mittenregel

Mittenregel is a German-language term that can refer to several heuristic principles centered on the midpoint of an interval or data set. In mathematical literature the closest standard term is Mittelpunktsregel, commonly translated as the midpoint rule. In numerical analysis, the midpoint rule provides a simple method for approximating definite integrals and other quantities that depend on a function’s average value over an interval.

For a definite integral ∫_a^b f(x) dx, the simplest midpoint rule uses a single subinterval and estimates the integral by (b−a) f((a+b)/2). The composite midpoint rule generalizes this to partitions a = x0 < x1 < ... < xn = b, with Δx = (b−a)/n, and estimates the integral by Δx ∑_{i=1}^n f((x_{i−1}+x_i)/2). The midpoint rule is second-order accurate, with the leading error term proportional to the maximum of |f''(x)| on [a,b] and to (b−a)Δx^2, under suitable smoothness assumptions.

Outside formal mathematics, the term Mittenregel can appear as an informal reference to using a midpoint or central value as a representative estimate in estimation or measurement contexts. However, this usage is ambiguous, and the standard mathematical name Mittelpunktsregel is preferred in technical writing.

See also: numerical integration, midpoint rule, Mittelpunktsregel.