Editing Gaussian quadrature (section)

=== Error estimates ===
The error of a Gaussian quadrature rule can be stated as follows.<ref>{{harvnb|Stoer|Bulirsch|2002|loc=Thm&nbsp;3.6.24}}</ref> For an integrand which has {{math|2''n''}} continuous derivatives,
<math display="block"> \int_a^b \omega(x)\,f(x)\,dx - \sum_{i=1}^n w_i\,f(x_i) = \frac{f^{(2n)}(\xi)}{(2n)!} \, (p_n, p_n) </math>
for some {{mvar|ξ}} in {{math|(''a'', ''b'')}}, where {{mvar|p<sub>n</sub>}} is the monic (i.e. the leading coefficient is {{math|1}}) orthogonal polynomial of degree {{mvar|n}} and where
<math display="block"> (f,g) = \int_a^b \omega(x) f(x) g(x) \, dx.</math>

In the important special case of {{math|1=''ω''(''x'') = 1}}, we have the error estimate<ref>{{harvnb|Kahaner|Moler|Nash|1989|loc=§5.2}}</ref>
<math display="block"> \frac{\left(b - a\right)^{2n+1} \left(n!\right)^4}{(2n + 1)\left[\left(2n\right)!\right]^3} f^{(2n)} (\xi), \qquad a < \xi < b.</math>

Stoer and Bulirsch remark that this error estimate is inconvenient in practice, since it may be difficult to estimate the order {{math|2''n''}} derivative, and furthermore the actual error may be much less than a bound established by the derivative. Another approach is to use two Gaussian quadrature rules of different orders, and to estimate the error as the difference between the two results. For this purpose, Gauss–Kronrod quadrature rules can be useful.