arithmetic-geometric mean

If $x$ and $y$ are non-negative real numbers, we can form their arithmeticmean $a_{0}=(x+y)/2$ as well as their geometric mean $g_{0}=\\sqrt{xy}$ .This procedure can be repeated to form a sequence of arithmetic andgeometic means $a_{n+1}=(a_{n}+g_{n})/2$ and $g_{n+1}=\\sqrt{a_{n}g_{n}}$ .By the arithmetic-geometric means inequality we have $a_{n}\\geq a_{n+1}\\geq g_{n+1}\\geq g_{n}$ (with equality holding only when $a_{n}=g_{n}$ ),hence these sequences converge to a number between $x$ and $y$ ,with the rate of convergence being superlinear.The arithmetic-geometric mean $M(x,y)$ of $x$ and $y$ is defined as this limit

M(x,y)=\\lim_{n\\to\\infty}a_{n},g_{n}.

The origin of the name is obvious from the construction. Alternative notationsfor $M(x,y)$ are $\\operatorname{agm}(x,y)$ or $\\operatorname{AGM}(x,y)$ .

The AGM lies between the arithmetic and geometricmeans of $x$ and $y$ ,

\\frac{x+y}{2}\\geq M(x,y)\\geq\\sqrt{xy},

with equality holding only in case of equality $x=y$ . The AGM is also ahomogeneous function of degree $1$ , namely $M(\\alpha x,\\alpha y)=\\alpha M(x,y)$ for $\\alpha>0$ . It is also symmetric $M(x,y)=M(y,x)$ .These properties are obvious from the construction.

The AGM can be used to numerically evaluate elliptic integrals of thefirst and second kinds. For example,

M(x,y)=\\frac{\\pi}{4}\\frac{x+y}{K\\left(\\frac{|x-y|}{x+y}\\right)},

(1)

where $K(k)$ is the elliptic integral of the first kind as function ofthe modulus $k$ .

As a numerical method, the arithmetic-geometric mean has much to recommend it.By its nature, it automatically provides upper and lower bounds for theanswer, so one does not have to separately estimate error. To computethe arithmetic-geometric mean to a certain accuracy, we only need to carryout the computation until the difference between $a_{n}$ and $g_{n}$ is smallerthan the desired accuracy.

Because convergence is superlinear, only a few iterations are necessarry toobtain the answer. For instance, if we compute $M(1,k)$ with $k$ less thana billion, we already obtain at least fifteen-place accuracy after eightiterations, as the following computation of $M(1,123456789)$ shows:

The fact that relatively few iterations are necessarry to obtain a highlyaccurate result also means that one does not have to worry much about thecumulative effect of roundoff errors in the various steps of the computation.