 1 to n s = s + x[i] else divide and conquer: recursively sum two halves of the array m = floor(n / 2) s = pairwise(x[1...m]) + pairwise(x[m+1...n]) end if
For some sufficiently small N, this algorithm switches to a naive loop-based summation as a base case, whose error bound is O(Nε). The entire sum has a worst-case error that grows asymptotically as O(ε log n) for large n, for a given condition number (see below). In an algorithm of this sort (as for divide and conquer algorithms in general), it is desirable to use a larger base case in order to amortize the overhead of the recursion. If N = 1, then there is roughly one recursive subroutine call for every input, but more generally there is one recursive call for (roughly) every N/2 inputs if the recursion stops at exactly n = N. By making N sufficiently large, the overhead of recursion can be made negligible (precisely this technique of a large base case for recursive summation is employed by high-performance FFT implementations). Regardless of N, exactly n−1 additions are performed in total, the same as for naive summation, so if the recursion overhead is made negligible then pairwise summation has essentially the same computational cost as for naive summation. A variation on this idea is to break the sum into b blocks at each recursive stage, summing each block recursively, and then summing the results, which was dubbed a "superblock" algorithm by its proposers. The above pairwise algorithm corresponds to b = 2 for every stage except for the last stage which is b = N. Dalton, Wang & Blainey (2014) describe a iterative, "shift-reduce" formulation for pairwise summation. It can be unrolled and sped up using SIMD instructions. The non-unrolled version is:
Accuracy Suppose that one is summing n values xi, for i = 1, ..., n. The exact sum is:
S n = ∑ i = 1 n x i {\displaystyle S_{n}=\sum _{i=1}^{n}x_{i}}
(computed with infinite precision). With pairwise summation for a base case N = 1, one instead obtains S n + E n {\displaystyle S_{n}+E_{n}} , where the error E n {\displaystyle E_{n}} is bounded above by:
| E n | ≤ ε log 2 n 1 − ε log 2 n ∑ i = 1 n | x i | {\displaystyle |E_{n}|\leq {\frac {\varepsilon \log _{2}n}{1-\varepsilon \log _{2}n}}\sum _{i=1}^{n}|x_{i}|}
where ε is the machine precision of the arithmetic being employed (e.g. ε ≈ 10−16 for standard double precision floating point). Usually, the quantity of interest is the relative error | E n | / | S n | {\displaystyle |E_{n}|/|S_{n}|} , which is therefore bounded above by:
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