B10 - numerical integration

discrete riemann integral

an riemann integral can be numerically represented as:

\begin{aligned} \int_{a}^{b} f (x) d x & = lim_{N \to \infty} \sum_{k = 1}^{N} f (a + f Δ x) Δ x \\ (right-sided rectangle) & = lim_{N \to \infty} \sum_{k = 1}^{N} f (x_{k}) Δ z \\ (left-sided rectangle) & = lim_{N \to \infty} \sum_{k = 0}^{N - 1} f (x_{k}) Δ z \end{aligned}

the local error $\sim Δ x^{2}$ , and the global error $\sim Δ x$
the trapezoidal integration averages the right and the left-sided values
so, the local error $\sim Δ x^{3}$ , and the global error $\sim Δ x^{2}$

simpson's rule

considering three datapoints: $f_{0}, f_{1}, f_{2}$
according to simpson's rule:

\int_{x_{0}}^{x_{2}} f (x) d x = \frac{Δ x}{3} (f_{0} + 4 f_{1} + f_{2}) - \frac{Δ x^{5}}{90} f^{(4)} (x)

here, the local error $\sim Δ x^{5}$ , and the global error $\sim Δ x^{4}$