The Aperiodical 

You're reading: Posts By Philipp Reinhard

Taming the AGM

By Philipp Reinhard. Posted February 21, 2018

This post is in response to Peter’s post introducing the Approximate Geometric Mean.

The approximate geometric mean $(\mathrm{A}\mathrm{G}\mathrm{M})$ is a nice approximation of the geometric mean $(\mathrm{G}\mathrm{M})$, but it has some quirks as we will see. After a discussion at the MathsJam gathering, I was intrigued to find out how good an approximation it is.

To get a better understanding, we first have to look again at its definition. For $A=a\cdot {10}^x$ and $B=b\cdot {10}^y$, we set

$$\mathrm{A}\mathrm{G}\mathrm{M}(A,B):=\mathrm{A}\mathrm{M}(a,b)\cdot {10}^{\mathrm{A}\mathrm{M}(x,y)}$$

where $\mathrm{A}\mathrm{M}$ stands for the arithmetic mean. This makes also sense when $a$ and $b$ are not just integers between 1 and 10, but any real numbers. Note that we won’t consider negative $A$ and $B$ (i.e. negative $a$ and $b$), as the geometric mean runs into issues if we do so. The values of $x$ and $y$ may be negative, though. The $\mathrm{A}\mathrm{G}\mathrm{M}$ looks like a mix between the $\mathrm{A}\mathrm{M}$ and the $\mathrm{G}\mathrm{M}$, so what can possibly go wrong?