Pairwise summation: Difference between revisions

In particular, pairwise summation of a sequence of ''n'' numbers ''x_n'' works by [[recursion (computer science)|recursively]] breaking the sequence into two halves, summing each half, and adding the two sums: a [[divide and conquer algorithm]]. Its roundoff errors grow [[Big O notation|asymptotically]] as at most ''O''(ε&nbsp;log&nbsp;''n''), where ε is the [[machine precision]] (assuming a fixed [[condition number]], as discussed below).<ref name=Higham93/> In comparison, the naive technique of accumulating the sum in sequence (adding each ''x_i'' one at a time for ''i''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;''n'') has roundoff errors that grow at worst as ''O''(ε''n'').<ref name=Higham93/> [[Kahan summation]] has a [[error bound|worst-case error]] of roughly ''O''(ε), independent of ''n'', but requires several times more arithmetic operations.<ref name=Higham93/> 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 <math>O(\varepsilon \sqrt{\log n})</math> for pairwise summation.<ref name=Tasche>Manfred Tasche and Hansmartin Zeuner ''Handbook of Analytic-Computational Methods in Applied Mathematics'' Boca Raton, FL: CRC Press, 2000).</ref>

 

 

==The algorithm==