View source for Arithmetic–geometric mean

{{Short description|Mathematical function of two positive real arguments}}
{{about|the particular type of mean|the similarly named inequality|Inequality of arithmetic and geometric means}}

[[File:Generalized means + agm.png|400px|thumb|right|Plot of the arithmetic–geometric mean <math>\operatorname{agm}(1,x)</math> among several [[generalized mean]]s.]]
In [[mathematics]], the '''arithmetic–geometric mean''' (AGM or agM<ref name="Cox" />) of two [[positive real numbers]] {{math|''x''}} and {{math|''y''}} is the mutual limit of a sequence of [[arithmetic mean]]s and a sequence of [[geometric mean]]s. The arithmetic–geometric mean is used in fast [[algorithm]]s for [[exponential function|exponential]], [[trigonometric functions]], and other [[special functions]], as well as some [[mathematical constant]]s, in particular, [[computing π|computing {{mvar|&pi;}}]].

The AGM is defined as the limit of the interdependent [[sequence]]s <math>a_i</math> and <math>g_i</math>.  Assuming <math> x \geq y \geq 0</math>, we write:<math display=block>\begin{align}
 a_0 &= x,\\
 g_0 &= y\\
 a_{n+1} &= \tfrac12(a_n + g_n),\\
 g_{n+1} &= \sqrt{a_n g_n}\, .
\end{align}</math>These two sequences [[limit of a sequence|converge]] to the same number, 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'')}} or {{math|AGM(''x'', ''y'')}}.

The arithmetic–geometric mean can be extended to [[Complex number|complex numbers]] and, when the [[Branch point#Branch cuts|branches]] of the square root are allowed to be taken inconsistently, it is a [[multivalued function]].<ref name="Cox">{{cite journal |last=Cox |first=David |date=January 1984 |title=The Arithmetic-Geometric Mean of Gauss|url=https://www.researchgate.net/publication/248675540 |journal=L'Enseignement Mathématique|volume=30|issue=2|pages=275–330}}</ref>

==Example==
To find the arithmetic–geometric mean of {{math|''a''<sub>0</sub> {{=}} 24}} and {{math|''g''<sub>0</sub> {{=}} 6}}, iterate as follows:<math display=block>\begin{array}{rcccl}
 a_1 & = & \tfrac12(24 + 6) & = & 15\\
 g_1 & = & \sqrt{24 \cdot 6} & = & 12\\
 a_2 & = & \tfrac12(15 + 12) & = & 13.5\\
 g_2 & = & \sqrt{15 \cdot 12} & = & 13.416\ 407\ 8649\dots\\
 & & \vdots & &
\end{array}</math>The first five iterations give the following values:

{| class="wikitable plainrowheaders" style="margin-left:2em;"
|-
! scope="col" | {{math|''n''}}
! scope="col" | {{math|''a''<sub>''n''</sub>}}
! scope="col" | {{math|''g''<sub>''n''</sub>}}
|-
! scope="row" | 0
| 24
| 6
|-
! scope="row" | 1
| {{underline|1}}5
| {{underline|1}}2
|-
! scope="row" | 2
| {{underline|13}}.5
| {{underline|13}}.416 407 864 998 738 178 455 042...
|-
! scope="row" | 3
| {{underline|13.458}} 203 932 499 369 089 227 521...
| {{underline|13.458}} 139 030 990 984 877 207 090...
|-
! scope="row" | 4
| {{underline|13.458 171 481 7}}45 176 983 217 305...
| {{underline|13.458 171 481 7}}06 053 858 316 334...
|-
! scope="row" | 5
| {{underline|13.458 171 481 725 615 420 766 8}}20...
| {{underline|13.458 171 481 725 615 420 766 8}}06...
|}

The number of digits in which {{math|''a''<sub>''n''</sub>}} and {{math|''g''<sub>''n''</sub>}} agree (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 {{val|13.4581714817256154207668131569743992430538388544}}.<ref>[https://www.wolframalpha.com/input/?i=agm%2824%2C+6%29 agm(24, 6)] at [[Wolfram Alpha]]</ref>

== History ==
The first algorithm based on this sequence pair appeared in the works of [[Joseph-Louis Lagrange|Lagrange]]. Its properties were further analyzed by [[Gauss]].<ref name="Cox"/>

==Properties==
Both the geometric mean and arithmetic mean of two positive numbers {{mvar|x}} and {{mvar|y}} are between the two numbers. (They are ''strictly'' between when {{math|''x'' ≠ ''y''}}.) The geometric mean of two positive numbers is [[Inequality of arithmetic and geometric means|never greater than the arithmetic mean]].<ref>{{cite book |last=Bullen |first=P. S. |contribution=The Arithmetic, Geometric and Harmonic Means |date=2003 |url=https://link.springer.com/10.1007/978-94-017-0399-4_2 |title=Handbook of Means and Their Inequalities |pages=60–174 |access-date=2023-12-11 |place=Dordrecht |publisher=Springer Netherlands |language=en |doi=10.1007/978-94-017-0399-4_2 |isbn=978-90-481-6383-0}}</ref> So the geometric means are an increasing sequence {{math|''g''{{sub|0}} ≤ ''g''{{sub|1}} ≤ ''g''{{sub|2}} ≤ ...}}; the arithmetic means are a decreasing sequence {{math|''a''{{sub|0}} ≥ ''a''{{sub|1}} ≥ ''a''{{sub|2}} ≥ ...}}; and {{math|''g<sub>n</sub>'' ≤ ''M''(''x'', ''y'') ≤ ''a<sub>n</sub>''}} for any {{mvar|n}}. These are strict inequalities if {{math|''x'' ≠ ''y''}}.

