The Common Mean and APL

by Joseph De Kerf

Transcribed from the printed issue; not yet reviewed. Read it in the PDF of the issue, page 21

The classical definitions of the mean of two non-negative real numbers are the harmonic mean \(h(x,y)\), the geometric mean \(g(x,y)\) and the arithmetic mean \(a(x,y)\):

\[\begin{align*} h(x,y) &= 2xy/(x+y) \\ g(x,y) &= \sqrt{xy} \\ a(x,y) &= (x+y)/2 \end{align*}\]

with \(\min(x,y) \le h(x,y) \le g(x,y) \le a(x,y) \le \max(x,y)\). For example, let \(x=1\) and \(y=99\). We obtain respectively:

\[\begin{align*} h(x,y) &= \hphantom{0}1.98000000 \\ g(x,y) &= \hphantom{0}9.94987437 \\ a(x,y) &= 50.00000000 \end{align*}\]

As we see, there may be a serious gap between the geometric mean \(g(x,y)\) and the arithmetic mean \(a(x,y)\). This gap may be filled by the concept of common mean [1] — a not very familiar concept from the literature. For convenience, let \(x_0\) be the smaller of two non-negative real numbers \(x_0\) and \(y_0\). The geometric mean \(x_1\) and arithmetic mean \(y_1\) are:

\[\begin{equation*} x_1 = \sqrt{x_0 y_0} \quad\text{and}\quad y_1 = (x_0+y_0)/2 \end{equation*}\]

If this procedure of forming alternatively geometric and arithmetic means is repeated indefinitely:

\[\begin{equation*} x_{i+1} = \sqrt{x_i y_i} \quad\text{and}\quad y_{i+1} = (x_i y_i)/2 \quad\text{with } i = 1,2,3\ldots \end{equation*}\]

one obtains:

\[\begin{equation*} x_0 \le x_1 \le x_2 \le \ldots \le x_i \le \ldots \le y_i \le \ldots \le y_2 \le y_1 \le y_0 \end{equation*}\]

\(x_i\) and \(y_i\) converging to the same value. We define this as the common mean \(c(x_0,y_0)\) of the numbers \(x_0\) and \(y_0\).

For the example \(x=1\) and \(y=99\) for instance, with an accuracy of 10 digits, we get successively:


9.94987437	and	50.00000000
22.30456721	and	29.97493719
25.85687532	and	26.13975220
25.99792902	and	25.99831376
25.99812139	and	25.99812139

such that \(c(x,y) = 25.99812139\).

\[\begin{alignat*}{2} &\text{Finally, we have:}\qquad & g(x,y) &= \hphantom{0}9.94987437 \\ && c(x,y) &= 25.99812139 \\ && a(x,y) &= 50.00000000 \end{alignat*}\]

with \(9.94987437 < 25.99812139 < 50.00000000\).

In fact the order in which \(x\) and \(y\) are treated and the order in which the sequences of geometric and arithmetic means are calculated is not relevant and \(c(x,y) = c(y,x)\) (commutativity). In addition \(c(x,x) = x\) (idempotency). Finally, \(c(x,y) = 0\) if and only if \(x = 0\) or \(y = 0\) (or both).

Programming the algorithm for calculating the common mean can be somewhat complicated in most programming languages. In APL however, it is very simple. A function to do the job is:

    ∇ R←X CMEAN Y
[1]    R←X,Y
[2]   LAB:R←(0.5×+/R),(×/R)*0.5
[3]    →(≠/R)/LAB
[4]    R←0.5×+/R
    ∇

which for the chosen example gives:

    1 CMEAN 99
25.99812139

Accuracy is determined by the current, i.e. the default value of comparison tolerance ⎕CT. It may be changed by defining the comparison tolerance as a global or local variable.

Note: a special case is the common mean of the numbers 1 and \(1\sqrt{2}\):

      1 CMEAN ÷ 2*0.5
0.8472130848

which is known in the literature as the "ubiquitous constant U" since it turns up all over the place. Finally, the common mean is very useful in the design of simple and efficient algorithms for calculating the complete elliptic integrals of the first kind \(K(p)\) and of the second kind \(E(p)\). More details may be found in [1].

Reference¶

[1] J Spanier and K B Oldham: An Atlas of Functions. Hemisphere Publishing Corporation, New York. New York 1987.