Arithmetic-geometric mean

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In mathematics, 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:

and

These two 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).

M(x, y) is sometimes denoted agm(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

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

The reciprocal of the arithmetic-geometric mean of 1 and the Square root of 2 is called Gauss's constant.

named after Carl Friedrich Gauss.

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.

Arithmetic-geometric mean

Implementation

See also