and
Both of these sequences converge to the same number, which we call the arithmeticgeometric 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).
(define agmean (lambda (a b epsilon) (letrec ((ratiodiff ; 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 (< (ratiodiff 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 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
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