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| {{mergefrom|AGM method|date=September 2012}}
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| In [[mathematics]], the '''arithmetic–geometric mean (AGM)''' of two positive [[real number]]s {{math|''x''}} and {{math|''y''}} is defined as follows:
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| 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''}}:
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| :<math>\begin{align}
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| a_1 &= \frac{1}{2}(x + y)\\
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| g_1 &= \sqrt{xy}
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| \end{align}</math>
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| 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:
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| :<math>\begin{align}
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| a_{n+1} &= \frac{1}{2}(a_n + g_n)\\
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| g_{n+1} &= \sqrt{a_n g_n}
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| \end{align}</math>
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| 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'')}}.
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| This can be used for algorithmic purposes as in the [[AGM method]].
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| ==Example==
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| 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:
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| :<math>\begin{align} | |
| a_1 &= \frac{1}{2}(24 + 6) = 15\\
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| g_1 &= \sqrt{24 \times 6} = 12
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| \end{align}</math>
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| and then iterate as follows:
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| :<math>\begin{align}
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| a_2 &= \frac{1}{2}(15 + 12) = 13.5\\
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| g_2 &= \sqrt{15 \times 12} = 13.41640786500\dots\\
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| \dots
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| \end{align}</math>
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| The first five iterations give the following values:
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| :{| class="wikitable"
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| |-
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| ! {{math|''n''}}
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| ! {{math|''a''<sub>''n''</sub>}}
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| ! {{math|''g''<sub>''n''</sub>}}
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| |-
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| | 0
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| | 24
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| | 6
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| |-
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| | 1
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| | {{underline|1}}5
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| | {{underline|1}}2
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| |-
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| | 2
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| | {{underline|13}}.5
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| | {{underline|13}}.416407864998738178455042…
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| |-
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| | 3
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| | {{underline|13.458}}203932499369089227521…
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| | {{underline|13.458}}139030990984877207090…
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| |-
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| | 4
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| | {{underline|13.4581714817}}45176983217305…
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| | {{underline|13.4581714817}}06053858316334…
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| |-
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| | 5
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| | {{underline|13.4581714817256154207668}}20…
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| | {{underline|13.4581714817256154207668}}06…
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| |}
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| 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>
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| == History ==
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| 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>
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| ==Properties==
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| 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'', ''y'') ≤ ''a<sub>n</sub>''}}. These are strict inequalities if {{math|''x'' ≠ ''y''}}.
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| {{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''}}.
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| If {{math|''r'' ≥ 0}}, then {{math|''M''(''rx'',''ry'') {{=}} ''r M''(''x'',''y'')}}.
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| There is an integral-form expression for {{math|''M''(''x'',''y'')}}:
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| :<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}}\\
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| &=\frac{\pi}{4} (x + y) \bigg/ K\left( \frac{x - y}{x + y} \right)
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| \end{align}</math>
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| where {{math|''K''(''k'')}} is the [[elliptic integral|complete elliptic integral of the first kind]]:
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| :<math>K(k) = \int_0^{\pi/2}\frac{d\theta}{\sqrt{1 - k^2\sin^2(\theta)}} </math> | |
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| 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>
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| == Related concepts ==
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| The reciprocal of the arithmetic–geometric mean of 1 and the [[square root of 2]] is called [[Gauss's constant]], after [[Carl Friedrich Gauss]].
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| :<math>\frac{1}{M(1, \sqrt{2})} = G = 0.8346268\dots</math>
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| 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]].
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| 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>
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| ==Proof of existence==
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| From [[inequality of arithmetic and geometric means]] we can conclude that:
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| :<math>g_n \leqslant a_n</math>
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| and thus
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| :<math>g_{n + 1} = \sqrt{g_n \cdot a_n} \geqslant \sqrt{g_n \cdot g_n} = g_n</math>
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| that is, the sequence {{math|''g<sub>n</sub>''}} is nondecreasing.
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| 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:
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| :<math>\lim_{n\to \infty}g_n = g</math>
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| However, we can also see that:
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| :<math>a_n = \frac{g_{n + 1}^2}{g_n}</math>
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| and so:
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| :<math>\lim_{n\to \infty}a_n = \lim_{n\to \infty}\frac{g_{n + 1}^2}{g_{n}} = \frac{g^2}{g} = g</math>
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| [[Q.E.D.]]
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| ==Proof of the integral-form expression==
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| This proof is given by Gauss.<ref name="BerggrenBorwein2004" />
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| Let
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| :<math>I(x,y) = \int_0^{\pi/2}\frac{d\theta}{\sqrt{x^2\cos^2\theta+y^2\sin^2\theta}},</math>
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| Changing the variable of integration to <math>\theta'</math>, where
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| :<math> \sin\theta = \frac{2x\sin\theta'}{(x+y)+(x-y)\sin^2\theta'}, </math>
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| gives
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| :<math>
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| \begin{align}
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| 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'}}\\
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| &= I\bigl(\tfrac12(x+y),\sqrt{xy}\bigr).
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| \end{align}
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| </math>
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| Thus, we have
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| :<math>
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| \begin{align}
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| I(x,y) &= I(a_1, g_1) = I(a_2, g_2) = \cdots\\
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| &= I\bigl(M(x,y),M(x,y)\bigr) = \pi/\bigr(2M(x,y)\bigl).
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| \end{align}
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| </math>
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| The last equality comes from observing that <math>I(z,z) = \pi/(2z)</math>.
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| Finally, we obtain the desired result
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| :<math>M(x,y) = \pi/\bigl(2 I(x,y) \bigr). </math>
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| ==See also==
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| * [[Generalized mean]]
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| * [[Inequality of arithmetic and geometric means]]
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| * [[Gauss–Legendre algorithm]]
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| ==External links==
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| * [http://arithmeticgeometricmean.blogspot.de/ Arithmetic-Geometric Mean Calculator]
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| * [http://planetmath.org/convergenceofarithmeticgeometricmean/ Proof of convergence rate in PlanetMath]
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| ==References==
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| {{More footnotes|date=October 2008}}
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| *{{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}}
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| * [[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. ISBN 0-471-31515-X {{MR|1641658}}
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| * [[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.
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| *{{SpringerEOM|author=M. Hazewinkel|title=Arithmetic–geometric mean process|urlname=a/a130280}}
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| *{{mathworld|urlname=Arithmetic-GeometricMean|title=Arithmetic–Geometric mean}}
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| <references />
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| {{DEFAULTSORT:Arithmetic-Geometric Mean}}
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| [[Category:Means]]
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| [[Category:Special functions]]
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| [[Category:Elliptic functions]]
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| [[Category:Articles containing proofs]]
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