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Fast sumation functions.


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.:

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).
  Source Edit