Arithmetic-Geometric Mean

The arithmetic-geometric mean (AGM) $M(a,b)$ of two numbers $a$ and $b$ is defined by starting with $a_0\equiv a$ and $b_0\equiv b$, then iterating

$$a_{n+1} = {\textstyle{1\over 2}}(a_n+b_n)$$ (1)

$$b_{n+1} = \sqrt{a_nb_n}$$ (2)

until $a_n=b_n$. $a_n$ and $b_n$ converge towards each other since

$$a_{n+1}-b_{n+1} = {\textstyle{1\over 2}}(a_n+b_n)-\sqrt{a_nb_n}$$

$$= {a_n-2\sqrt{a_nb_n}+b_n\over 2}.$$ (3)

But $\sqrt{b_n}<\sqrt{a_n}$, so

$$2b_n<2\sqrt{a_nb_n}.$$ (4)

Now, add $a_n-b_n-2\sqrt{a_nb_n}$ to each side

$$a_n+b_n-2\sqrt{a_nb_n}<a_n-b_n,$$ (5)

so

$$a_{n+1}-b_{n+1}<{\textstyle{1\over 2}}(a_n-b_n).$$ (6)

The AGM is very useful in computing the values of complete Elliptic Integrals and can also be used for finding the Inverse Tangent. The special value $1/M(1,\sqrt{2}\,)$ is called Gauss's Constant.

The AGM has the properties

$$\lambda M(a,b)=M(\lambda a,\lambda b)$$ (7)

$$M(a,b)=M\left({{\textstyle{1\over 2}}(a+b), \sqrt{ab}\,}\right)$$ (8)

$$M(1,\sqrt{1-x^2}\,)=M(1+x,1-x)$$ (9)

$$M(1,b)={1+b\over 2}M\left({1, {2\sqrt{b}\over 1+b}}\right).$$ (10)

The Legendre form is given by

$$M(1,x)=\prod_{n=0}^\infty {\textstyle{1\over 2}}(1+k_n),$$ (11)

where $k_0\equiv x$ and

$$k_{n+1}\equiv {2\sqrt{k_n}\over 1+k_n}.$$ (12)

Solutions to the differential equation

$$(x^3-x){d^2y\over dx^2}+(3x^2-1){dy\over dx}+xy=0$$ (13)

are given by $[M(1+x,1-x)]^{-1}$ and $[M(1,x)]^{-1}$.

A generalization of the Arithmetic-Geometric Mean is

$$I_p(a,b)=\int_0^\infty {x^{p-2}\,dx\over (x^p+a^p)^{1/p}(x^p+b^p)^{(p-1)/p}},$$ (14)

which is related to solutions of the differential equation

$$x(1-x^p)Y''+[1-(p+1)x^p]Y'-(p-1)x^{p-1}Y=0.$$ (15)

When $p=2$ or $p=3$, there is a modular transformation for the solutions of (15) that are bounded as $x\to 0$. Letting $J_p(x)$ be one of these solutions, the transformation takes the form

$$J_p(\lambda)=\mu J_p(x),$$ (16)

where

$$\lambda = {1-u\over 1+(p-1)u}$$ (17)

$$\mu = {1+(p-1)u\over p}$$ (18)

and

$$x^p+u^p=1.$$ (19)

The case $p=2$ gives the Arithmetic-Geometric Mean, and $p=3$ gives a cubic relative discussed by Borwein and Borwein (1990, 1991) and Borwein (1996) in which, for $a,b>0$ and $I(a,b)$ defined by

$$I(a,b)=\int_0^\infty {t\,dt\over [(a^3+t^3)(b^3+t^3)^2]^{1/3}},$$ (20)

$$I(a,b)=I\left({{a+2b\over 3}, \left[{{b\over 3} (a^2+ab+b^2)}\right]}\right).$$ (21)

For iteration with $a_0=a$ and $b_0=b$ and

$$a_{n+1} = {a_n+2b_n\over 3}$$ (22)

$$b_{n+1} = {b_n\over 3} ({a_n}^2+a_nb_n+{b_n}^2),$$ (23)

$$\lim_{n\to\infty} a_n=\lim_{n\to\infty} b_n={I(1,1)\over I(a,b)}.$$ (24)

Modular transformations are known when $p=4$ and $p=6$, but they do not give identities for $p=6$ (Borwein 1996).

See also Arithmetic-Harmonic Mean

References

Abramowitz, M. and Stegun, C. 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 ad 598-599, 1972.

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. ``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, L. C., 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.