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<p><b>New page</b></p><div>{{mergefrom|AGM method|date=September 2012}}<br />
In [[mathematics]], the '''arithmetic–geometric mean (AGM)''' of two positive [[real number]]s {{math|''x''}} and {{math|''y''}} is defined as follows:<br />
<br />
First compute the [[arithmetic mean]] of {{math|''x''}} and {{math|''y''}} and call it {{math|''a''<sub>1</sub>}}. Next compute the [[geometric mean]] of {{math|''x''}} and {{math|''y''}} and call it {{math|''g''<sub>1</sub>}}; this is the [[square root]] of the product {{math|''xy''}}:<br />
<br />
:<math>\begin{align}<br />
a_1 &= \frac{1}{2}(x + y)\\<br />
g_1 &= \sqrt{xy}<br />
\end{align}</math><br />
<br />
Then iterate this operation with {{math|''a''<sub>1</sub>}} taking the place of {{math|''x''}} and {{math|''g''<sub>1</sub>}} taking the place of {{math|''y''}}. In this way, two [[sequence]]s {{math|(''a''<sub>''n''</sub>)}} and {{math|(''g''<sub>''n''</sub>)}} are defined:<br />
<br />
:<math>\begin{align}<br />
a_{n+1} &= \frac{1}{2}(a_n + g_n)\\<br />
g_{n+1} &= \sqrt{a_n g_n}<br />
\end{align}</math><br />
<br />
These two sequences [[limit of a sequence|converge]] to the same number, which is the '''arithmetic–geometric mean''' of {{math|''x''}} and {{math|''y''}}; it is denoted by {{math|''M''(''x'', ''y'')}}, or sometimes by {{math|agm(''x'', ''y'')}}.<br />
<br />
This can be used for algorithmic purposes as in the [[AGM method]].<br />
<br />
==Example==<br />
To find the arithmetic–geometric mean of {{math|''a''<sub>0</sub> {{=}} 24}} and {{math|''g''<sub>0</sub> {{=}} 6}}, first calculate their arithmetic mean and geometric mean, thus:<br />
<br />
:<math>\begin{align}<br />
a_1 &= \frac{1}{2}(24 + 6) = 15\\<br />
g_1 &= \sqrt{24 \times 6} = 12<br />
\end{align}</math><br />
<br />
and then iterate as follows:<br />
<br />
:<math>\begin{align}<br />
a_2 &= \frac{1}{2}(15 + 12) = 13.5\\<br />
g_2 &= \sqrt{15 \times 12} = 13.41640786500\dots\\<br />
\dots<br />
\end{align}</math><br />
<br />
The first five iterations give the following values:<br />
<br />
:{| class="wikitable"<br />
|-<br />
! {{math|''n''}}<br />
! {{math|''a''<sub>''n''</sub>}}<br />
! {{math|''g''<sub>''n''</sub>}}<br />
|-<br />
| 0<br />
| 24<br />
| 6<br />
|-<br />
| 1<br />
| {{underline|1}}5<br />
| {{underline|1}}2<br />
|-<br />
| 2<br />
| {{underline|13}}.5<br />
| {{underline|13}}.416407864998738178455042…<br />
|-<br />
| 3<br />
| {{underline|13.458}}203932499369089227521…<br />
| {{underline|13.458}}139030990984877207090…<br />
|-<br />
| 4<br />
| {{underline|13.4581714817}}45176983217305…<br />
| {{underline|13.4581714817}}06053858316334…<br />
|-<br />
| 5<br />
| {{underline|13.4581714817256154207668}}20…<br />
| {{underline|13.4581714817256154207668}}06…<br />
|}<br />
<br />
As can be seen, the number of digits in agreement (underlined) approximately doubles with each iteration. The arithmetic–geometric mean of 24 and 6 is the common limit of these two sequences, which is approximately 13.4581714817256154207668131569743992430538388544.<ref>[http://www.wolframalpha.com/input/?i=agm%2824%2C+6%29 agm(24, 6) at WolframAlpha]</ref><br />
<br />
== History ==<br />
