Editing Trapezoidal rule (section)

== Applicability and alternatives ==
The trapezoidal rule is one of a family of formulas for [[numerical integration]] called [[Newton–Cotes formulas]], of which the [[midpoint rule]] is similar to the trapezoid rule. [[Simpson's rule]] is another member of the same family, and in general has faster convergence than the trapezoidal rule for functions which are twice continuously differentiable, though not in all specific cases. However, for various classes of rougher functions (ones with weaker smoothness conditions), the trapezoidal rule has faster convergence in general than Simpson's rule.<ref name="cun02">{{Harv|Cruz-Uribe|Neugebauer|2002}}</ref>

Moreover, the trapezoidal rule tends to become extremely accurate when [[periodic function]]s are integrated over their periods, which can be [[#Periodic and peak functions|analyzed in various ways]].<ref name="rs90">{{Harv|Rahman|Schmeisser|1990}}</ref><ref name="w02">{{Harv|Weideman|2002}}</ref> A similar effect is available for peak functions.<ref name=":0" /><ref name="w02" />

For non-periodic functions, however, methods with unequally spaced points such as [[Gaussian quadrature]] and [[Clenshaw–Curtis quadrature]] are generally far more accurate; Clenshaw–Curtis quadrature can be viewed as a change of variables to express arbitrary integrals in terms of periodic integrals, at which point the trapezoidal rule can be applied accurately.