Question

Which one of the following is a correct algorithm for the above numerical sequence?
1 point
Algorithm:
Algorithm:
Algorithm:
Algorithm:

Answer 1

Answer:

In numerical analysis, the Kahan summation algorithm, also known as compensated summation,[1] significantly reduces the numerical error in the total obtained by adding a sequence of finite-precision floating-point numbers, compared to the obvious approach. This is done by keeping a separate running compensation (a variable to accumulate small errors).

In particular, simply summing n numbers in sequence has a worst-case error that grows proportional to n, and a root mean square error that grows as {\displaystyle {\sqrt {n}}}{\sqrt {n}} for random inputs (the roundoff errors form a random walk).[2] With compensated summation, the worst-case error bound is effectively independent of n, so a large number of values can be summed with an error that only depends on the floating-point precision.[2]

The algorithm is attributed to William Kahan.[3] Similar, earlier techniques are, for example, Bresenham's line algorithm, keeping track of the accumulated error in integer operations (although first documented around the same time[4]) and the delta-sigma modulation[5] (integrating, not just summing the error).