sum of small double numbers c++

Question

Assume that we have an array of small (about 10^(-15) ) double numbers in c++. If we calculate the sum of numbers in this array sequentially, for example

double sum = 0;
for (int i = 0; i < n; i++) sum+=array[i];

we get some value x.

But if we divide an array into some parts and then calculate the sum in each part and after this we add all the partial sums together we get some value x2, which is close to x but not exactly x. So I have lost accruacy in calculating sum.

Does someone know how to calculate the sum of small double numbers by partitioning these numbers into some parts without loosing accuracy?

Answer 1

Using Kahan Summation:

#include <numeric>
#include <iostream>
#include <vector>

struct KahanAccumulation
{
    double sum;
    double correction;
};

KahanAccumulation KahanSum(KahanAccumulation accumulation, double value)
{
    KahanAccumulation result;
    double y = value - accumulation.correction;
    double t = accumulation.sum + y;
    result.correction = (t - accumulation.sum) - y;
    result.sum = t;
    return result;
}

int main()
{
    std::vector<double> numbers = {0.01, 0.001, 0.0001, 0.000001, 0.00000000001};
    KahanAccumulation init = {0};
    KahanAccumulation result =
        std::accumulate(numbers.begin(), numbers.end(), init, KahanSum);

    std::cout << "Kahan Sum: " << result.sum << std::endl;
    return 0;
}

Output:

Kahan Sum: 0.011101

Code here.

Answer 2

The absolute size of the numbers is not the issue.

If you want a more accurate summation, have you considered a compensated sum? http://en.wikipedia.org/wiki/Kahan_summation_algorithm

However, if you really mean without losing any accuracy, your result will not necessarily fit in a double. If this is really what you want, you could look Algorithm 908 at http://dl.acm.org/citation.cfm?id=1824815 or similar.

Answer 3

The trick in those cases is to first order the array from smaller to higher, and then sum then in the cycle you've made. That way, the accuracy is best.

You can also check Kahan summation algorithm