Functions for summing floating point numbers more accurately than
sum function and its counterparts in the
vector package and elsewhere.
When used with floating point numbers, in the worst case, the
sum function accumulates numeric error at a rate
proportional to the number of values being summed. The algorithms
in this module implement different methods of /compensated
summation/, which reduce the accumulation of numeric error so that
it either grows much more slowly than the number of inputs
(e.g. logarithmically), or remains constant.
- class Summation s where
- sumVector :: (Vector v Double, Summation s) => (s -> Double) -> v Double -> Double
- data KBNSum = KBNSum !Double !Double
- kbn :: KBNSum -> Double
- data KB2Sum = KB2Sum !Double !Double !Double
- kb2 :: KB2Sum -> Double
- data KahanSum = KahanSum !Double !Double
- kahan :: KahanSum -> Double
- pairwiseSum :: Vector v Double => v Double -> Double
Summation type class
A class for summation of floating point numbers.
The identity for summation.
Add a value to a sum.
O(n) Sum a vector of values.
Most of these summation algorithms are intended to be used via the
Summation typeclass interface. Explicit type annotations should
not be necessary, as the use of a function such as
to extract the final sum out of a
Summation instance gives the
compiler enough information to determine the precise type of
summation algorithm to use.
As an example, here is a (somewhat silly) function that manually computes the sum of elements in a list.
sillySumList :: [Double] -> Double sillySumList = loop
zerowhere loop s  =
kbns loop s (x:xs) =
seqs' loop s' xs where s' =
In most instances, you can simply use the much more general
function instead of writing a summation function by hand.
Kahan-Babuška-Neumaier summation. This is a little more computationally costly than plain Kahan summation, but is always at least as accurate.
Order-2 Kahan-Babuška summation
Second-order Kahan-Babuška summation. This is more computationally costly than Kahan-Babuška-Neumaier summation, running at about a third the speed. Its advantage is that it can lose less precision (in admittedly obscure cases).
This method compensates for error in both the sum and the first-order compensation term, hence the use of "second order" in the name.
Less desirable approaches
Kahan summation. This is the least accurate of the compensated summation methods. In practice, it only beats naive summation for inputs with large magnitude. Kahan summation can be less accurate than naive summation for small-magnitude inputs.
This summation method is included for completeness. Its use is not
recommended. In practice,
KBNSum is both 30% faster and more
O(n) Sum a vector of values using pairwise summation.
This approach is perhaps 10% faster than
KBNSum, but has poorer
bounds on its error growth. Instead of having roughly constant
error regardless of the size of the input vector, in the worst case
its accumulated error grows with O(log n).
- Kahan, W. (1965), Further remarks on reducing truncation errors. Communications of the ACM 8(1):40.
- Neumaier, A. (1974), Rundungsfehleranalyse einiger Verfahren zur Summation endlicher Summen. Zeitschrift für Angewandte Mathematik und Mechanik 54:39–51.
- Klein, A. (2006), A Generalized Kahan-Babuška-Summation-Algorithm. Computing 76(3):279-293.
- Higham, N.J. (1993), The accuracy of floating point summation. SIAM Journal on Scientific Computing 14(4):783–799.