A función f (x ) (azul) é aproximada por unha función cuadrática P (x ) (vermello). Na análise numérica , a regra ou método de Simpson (chamada así na honra de Thomas Simpson ) é un método de integración numérica que se utiliza para obter a aproximación da integral :
∫
a
b
f
(
x
)
d
x
≈
b
−
a
6
[
f
(
a
)
+
4
f
(
a
+
b
2
)
+
f
(
b
)
]
{\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{6}}\left[f(a)+4f\left({\frac {a+b}{2}}\right)+f(b)\right]}
Consideramos o polinomio interpolante de orde dous
P
2
(
x
)
{\displaystyle \mathrm {P_{2}(x)} }
f
(
x
)
{\displaystyle \mathrm {f(x)} }
0 = a, x1 = b e m = (a+b)/2. A expresión dese polinomio interpolante, expresado a través da Interpolación polinómica de Lagrange é:
P
2
(
x
)
=
f
(
a
)
(
x
−
m
)
(
x
−
b
)
(
a
−
m
)
(
a
−
b
)
+
f
(
m
)
(
x
−
a
)
(
x
−
b
)
(
m
−
a
)
(
m
−
b
)
+
f
(
b
)
(
x
−
a
)
(
x
−
m
)
(
b
−
a
)
(
b
−
m
)
.
{\displaystyle P_{2}(x)=f(a){\frac {(x-m)(x-b)}{(a-m)(a-b)}}+f(m){\frac {(x-a)(x-b)}{(m-a)(m-b)}}+f(b){\frac {(x-a)(x-m)}{(b-a)(b-m)}}.}
Así, a integral buscada pódese aproximar como:
∫
a
b
f
(
x
)
d
x
≈
∫
a
b
P
2
(
x
)
d
x
=
h
3
[
f
(
a
)
+
4
f
(
m
)
+
f
(
b
)
]
.
{\displaystyle \int _{a}^{b}f(x)\,dx\approx \int _{a}^{b}P_{2}(x)\,dx={\frac {h}{3}}\left[f(a)+4f(m)+f(b)\right].}
O erro de aproximar a integral mediante o método de Simpson é
−
h
5
90
f
(
4
)
(
ξ
)
,
{\displaystyle -{\frac {h^{5}}{90}}f^{(4)}(\xi ),}
onde
h
=
(
b
−
a
)
/
2
{\displaystyle h=(b-a)/2}
ξ
∈
[
a
,
b
]
{\displaystyle \xi \in [a,b]}
No caso de que o intervalo [a,b] non sexa o suficientemente pequeno, o erro ao calcular a integral pode ser moi grande. Para iso, recórrese á fórmula composta de Simpson.
Dividiremos o intervalo [a,b] en n subintervalos iguais, de xeito que
x
i
=
a
+
i
h
{\displaystyle x_{i}=a+ih}
h
=
(
b
−
a
)
/
2
n
{\displaystyle h=(b-a)/2n}
i
=
0
,
1
,
.
.
.
,
n
{\displaystyle i=0,1,...,n}
Aplicando a Regra de Simpson a cada subintervalo, temos:
∫
x
j
−
1
x
j
f
(
x
)
d
x
=
x
j
−
x
j
−
1
6
[
f
(
x
j
−
1
)
+
4
f
(
x
j
−
1
+
x
j
2
)
+
f
(
x
j
)
]
j
=
1
,
.
.
.
,
n
.
{\displaystyle \int _{x_{j-1}}^{x_{j}}f(x)\,dx={\frac {x_{j}-x_{j-1}}{6}}\left[f(x_{j-1})+4f\left({\frac {x_{j-1}+x_{j}}{2}}\right)+f(x_{j})\right]j=1,...,n.}
Sumando as integrais de todos os subintervalos, chegamos a que:
∫
a
b
f
(
x
)
d
x
≈
h
3
[
f
(
x
0
)
+
2
∑
j
=
1
n
/
2
−
1
f
(
x
2
j
)
+
4
∑
j
=
1
n
/
2
f
(
x
2
j
−
1
)
+
f
(
x
n
)
]
,
{\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {h}{3}}{\bigg [}f(x_{0})+2\sum _{j=1}^{n/2-1}f(x_{2j})+4\sum _{j=1}^{n/2}f(x_{2j-1})+f(x_{n}){\bigg ]},}
O máximo erro vén dado pola expresión
−
h
4
180
(
b
−
a
)
f
(
4
)
(
ξ
)
.
{\displaystyle -{\frac {h^{4}}{180}}(b-a)f^{(4)}(\xi ).}
![{\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{6}}\left[f(a)+4f\left({\frac {a+b}{2}}\right)+f(b)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c88a511d85fc5ad91c6ac0fc8d1869faf37db32)
![{\displaystyle \int _{a}^{b}f(x)\,dx\approx \int _{a}^{b}P_{2}(x)\,dx={\frac {h}{3}}\left[f(a)+4f(m)+f(b)\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ffd8b89cdaf6d99421839e81436fd70b522f79)
![{\displaystyle \xi \in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc05d1fb45b90e25c99bc6a57473d508d3e9c23)
![{\displaystyle \int _{x_{j-1}}^{x_{j}}f(x)\,dx={\frac {x_{j}-x_{j-1}}{6}}\left[f(x_{j-1})+4f\left({\frac {x_{j-1}+x_{j}}{2}}\right)+f(x_{j})\right]j=1,...,n.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0e0d7e9bbf91db03e28ee019793b8b42c5ab892)
![{\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {h}{3}}{\bigg [}f(x_{0})+2\sum _{j=1}^{n/2-1}f(x_{2j})+4\sum _{j=1}^{n/2}f(x_{2j-1})+f(x_{n}){\bigg ]},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b2070b3485a1e013e6c1d72e5fea9581a65094)
