composite quadrature rule

Applying a simple quadrature rule (midpoint, trapezoidal, or Simpson's) separately on each of many small panels covering [a,b], instead of once across the whole interval.

A composite quadrature rule partitions $[a,b]$ into $n$ subintervals of width $h=\frac{b-a}{n}$ and applies a single-panel rule to each one, summing the results. Composite trapezoidal and midpoint rules have error $O(h^2)$ ; composite Simpson's rule has error $O(h^4)$ , one order better because a single Simpson panel is already exact for cubics, not just parabolas.

Examples

Composite Simpson's rule on $[0,4]$ with $n=4$ subintervals applies the single-panel rule to $[0,1],[1,2],[2,3],[3,4]$ and sums the four results.