Newton-Cotes Quadrature

Newton-Cotes formulas approximate integrals by integrating polynomial interpolants through equally-spaced points, providing simple and widely-used quadrature rules.

DefinitionNewton-Cotes Formula

A Newton-Cotes formula of order $n$ approximates: $\int_a^b f(x) \, dx \approx \sum_{i=0}^n w_i f(x_i)$ where $x_i = a + ih$ with $h = (b-a)/n$ are equally-spaced nodes, and weights $w_i$ are chosen so the formula is exact for polynomials of degree $\leq n$ .

The weights satisfy $\sum_{i=0}^n w_i = b - a$ (exactness for constant functions).

The most common Newton-Cotes rules are:

ExampleFundamental Quadrature Rules

Trapezoidal Rule ( $n=1$ ): $\int_a^b f(x) \, dx \approx \frac{b-a}{2}[f(a) + f(b)]$ Exact for linear functions. Error: $-\frac{(b-a)^3}{12}f''(\xi)$ for some $\xi \in (a,b)$ .

Simpson's Rule ( $n=2$ ): $\int_a^b f(x) \, dx \approx \frac{b-a}{6}\left[f(a) + 4f\left(\frac{a+b}{2}\right) + f(b)\right]$ Exact for cubics (surprisingly, not just quadratics). Error: $-\frac{(b-a)^5}{2880}f^{(4)}(\xi)$ .

Simpson's 3/8 Rule ( $n=3$ ): $\int_a^b f(x) \, dx \approx \frac{3(b-a)}{8}\left[f(a) + 3f\left(\frac{2a+b}{3}\right) + 3f\left(\frac{a+2b}{3}\right) + f(b)\right]$

Remark

Composite Rules: For better accuracy, subdivide $[a,b]$ into $m$ subintervals and apply the rule on each:

Composite Trapezoidal: $\int_a^b f(x) \, dx \approx \frac{h}{2}\left[f(x_0) + 2\sum_{i=1}^{m-1} f(x_i) + f(x_m)\right]$ where $h = (b-a)/m$ . Error: $O(h^2)$ .

Composite Simpson: $\int_a^b f(x) \, dx \approx \frac{h}{3}\left[f(x_0) + 4\sum_{i \text{ odd}} f(x_i) + 2\sum_{i \text{ even} \neq 0,m} f(x_i) + f(x_m)\right]$ Error: $O(h^4)$ .

The composite trapezoidal rule is simple but requires many points for high accuracy. Simpson's rule is more efficient, achieving quartic convergence with only twice the work. However, for very smooth functions, adaptive methods or Gaussian quadrature (using optimally-chosen non-uniform nodes) outperform fixed Newton-Cotes formulas.

Higher-order Newton-Cotes formulas exist but are rarely used: they can have negative weights (numerical instability) and don't improve as rapidly as Gaussian quadrature. The trapezoidal and Simpson rules remain workhorses due to their simplicity, reliability, and sufficient accuracy for most applications.

Practical implementations use adaptive refinement: subdividing intervals where error estimates indicate insufficient accuracy, concentrating computational effort where the integrand varies rapidly.