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