Order of Magnitude

Order of magnitude refers to the scale of a quantity expressed using big-O notation. We write $f(x)=O(x^n)$ to mean that $f(x)$ grows no faster than $x^n$ as $x\to 0$ (or $x\to \infty$, depending on context).

Formally, $f(x)=O(x^n)$ means there exists a constant $C>0$ such that

$$|f(x)|\le C|x^n|$$

for all sufficiently small (or large) $x$.

In numerical methods, big-O notation is used to describe how errors scale with step size $dt$ or $dx$. For example, saying a method has error $O(d{t}^2)$ means that halving the step size reduces the error by a factor of roughly 4.

This notation underpins the definitions of Local truncation error, Global truncation error, and Order of a method.

Order of a method | Order of convergence | Local truncation error | Global truncation error