Editing Simpson's rule (section)

{{short description|Method for numerical integration}}
{{For-text|Simpson's voting rule|[[Minimax Condorcet]]|the rule in naval architecture|[[Simpson's rules (ship stability)]]}}
{{Use dmy dates|date=January 2020}}
[[Image:simpsons method illustration.svg|thumb|right|Simpson's rule can be derived by approximating the integrand ''f ''(''x'') <span style="color:blue;">(in blue)</span> by the quadratic interpolant ''P''(''x'') <span style="color:red;">(in red)</span>.]]

[[File:Simpson's One-Third Rule.gif|thumb|right|An animation showing how Simpson's rule approximates the function with a parabola and the reduction in error with decreased step size]]
[[Image:simpsonsrule2.gif|thumb|right|An animation showing how Simpson's rule approximation improves with more subdivisions.]]

In [[numerical integration]], '''Simpson's rules''' are several [[approximation]]s for [[definite integral]]s, named after [[Thomas Simpson]] (1710–1761).

The most basic of these rules, called '''Simpson's 1/3 rule''', or just '''Simpson's rule''', reads
<math display="block">\int_a^b f(x) \, dx \approx \frac{b - a}{6} \left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right].</math>

In German and some other languages, it is named after [[Johannes Kepler]], who derived it in 1615 after seeing it used for wine barrels (barrel rule, {{lang|de|Keplersche Fassregel}}). The approximate equality in the rule becomes exact if {{math|''f''}} is a polynomial up to and including 3rd degree.

If the 1/3 rule is applied to ''n'' equal subdivisions of the integration range [''a'',&nbsp;''b''], one obtains the [[#Composite Simpson's 1/3 rule|'''composite Simpson's 1/3 rule''']]. Points inside the integration range are given alternating weights 4/3 and 2/3.

[[#Simpson's 3/8 rule|'''Simpson's 3/8 rule''']], also called '''Simpson's second rule''', requires one more function evaluation inside the integration range and gives lower error bounds, but does not improve the order of the error.

If the 3/8 rule is applied to ''n'' equal subdivisions of the integration range [''a'',&nbsp;''b''], one obtains the [[#Composite Simpson's 3/8 rule|'''composite Simpson's 3/8 rule''']].

Simpson's 1/3 and 3/8 rules are two special cases of closed [[Newton–Cotes formulas]].

In naval architecture and ship stability estimation, there also exists ''Simpson's third rule'', which has no special importance in general [[numerical analysis]], see [[Simpson's rules (ship stability)]].