arithmetic-geometric mean

If x and y are non-negative real numbers, we can form their arithmetic meanMathworldPlanetmath a0=(x+y)/2 as well as their geometric meanMathworldPlanetmath g0=xy. This procedure can be repeated to form a sequence of arithmetic and geometic means an+1=(an+gn)/2 and gn+1=angn. By the arithmetic-geometric means inequality we have anan+1gn+1gn (with equality holding only when an=gn), hence these sequences converge to a number between x and y, with the rate of convergence being superlinear. The arithmetic-geometric meanDlmfDlmfMathworldPlanetmath M(x,y) of x and y is defined as this limit


The origin of the name is obvious from the construction. Alternative notationsDlmfDlmfDlmfDlmfDlmf for M(x,y) are agm(x,y) or AGM(x,y).

The AGM lies between the arithmetic and geometric means of x and y,


with equality holding only in case of equality x=y. The AGM is also a homogeneous function of degree 1, namely M(αx,αy)=αM(x,y) for α>0. It is also symmetricPlanetmathPlanetmath M(x,y)=M(y,x). These properties are obvious from the construction.

The AGM can be used to numerically evaluate elliptic integralsMathworldPlanetmath of the first and second kinds. For example,

M(x,y)=π4x+yK(|x-y|x+y), (1)

where K(k) is the elliptic integral of the first kind as functionMathworldPlanetmath of the 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 the answer, so one does not have to separately estimate error. To compute the arithmetic-geometric mean to a certain accuracy, we only need to carry out the computation until the difference between an and gn is smaller than the desired accuracy.

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

n gn an
0 1.0 123456789.0
1 11111.111060555555 61728395.0
2 828173.3227017411 30869753.055530276
3 5056234.365511624 15848963.189116009
4 8951875.352937901 10452598.777313817
5 9673177.418448625 9702237.06512586
6 9687696.345716598 9687707.241787244
7 9687701.793750389 9687701.793751922
8 9687701.793751154 9687701.793751154

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

Title arithmetic-geometric mean
Canonical name ArithmeticgeometricMean
Date of creation 2013-03-22 14:23:46
Last modified on 2013-03-22 14:23:46
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 7
Author rspuzio (6075)
Entry type Definition
Classification msc 26E60
Classification msc 33E05
Synonym agm
Synonym AGM
Related topic EllipticIntegralsAndJacobiEllipticFunctions