The first algorithm based on this sequence pair appeared in the works of [[Adrien-Marie Legendre|Legendre]]. Its properties were further analyzed by [[Gauss]].<ref name="BerggrenBorwein2004">{{cite book|editor=J.L. Berggren, Jonathan M. Borwein, Peter Borwein|title=Pi: A Source Book|url=http://books.google.com/books?id=QlbzjN_5pDoC&pg=PA481|year=2004|publisher=Springer|isbn=978-0-387-20571-7|page=481|chapter=The Arithmetic-Geometric Mean of Gauss|author=David A. Cox}} first published in ''[[L'Enseignement Mathématique]]'', t. 30 (1984), p. 275-330</ref><br />
<br />
==Properties==<br />
The geometric mean of two positive numbers is never bigger than the arithmetic mean (see [[inequality of arithmetic and geometric means]]); as a consequence, {{math|(''g<sub>n</sub>'')}} is an increasing sequence, {{math|(''a<sub>n</sub>'')}} is a decreasing sequence, and {{math|''g<sub>n</sub>'' ≤ ''M''(''x'',&nbsp;''y'') ≤ ''a<sub>n</sub>''}}. These are strict inequalities if {{math|''x'' ≠ ''y''}}.<br />
<br />
{{math|''M''(''x'', ''y'')}} is thus a number between the geometric and arithmetic mean of {{math|''x''}} and {{math|''y''}}; in particular it is between {{math|''x''}} and {{math|''y''}}.<br />
<br />
If {{math|''r'' ≥ 0}}, then {{math|''M''(''rx'',''ry'') {{=}} ''r M''(''x'',''y'')}}.<br />
<br />
There is an integral-form expression for {{math|''M''(''x'',''y'')}}:<br />
<br />
:<math>\begin{align}M(x,y) &= \frac\pi2\bigg/\int_0^{\pi/2}\frac{d\theta}{\sqrt{x^2\cos^2\theta+y^2\sin^2\theta}}\\<br />
&=\frac{\pi}{4} (x + y) \bigg/ K\left( \frac{x - y}{x + y} \right)<br />
\end{align}</math><br />
<br />
where {{math|''K''(''k'')}} is the [[elliptic integral|complete elliptic integral of the first kind]]:<br />
<br />
:<math>K(k) = \int_0^{\pi/2}\frac{d\theta}{\sqrt{1 - k^2\sin^2(\theta)}} </math><br />
<br />
Indeed, since the arithmetic–geometric process converges so quickly, it provides an effective way to compute elliptic integrals via this formula. In engineering, it is used for instance in [[elliptic filter]] design.<ref name="Dimopoulos2011">{{cite book|author=Hercules G. Dimopoulos|title=Analog Electronic Filters: Theory, Design and Synthesis|url=http://books.google.com/books?id=6W1eX4QwtyYC&pg=PA147|year=2011|publisher=Springer|isbn=978-94-007-2189-0|pages=147–155}}</ref><br />
<br />
== Related concepts ==<br />
The reciprocal of the arithmetic–geometric mean of 1 and the [[square root of 2]] is called [[Gauss's constant]], after [[Carl Friedrich Gauss]].<br />
<br />
:<math>\frac{1}{M(1, \sqrt{2})} = G = 0.8346268\dots</math><br />
<br />
The [[geometric–harmonic mean]] can be calculated by an analogous method, using sequences of geometric and [[harmonic mean|harmonic]] means. The arithmetic–harmonic mean can be similarly defined, but takes the same value as the [[geometric mean]].<br />
<br />
The arithmetic-geometric mean can be used to compute [[Elliptic integral#Complete elliptic integral of the first kind|complete elliptic integrals of the first kind]]. A modified arithmetic-geometric mean can be used to efficiently compute [[Elliptic integral#Complete elliptic integral of the second kind|complete elliptic integrals of the second kind]].<ref>{{Citation |last=Adlaj |first=Semjon |title=An eloquent formula for the perimeter of an ellipse |url=http://www.ams.org/notices/201208/rtx120801094p.pdf |journal=Notices of the AMS |volume=59 |issue=8 |pages=1094–1099 |date=September 2012 |doi=10.1090/noti879 |accessdate=2013-12-14}}</ref><br />
<br />
==Proof of existence==<br />
From [[inequality of arithmetic and geometric means]] we can conclude that:<br />
<br />
:<math>g_n \leqslant a_n</math><br />
<br />
and thus<br />
<br />
:<math>g_{n + 1} = \sqrt{g_n \cdot a_n} \geqslant \sqrt{g_n \cdot g_n} = g_n</math><br />
