Iterative [[Quadrature|Numerical Integration]] are methods of [[Quadrature]] that work iteratively, typically to numerically solve [[Differential Equations]]. For a given [[Set|set]] of [[Differential Equations]], $\huge \begin{cases} y'(t) = f(t,y) \\ y(t_{0}) = y_{0} \end{cases} $ As well as a finite step size $h\in\R$, There are many iterative methods of approximating $y(t)$, which we denote $z_i, t_i$. For a given method, $z_i \approx y_i$ (assuming its correct). ## Error When discussing error in relation to these forms of methods, there is a distinction between *local error* and *global error* Suppose $z_{i}=y_{i}$, the local error is $z_{i+1} -y_{i+1}$. Global Error is a measure of the long term error of a given method. At step $i$, $ \huge \epsilon _{i} = (z_{i}-y_{i}) $ Let $w_{i}=y_{i}$, let $w_{i+1}$ be the next iteration of our method $M$ on $w_{i}$.