# Iterating Arithmetic, Harmonic and Geometric Means

Starting with a data set $X_{0}$, compute its arithmetic, geometric and harmonic means, $A(X_{0}), G(X_{0})$ and $H(X_{0})$ respectively. Let $X_{1} = \{A(X_{0}),G(X_{0}),H(X_{0})\}$, and compute $A(X_{1}), G(X_{1})$ and $H(X_{1})$. Extending this, we recursively define $X_{n} = \{A(X_{n-1}),G(X_{n-1}),H(X_{n-1})\}$. Is there anything interesting about the behavior of the $\{X_{n}\}$ with regards to convergence, does a closed form representation exist, and how does it depend on the initial data set $X_{0}$?

Any insight would be appreciated. Thanks!

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My guess is it "converges" to $\{x\}$ for some $x$, under suitable interpretation. – anon Jan 29 '12 at 23:46
C++ seems to confirm your suspicion. I'd be curious as to the mathematical derivation, though... – Isaac Solomon Jan 30 '12 at 1:00
Convergence is easy to show, when each element of the initial data set is nonnegative. The real question is, what will be the limit (in terms of the initial data)? There is an evidence that identifying the resulting limit is egregiously difficult. Even in a simpler case, where $X_0 = \{x, y\}$ and $X_{n+1} = \{A(X_n), G(X_n)\}$, the resulting limit is called the 'Arithmetic-Geometric Mean', denoted as $\mathrm{AGM}(x, y)$, which turns out to be closely related to the function so called elliptic integral. You may refer to Landen transform, for the detail. – Sangchul Lee Jan 30 '12 at 3:00