<br />
that is, the sequence {{math|''g<sub>n</sub>''}} is nondecreasing.<br />
<br />
Furthermore, it is easy to see that it is also bounded above by the larger of {{math|''x''}} and {{math|''y''}} (which follows from the fact that both arithmetic and geometric means of two numbers both lie between them). Thus by the [[monotone convergence theorem]] the sequence is convergent, so there exists a {{math|''g''}} such that:<br />
<br />
:<math>\lim_{n\to \infty}g_n = g</math><br />
<br />
However, we can also see that:<br />
<br />
:<math>a_n = \frac{g_{n + 1}^2}{g_n}</math><br />
<br />
and so:<br />
<br />
:<math>\lim_{n\to \infty}a_n = \lim_{n\to \infty}\frac{g_{n + 1}^2}{g_{n}} = \frac{g^2}{g} = g</math><br />
<br />
[[Q.E.D.]]<br />
<br />
==Proof of the integral-form expression==<br />
This proof is given by Gauss.<ref name="BerggrenBorwein2004" /><br />
Let<br />
<br />
:<math>I(x,y) = \int_0^{\pi/2}\frac{d\theta}{\sqrt{x^2\cos^2\theta+y^2\sin^2\theta}},</math><br />
<br />
Changing the variable of integration to <math>\theta'</math>, where<br />
<br />
:<math> \sin\theta = \frac{2x\sin\theta'}{(x+y)+(x-y)\sin^2\theta'}, </math><br />
<br />
gives<br />
<br />
:<math><br />
\begin{align}<br />
I(x,y) &= \int_0^{\pi/2}\frac{d\theta'}{\sqrt{\bigl(\frac12(x+y)\bigr)^2\cos^2\theta'+\bigl(\sqrt{xy}\bigr)^2\sin^2\theta'}}\\<br />
&= I\bigl(\tfrac12(x+y),\sqrt{xy}\bigr).<br />
\end{align}<br />
</math><br />
<br />
Thus, we have<br />
<br />
:<math><br />
\begin{align}<br />
I(x,y) &= I(a_1, g_1) = I(a_2, g_2) = \cdots\\<br />
&= I\bigl(M(x,y),M(x,y)\bigr) = \pi/\bigr(2M(x,y)\bigl).<br />
\end{align}<br />
</math><br />
The last equality comes from observing that <math>I(z,z) = \pi/(2z)</math>.<br />
Finally, we obtain the desired result<br />
<br />
:<math>M(x,y) = \pi/\bigl(2 I(x,y) \bigr). </math><br />
<br />
==See also==<br />
* [[Generalized mean]]<br />
* [[Inequality of arithmetic and geometric means]]<br />
* [[Gauss–Legendre algorithm]]<br />
<br />
==External links==<br />
* [http://arithmeticgeometricmean.blogspot.de/ Arithmetic-Geometric Mean Calculator]<br />
* [http://planetmath.org/convergenceofarithmeticgeometricmean/ Proof of convergence rate in PlanetMath]<br />
<br />
==References==<br />
{{More footnotes|date=October 2008}}<br />
*{{cite journal|last = Adlaj|first = Semjon|title = An eloquent formula for the perimeter of an ellipse|journal = Notices of the AMS|volume = 59|issue = 8|pages = 1094–1099|date = September 2012|url = http://www.ams.org/notices/201208/rtx120801094p.pdf}}<br />
* [[Jonathan Borwein]], [[Peter Borwein]], ''Pi and the AGM. A study in analytic number theory and computational complexity.'' Reprint of the 1987 original. Canadian Mathematical Society Series of Monographs and Advanced Texts, 4. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1998. xvi+414 pp.&nbsp;ISBN 0-471-31515-X {{MR|1641658}}<br />
* [[Zoltán Daróczy]], [[Zsolt Páles]], ''Gauss-composition of means and the solution of the Matkowski–Suto problem.'' Publ. Math. Debrecen 61/1-2 (2002), 157–218.<br />
*{{SpringerEOM|author=M. Hazewinkel|title=Arithmetic–geometric mean process|urlname=a/a130280}}<br />
*{{mathworld|urlname=Arithmetic-GeometricMean|title=Arithmetic–Geometric mean}}<br />
<references /><br />
<br />
{{DEFAULTSORT:Arithmetic-Geometric Mean}}<br />
[[Category:Means]]<br />
[[Category:Special functions]]<br />
[[Category:Elliptic functions]]<br />
[[Category:Articles containing proofs]]</div>en>Trappist the monk