Editing Newton–Cotes formulas (section)

== Instability for high degree ==
A Newton–Cotes formula of any degree {{mvar|n}} can be constructed. However, for large {{mvar|n}} a Newton–Cotes rule can sometimes suffer from catastrophic [[Runge's phenomenon]]<ref>{{cite book |author-link1= Alfio Quarteroni|last1=Quarteroni |first1=Alfio |last2=Sacco |first2=Riccardo |last3=Saleri |first3=Fausto |title=Numerical Mathematics |date=2006 |publisher=Springer |isbn=978-3-540-34658-6 |pages=390–391 |edition=Second}}</ref> where the error grows exponentially for large {{mvar|n}}. Methods such as Gaussian quadrature and Clenshaw–Curtis quadrature with unequally spaced points (clustered at the ''endpoints'' of the integration interval) are stable and much more accurate, and are normally preferred to Newton–Cotes. If these methods cannot be used, because the integrand is only given at the fixed equidistributed grid, then Runge's phenomenon can be avoided by using a composite rule, as explained below.

Alternatively, stable Newton–Cotes formulas can be constructed using least-squares approximation instead of interpolation. This allows building numerically stable formulas even for high degrees.<ref>{{cite web|url=http://www.holoborodko.com/pavel/numerical-methods/numerical-integration/stable-newton-cotes-formulas/|title=Stable Newton-Cotes Formulas|author=Pavel Holoborodko|date=2011-03-24|access-date=2015-08-17}}</ref><ref>{{cite web|url=http://www.holoborodko.com/pavel/numerical-methods/numerical-integration/stable-newton-cotes-formulas-open-type/|title=Stable Newton-Cotes Formulas (Open Type)|author=Pavel Holoborodko|date=2012-05-20|access-date=2015-08-18}}</ref>