The previous post made use of both the arithmetic and geometric means. It also showed how both of these means correspond to different points along a continuum of means. This post combines those ideas.

Let *a* and *b* be two positive numbers. Then the arithmetic and geometric means are defined by

*A*(*a*, *b*) = (*a* + *b*)/2

*G(a, b*) = √(*ab*)

The **arithmetic-geometric mean** (AGM) of *a* and *b* is the limit of the sequence that takes the arithmetic and geometric mean of the arithmetic and geometric mean over and over. Specifically, the sequence starts with

*a*_{0} = *A*(*a*, *b*)

*b*_{0} = *G*(*a*, *b*)

and continues

*a*_{n+1} = *A*(*a*_{n}, *b*_{n})

*b*_{n+1} = *G*(*a*_{n}, *b*_{n})

AGM(*a*, *b*) is defined as the common limit of the *a*‘s and the *b*‘s [1].

The arithmetic mean dominates the geometric mean. That is,

*G*(*a*, *b*) ≤ *A*(*a*, *b*)

with equality if and only if *a* = *b*. So the *a*‘s converge down to AGM(*a*, *b*) and the *b*‘s converge up to AGM(*a*, *b*). The AGM is between the arithmetic and geometric means.

There’s another way to create means between the arithmetic and geometric means, and that is by varying the parameter *r* in the family of means

The arithmetic mean corresponds to *r* = 1 and the geometric mean corresponds to the limit as *r* approaches 0. So if *r* is between 1 and 0, we have a mean that’s somewhere between the arithmetic and geometric mean. For a fixed argument, these means are an increasing function of *r* as discussed here.

So here’s the idea I wanted to get to. Since the AGM is between the arithmetic and geometric mean, as are the *r*-means for *r* between 0 and 1, is there some value of *r* where the AGM is well approximated by an *r*-mean? This question was inspired by the result I wrote about here that the perimeter of an ellipse is very well approximated by an *r*-mean of its axes.

We can simplify things a little by assuming one of our arguments is 1. This is because all the means mentioned here are homogeneous, i.e. you can pull out constants.

I looked at AGM(1, *x*) for *x* ranging from 1 to 100 and compared it to *r*-means for varying values of *r* and found that I got a good fit for *r* = 0.415. I have no theory behind this, just tinkering. The optimal value depends on how you measure the error, and probably depends on the range of *x*.

When I plot AGM(1, *x*) and *M*_{0.415}(1, *x*) it’s hard to tell the lines apart. Here’s what I get for their relative difference.

There’s a connection between the AGM and elliptic functions, and so maybe *r*-means provide useful approximations to elliptic functions.

[1] The sequence converges *very* rapidly. I intended to show a plot, but the convergence is so rapid that it’s hard to plot. If I start with *a* = 1 and *b* = 100, then *a* and *b* agree to 44 significant figures after just 7 iterations.

Very neat parameterisation. For anyone who else may be wondering, the final of the 3 classical Pythagorean means, the harmonic mean, is obtained by setting r=-1. So, the harmonic, geometric and arithmetic means correspond to r=-1, r=0 and r=1 respectively. (In the case of r=0 it’s actually defined in the limit as John mentions).

Despite sharing many useful properties, the AGM lacks a property that all the r-means share, which is that in general Mr(Mr(a,b),Mr(c,d)) = Mr(Mr(a,c),Mr(b,d)) for positive real a, b, c and d.

This is a curious discrepancy considering the AGM can be ‘built’ from two r-means.

What are your thoughts on Geothmetic Meandian?

https://xkcd.com/2435/

Hadn’t seen that one. :)