Truncation error in Simpson's rule

Numerical Integration: Simpson's rule

Truncation error in Simpson's rule

Without proof we give the estimate of the truncation error in Simpson's rule for a 'neat' function on the interval $[a,b]$ with a partition into an even number of $n$ sub-intervals.

Suppose that $M$ is the maximum of $|f''''|$ on $[a,b]$ . Then we get for the result $S$ of Simpson's rule with mesh size $h=\frac{b-a}{n}$ :

$\left|\int_a^bf(x)\,\dd x - S\right|\le \tfrac{1}{180}M(b-a)h^4$ In other words: the truncation error is a fourth power in the mesh size $h$ .