Parabolas approximating the area under a curve, with adaptive subdivision refining where the local error is large.
| Rule | Shape | Per-panel error | Composite error | Degree of precision | Eval cost |
|---|---|---|---|---|---|
| Midpoint | rectangle | +(h³/24)f″ | O(h²) | 1 | O(n) |
| Trapezoid | line | −(h³/12)f″ | O(h²) | 1 | O(n) |
| Simpson 1/3 | parabola | −(h⁵/90)f⁗ | O(h⁴) | 3 | O(n) |
| Boole | quartic | −(8h⁷/945)f⁽⁶⁾ | O(h⁶) | 5 | O(n) |
| Gauss n-point | optimal | f⁽²ⁿ⁾ term | degree 2n−1 exact | 2n−1 | O(n) |