Arithmetic-harmonic mean

I’ve written several times about the arithmetic-geometric mean and variations. Take the arithmetic and geometric mean of two positive numbers a and b. Then take the arithmetic and geometric of the means from the previous step. Repeat ad infinitum and the result converges to a limit. This limit is called the arthmetic-geometric mean or AGM.

What if instead of repeatedly taking the arithmetic and geometric means, we repeatedly take the arithmetic and harmonic means? That is, first define

$\begin{align*} a_0 &= a \\ b_0 &= b \\ \end{align*}$

and for integer n > 0 define

$\begin{align*} a_{n+1} &= (a_n + b_n)/2 \\ b_{n+1} &= 2a_nb_n/(a_n + b_n) \end{align*}$

and take the limit. We could call this the arithmetic-harmonic mean by analogy with the arithmetic-geometric mean. But it already has another name: the geometric mean! The iteration defined above converges to the geometric mean √(ab). Not only that, it converges quickly.

We can illustrate this with a little Python code.

    a, b = 4, 9

    for _ in range(5):
        a, b = (a + b)/2, 2*a*b/(a + b)
        print(a, b)

This prints the following.

    6.5 5.538461538461538
    6.019230769230769 5.980830670926518
    6.000030720078644 5.999969280078643
    6.000000000078643 5.999999999921357
    6.0 6.0

One thought on “Arithmetic-harmonic mean”

Andy

15 May 2023 at 12:34

If you set b_n=2/a_n, you get the familiar iteration for calculating the square root of 2: the first formula becomes a_n+1 = 1/2 (a_n + 2/a_n), and the second simplifies to b_n+1 = 4/(a_n + 2/(a_n)) = 2/(a_n+1) by the first equation.

Indeed for any x, setting b_n = x/a_n gives the square root of x.

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One thought on “Arithmetic-harmonic mean”