The composite Simpson's rule approximates a definite integral by applying Simpson's rule to successive
pairs of equal subintervals. For 
and 
, it is
![int_a^bf(x)dx approx h/3[f(x_0)+4sum_(j=1)^nf(x_(2j-1))+2sum_(j=1)^(n-1)f(x_(2j))+f(x_(2n))].](/images/equations/CompositeSimpsonsRule/NumberedEquation1.svg) |
(1)
|
If 
has a continuous fourth derivative, then for some
the error is
 |
(2)
|
where the error means the exact integral minus its approximation. Thus the error satisfies
 |
(3)
|
Here 
is big-O notation indicating that as 
, the error magnitude is bounded by a constant multiple
of 
for all sufficiently small positive 
.
