KahanSummen

KahanSummen refers to a method for computing the sum of a sequence of floating-point numbers that aims to mitigate the loss of precision that can occur with naive summation. Developed by William Kahan, this algorithm is particularly effective when dealing with numbers of vastly different magnitudes or when summing a large number of values. The core idea behind KahanSummen is to keep track of a running compensation for the small, "lost" parts of the sum due to floating-point arithmetic.

In standard summation, when adding a small number to a large sum, the smaller number might be rounded off entirely if it's much smaller than the current sum. This truncation error accumulates over many additions. KahanSummen addresses this by maintaining a variable, often called `c` for compensation or error. At each step, the algorithm adds the current number to the sum, and then corrects the sum by adding the difference between the current number and the accumulated error back to the sum. The error variable itself is then updated to reflect the new accumulated error.

The algorithm typically involves a few variables: the current sum, the running compensation (error), and the next number to be added. The process is iterative. While it adds a small overhead compared to naive summation, the improvement in accuracy can be significant, especially in scientific computing and financial applications where precision is paramount. Libraries in many programming languages offer implementations of KahanSummen or similar compensated summation algorithms.