Approximation Order of a Function

Topics: Numerical Analysis - Function

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Let $p(h)$ be a function that approximates another function $f(h)$$ (such as a Taylor polynomial).

Suppose that there exists a real constant $M>0$, and a natural number $n$ such that:

$$\frac{|f(h)-p(h)|}{|h^n|}\le M$$

…for a sufficiently small $h$. Then say that $p(h)$ approximates $f(h)$ with an approximation order of $O(h^n)$.

We denote this with:

$$f(h)=p(h)+O(h^n)$$

$$|f(h)-p(h)|\le M|h_n|$$

Notice that $O(h^n)$ takes the place of $M|h^n|$, the error bound.