Arithmetic-geometric mean

The arithmetic-geometric mean M(x, y) of two positive real numbers x and y is defined as follows: we first form the arithmetic mean of x and y and call it a₁, i.e. a₁ = (x+y) / 2. We then form the geometric mean of x and y and call it g₁, i.e. g₁ is the square root of xy. Now we can iterate this operation with a₁ taking the place of x and g₁ taking the place of y. In this way, two sequences (a_n) and (g_n) are defined:

<math>a_{n+1} = \frac{a_n + g_n}{2}</math>

and

<math>g_{n+1} = \sqrt{a_n g_n}</math>

Both of these sequences converge to the same number, which we call the arithmetic-geometric mean M(x, y) of x and y.

M(x, y) is a number between the geometric and arithmetic mean of x and y; in particular it is between x and y. If r > 0, then M(rx, ry) = r M(x, y).

Implementation

The following example code in the Scheme programming language computes the arithmetic-geometric mean of two positive real numbers:

(define agmean
  (lambda (a b epsilon)
    (letrec ((ratio-diff       ; determine whether two numbers
	      (lambda (a b)    ; are already very close together
		(abs (/ (- a b) b))))
	     (loop             ; actually do the computation
	      (lambda (a b)
		;; if they're already really close together,
		;; just return the arithmetic mean
		(if (< (ratio-diff a b) epsilon)
		    (/ (+ a b) 2)
		    ;; otherwise, do another step
		    (loop (sqrt (* a b)) (/ (+ a b) 2))))))
      ;; error checking
      (if (or (not (real? a))
	      (not (real? b))
	      (<= a 0)
	      (<= b 0))
	  (error 'agmean "~s and ~s must both be positive real numbers" a b)
	  (loop a b)))))

One can show that

<math>M(x,y) = \frac{\pi}{4} \cdot \frac{x + y}{K \left( \frac{x - y}{x + y} \right) }</math>

where K(x) is the complete elliptic integral of the first kind.

The geometric harmonic mean can be calculated by an analogous method, using sequences of geometric and harmonic means. The arithmetic harmonic mean is none other than the geometric mean.

See also: generalized mean