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Arithmetic-Geometric Mean

The arithmetic-geometric mean agm(a,b) of two numbers a and b (often also written AGM(a,b) or M(a,b)) is defined by starting with a_0=a and b_0=b, then iterating

a_(n+1)=1/2(a_n+b_n)
(1)
b_(n+1)=sqrt(a_nb_n)
(2)

until a_n=b_n to the desired precision.

a_n and b_n converge towards each other since

a_(n+1)-b_(n+1)=1/2(a_n+b_n)-sqrt(a_nb_n)
(3)
=(a_n-2sqrt(a_nb_n)+b_n)/2.
(4)

But sqrt(b_n)<sqrt(a_n), so

2b_n<2sqrt(a_nb_n).
(5)

Now, add a_n-b_n-2sqrt(a_nb_n) to each side

a_n+b_n-2sqrt(a_nb_n)<a_n-b_n,
(6)

so

a_(n+1)-b_(n+1)<1/2(a_n-b_n).
(7)
AGMReal
AGM

The top plots show agm(1,b) for 0<=b<=20 and agm(a,b) for 0<=a,b<=200, while the bottom two plots show agm(1,z) for complex values of z.

The AGM is very useful in computing the values of complete elliptic integrals and can also be used for finding the inverse tangent.

It is implemented in the Wolfram Language as ArithmeticGeometricMean[a, b].

agm(a,b) can be expressed in closed form in terms of the complete elliptic integral of the first kind K(k) as

agm(a,b)=((a+b)pi)/(4K((a-b)/(a+b))).
(8)
AGMReIm
AGMContours

The definition of the arithmetic-geometric mean also holds in the complex plane, as illustrated above for agm(1,z).

The Legendre form of the arithmetic-geometric mean is given by

agm(1,x)=product_(n=0)^infty1/2(1+k_n),
(9)

where k_0=x and

k_(n+1)=(2sqrt(k_n))/(1+k_n).
(10)

Special values of agm(a,b) are summarized in the following table. The special value

1/(agm(1,sqrt(2)))=0.83462684167407318628...
(11)

(OEIS A014549) is called Gauss's constant. It has the closed form

1/(agm(1,sqrt(2)))=2/piint_0^1(dt)/(sqrt(1-t^4))
(12)
=([Gamma(1/4)]^2)/(2pi^(3/2)sqrt(2))
(13)

where the above integral is the lemniscate function and the equality of the arithmetic-geometric mean to this integral was known to Gauss (Borwein and Bailey 2003, pp. 13-15).

agm(a,b)OEISvalue
agm(1,2)A0685211.4567910310469068692...
agm(1,3)A0848951.8636167832448965424...
agm(1,4)A0848962.2430285802876025701...
agm(1,5)A0848972.6040081905309402887...

The derivative of the AGM is given by

partial/(partialb)agm(a,b)=(agm(a,b))/((a-b)bpi)[2agm(a,b)E(k)-bpi]
(14)
=pi/(8kb)((a+b)E(k)-2bK(k))/([K(k)]^2),
(15)

where k=(a-b)/(a+b), K(k) is a complete elliptic integral of the first kind, and E(k) is the complete elliptic integral of the second kind.

A series expansion for agm(1,b) is given by

agm(1,b)=-pi/(2ln(1/4b))+(pi[1+ln(1/4b)]b^2)/(8[ln(1/4b)]^2)+O(b^4).
(16)

The AGM has the properties

lambdaagm(a,b)=agm(lambdaa,lambdab)
(17)
agm(a,b)=agm(1/2(a+b),sqrt(ab))
(18)
agm(1,sqrt(1-x^2))=agm(1+x,1-x)
(19)
agm(1,b)=(1+b)/2agm(1,(2sqrt(b))/(1+b)).
(20)

Solutions to the differential equation

(x^3-x)(d^2y)/(dx^2)+(3x^2-1)(dy)/(dx)+xy=0
(21)

are given by [agm(1+x,1-x)]^(-1) and [agm(1,x)]^(-1).

A generalization of the arithmetic-geometric mean is

I_p(a,b)=int_0^infty(x^(p-2)dx)/((x^p+a^p)^(1/p)(x^p+b^p)^((p-1)/p)),
(22)

which is related to solutions of the differential equation

x(1-x^p)Y^('')+[1-(p+1)x^p]Y^'-(p-1)x^(p-1)Y=0.
(23)

The case p=2 corresponds to the arithmetic-geometric mean via

I_2(a,b)=int_0^infty(dx)/(sqrt((x^2+a^2)(x^2+b^2)))
(24)
=pi/(2agm(a,b)).
(25)

The case p=3 gives the cubic relative

I_3(a,b)=int_0^infty(xdx)/([(a^3+x^3)(b^3+x^3)^2]^(1/3))
(26)
=(Gamma^3(1/3)_2F_1(1/3,1/3;2/3;(a/b)^3))/(2pibsqrt(3))-(4api^2_2F_1(2/3,2/3;4/3;(a/b)^3))/(3b^2Gamma^3(1/3))
(27)

discussed by Borwein and Borwein (1990, 1991) and Borwein (1996). For a,b>0, this function satisfies the functional equation

I_3(a,b)=I_3((a+2b)/3,[b/3(a^2+ab+b^2)]^(1/3)).
(28)

It therefore turns out that for iteration with a_0=a and b_0=b and

a_(n+1)=(a_n+2b_n)/3
(29)
b_(n+1)=[(b_n)/3(a_n^2+a_nb_n+b_n^2)]^(1/3),
(30)

so

lim_(n->infty)a_n=lim_(n->infty)b_n=(I_3(1,1))/(I_3(a,b)),
(31)

where

I_3(1,1)=(2pi)/(3sqrt(3)).
(32)

See also

Arithmetic Mean, Arithmetic-Harmonic Mean, Gauss's Constant, Geometric Mean, Lemniscate Function

Related Wolfram sites

http://functions.wolfram.com/EllipticFunctions/ArithmeticGeometricMean/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "The Process of the Arithmetic-Geometric Mean." §17.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 571 and 598-599, 1972.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Borwein, J. M. Problem 10281. "A Cubic Relative of the AGM." Amer. Math. Monthly 103, 181-183, 1996.Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.Borwein, J. M. and Borwein, P. B. "A Remarkable Cubic Iteration." In Computational Method & Function Theory: Proc. Conference Held in Valparaiso, Chile, March 13-18, 1989 (Ed. A. Dold, B. Eckmann, F. Takens, E. B. Saff, S. Ruscheweyh, L. C. Salinas, and R. S. Varga). New York: Springer-Verlag, 1990.Borwein, J. M. and Borwein, P. B. "A Cubic Counterpart of Jacobi's Identity and the AGM." Trans. Amer. Math. Soc. 323, 691-701, 1991.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 906-907, 1992.Sloane, N. J. A. Sequences A014549, A068521, A084895, A084896, and A084897 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Arithmetic-Geometric Mean

Cite this as:

Weisstein, Eric W. "Arithmetic-Geometric Mean." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Arithmetic-GeometricMean.html

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