Rundlauffehler

Rundlauffehler, often translated as rounding error, is the difference between a computed value produced with finite‑precision arithmetic and the exact mathematical result. It arises because computers represent real numbers with a limited number of digits and because each arithmetic operation rounds its exact result to this finite format. In floating‑point systems, every operation can introduce a small relative error on the order of the machine epsilon, the smallest distinguishable difference for the format (for double precision this is about 2.22×10−16).

Errors accumulate over sequences of operations. They are especially problematic in ill‑conditioned problems or when subtracting nearly equal numbers, where cancellation can magnify relative error. Rounding errors are distinct from truncation errors, which arise when a mathematical model is approximated rather than evaluated exactly.

Mitigation and analysis: Use higher precision or compensated summation methods, such as Kahan summation, or reformulate algorithms to be numerically stable. Rearranging computations, scaling inputs, and using algorithms with proven backward stability reduce sensitivity to rounding. Error analysis distinguishes forward error (difference between computed and exact values) from backward error (the size of input perturbation that would produce the computed result exactly).

Practical notes: In many applications, rounding error bounds are proportional to the number of operations times the machine epsilon, but the actual behavior depends on the algorithm and data. Understanding Rundlauffehler helps in choosing numerical methods and in interpreting results from finite‑precision computations.