Boole's rule

Remarks

The rule is the closed Newton–Cotes formula with five nodes: its weights are the integrals of the Lagrange basis polynomials of the five equally spaced nodes. By symmetry it is exact for all polynomials of degree at most $5$, one more than the interpolation degree, so its degree of exactness is $5$. The error constant can be read off from $f(x) = x^6$ on $[0, 4]$ with $h = 1$: the integral is $\frac{4^7}{7}$, the rule gives $\frac{4^7}{7} + \frac{128}{21}$, and $\frac{8}{945} \cdot 720 = \frac{128}{21}$. Check of the weights: $\frac{2h}{45}(7 + 32 + 12 + 32 + 7) = 4h$, the length of the interval. The composite rule on $4m$ subintervals has error $O(h^6)$ for $f \in C^6$.

Going deeper

For a student

Boole's rule is the next member after the trapezoidal rule, Simpson's rule and Simpson's $3/8$ rule in the closed Newton–Cotes family. It equals one step of Richardson extrapolation applied to Simpson's rule with steps $2h$ and $h$, as in the Romberg method. Higher Newton–Cotes rules eventually have negative weights and become unstable, so in practice one uses composite rules of low order or Gaussian rules instead.