Fast sumation functions.
Funcs
func sumKbn[T](x: openArray[T]): T
- Kahan (compensated) summation: O(1) error growth, at the expense of a considerable increase in computational expense. Source Edit
func sumPairs[T](x: openArray[T]): T
-
Pairwise (cascade) summation of x[i0:i0+n-1], with O(log n) error growth (vs O(n) for a simple loop) with negligible performance cost if the base case is large enough.
See, e.g.:
- http://en.wikipedia.org/wiki/Pairwise_summation Higham, Nicholas J. (1993), "The accuracy of floating point summation", SIAM Journal on Scientific Computing 14 (4): 783–799.
In fact, the root-mean-square error growth, assuming random roundoff errors, is only O(sqrt(log n)), which is nearly indistinguishable from O(1) in practice. See:
- Manfred Tasche and Hansmartin Zeuner, Handbook of Analytic-Computational Methods in Applied Mathematics (2000).