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'arithmetic-geometric mean'
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| Title of object: |
arithmetic-geometric mean |
| Canonical Name: |
ArithmeticGeometricMean |
| Type: |
Definition |
| Created on: |
2004-06-05 14:07:00 |
| Modified on: |
2007-04-15 13:36:27 |
| Classification: |
msc:26E60, msc:33E05 |
| Synonyms: |
arithmetic-geometric mean=agm
arithmetic-geometric mean=AGM |
Revision comment (for changes between this and next version):
| Changes for correction #11671 ('have to show that the limits are the same'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\def\sse{\subseteq}
\def\bigtimes{\mathop{\mbox{\Huge $\times$}}}
\def\impl{\Rightarrow}
\def\oo{\infty}
\def\agm{\operatorname{agm}}
\def\AGM{\operatorname{AGM}} |
Content:
If $x$ and $y$ are non-negative real numbers, we can form their arithmetic
mean $a_0 = (x+y)/2$ as well as their geometric mean $g_0 = \sqrt{xy}$.
This procedure can be repeated to form a sequence of arithmetic and
geometic 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 $a_n \ge a_{n+1} \ge g_{n+1} \ge g_n$ (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 \PMlinkname{linear}{LinearConvergence}.
The \emph{arithmetic-geometric mean} $M(x,y)$ of $x$ and $y$
is defined as this limit
\begin{equation*}
M(x,y) = \lim_{n\to\oo} a_n, g_n.
\end{equation*}
The origin of the name is obvious from the construction. Alternative notations
for $M(x,y)$ are $\agm(x,y)$ or $\AGM(x,y)$.
The AGM lies between the arithmetic and geometric
means of $x$ and $y$,
\begin{equation*}
\frac{x+y}{2} \ge M(x,y) \ge \sqrt{xy},
\end{equation*}
with equality holding only in case of equality $x=y$. The AGM is also a
homogeneous 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 the
first and second kinds. For example,
\begin{equation}
M(x,y) = \frac{\pi}{4} \frac{x+y}{K\left(\frac{|x-y|}{x+y}\right)},
\end{equation}
where $K(k)$ is the elliptic integral of the first kind as function of
the modulus $k$. |
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