Neumaiersummointi

Neumaier summation is a compensated summation technique used to reduce floating-point round-off errors when summing a sequence of real numbers. The method is named after Arnold Neumaier, who contributed to numerical analysis and error-aware accumulation. Unlike naive summation, which can lose small magnitudes through cancellation, Neumaier summation maintains a running total and an auxiliary correction that tracks accumulated rounding errors. The final result is obtained by adding the correction to the running total, yielding a more accurate sum in practice.

Algorithmically, the method uses two variables: s for the running sum and c for the compensation. Initialization is s = 0 and c = 0. For each input x, compute t = s + x. If abs(s) >= abs(x), update c = c + (s - t) + x; otherwise update c = c + (x - t) + s. Then set s = t. After processing all terms, the final sum is s + c. This approach is a form of compensated summation and is related to, but distinct from, Kahan summation.

Properties and usage: Neumaier summation provides improved numerical stability over naive summation and requires constant extra space with linear time in the number of terms. It performs well when summands vary greatly in magnitude or sign. However, like other floating-point methods, it does not guarantee exact results for all input sequences due to finite precision. It is used in numerical libraries and applications where accurate accumulation of many terms is important, such as scientific computing and simulations. Related methods include Kahan summation and other compensated summation techniques.