{{math|''M''(''x'', ''y'')}} is thus a number between {{math|''x''}} and {{math|''y''}}; it is also between the geometric and arithmetic mean of {{math|''x''}} and {{math|''y''}}.

If {{math|''r'' ≥ 0}} then {{math|''M''(''rx'', ''ry'') {{=}} ''r M''(''x'', ''y'')}}.

There is an integral-form expression for {{math|''M''(''x'', ''y'')}}:<ref>{{dlmf|first1=B. C.|last1=Carson|id=19.8.i|title=Elliptic Integrals|mode=cs1}}</ref><math display=block>\begin{align}
 M(x,y) &= \frac{\pi}{2} \left( \int_0^\frac{\pi}{2}\frac{d\theta}{\sqrt{x^2\cos^2\theta+y^2\sin^2\theta}} \right)^{-1}\\
&=\pi\left(\int_0^\infty \frac{dt}{\sqrt{t(t+x^2)(t+y^2)}}\right)^{-1}\\
        &= \frac{\pi}{4} \cdot \frac{x + y}{K\left( \frac{x - y}{x + y} \right)}
\end{align}</math>where {{math|''K''(''k'')}} is the [[elliptic integral|complete elliptic integral of the first kind]]:<math display="block">K(k) = \int_0^\frac{\pi}{2}\frac{d\theta}{\sqrt{1 - k^2\sin^2\theta}} </math>Since the arithmetic–geometric process converges so quickly, it provides an efficient way to compute elliptic integrals, which are used, for example, in [[elliptic filter]] design.<ref name="Dimopoulos2011">{{cite book |author-first=Hercules G. |author-last=Dimopoulos |title=Analog Electronic Filters: Theory, Design and Synthesis |url=https://books.google.com/books?id=6W1eX4QwtyYC&pg=PA147 |year=2011 |publisher=Springer |isbn=978-94-007-2189-0 |pages=147–155 }}</ref>


The arithmetic–geometric mean is connected to the [[Theta function|Jacobi theta function]] <math>\theta_3</math> by<ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} pages 35, 40</ref><math display="block">M(1,x)=\theta_3^{-2}\left(\exp \left(-\pi \frac{M(1,x)}{M\left(1,\sqrt{1-x^2}\right)}\right)\right)=\left(\sum_{n\in\mathbb{Z}}\exp \left(-n^2 \pi \frac{M(1,x)}{M\left(1,\sqrt{1-x^2}\right)}\right)\right)^{-2},</math>which upon setting <math>x=1/\sqrt{2}</math> gives<math display="block">M(1,1/\sqrt{2})=\left(\sum_{n\in\mathbb{Z}}e^{-n^2\pi}\right)^{-2}.</math>

== Related concepts ==
The reciprocal of the arithmetic–geometric mean of 1 and the [[square root of 2]] is [[Gauss's constant]].<math display=block>\frac{1}{M(1, \sqrt{2})} = G = 0.8346268\dots</math>In 1799, Gauss proved<ref group="note">By 1799, Gauss had two proofs of the theorem, but neither of them was rigorous from the modern point of view.</ref> that<math display="block">M(1,\sqrt{2})=\frac{\pi}{\varpi}</math>where <math>\varpi</math> is the [[lemniscate constant]].


In 1941, <math>M(1,\sqrt{2})</math> (and hence <math>G</math>) was proved [[Transcendental number|transcendental]] by [[Theodor Schneider]].<ref group="note">In particular, he proved that the [[beta function]] <math>\Beta (a,b)</math> is transcendental for all <math>a,b\in\mathbb{Q}\setminus\mathbb{Z}</math> such that <math>a+b\notin \mathbb{Z}_0^-</math>. The fact that <math>M(1,\sqrt{2})</math> is transcendental follows from <math>M(1,\sqrt{2})=\tfrac{1}{2}\Beta \left(\tfrac{1}{2},\tfrac{3}{4}\right).</math></ref><ref>{{cite journal |first=Theodor |last=Schneider |url=https://www.deepdyve.com/lp/de-gruyter/zur-theorie-der-abelschen-funktionen-und-integrale-mn0U50bvkB |title=Zur Theorie der Abelschen Funktionen und Integrale |year=1941 |journal=Journal für die reine und angewandte Mathematik |volume=183 |number=19 |pages=110–128 |doi=10.1515/crll.1941.183.110 |s2cid=118624331 }}</ref><ref>{{Cite journal |title=The Lemniscate Constants |last=Todd |first=John |journal=[[Communications of the ACM]] |volume=18 |number=1 <!-- |month=January -->|year=1975 |pages=14–19 |doi=10.1145/360569.360580 |s2cid=85873 |doi-access=free }}</ref> The set <math>\{\pi,M(1,1/\sqrt{2})\}</math> is [[Algebraic independence|algebraically independent]] over <math>\mathbb{Q}</math>,<ref>G. V. Choodnovsky: ''Algebraic independence of constants connected with the functions of analysis'', Notices of the AMS 22, 1975, p. A-486</ref><ref>G. V. Chudnovsky: ''Contributions to The Theory of Transcendental Numbers'', American Mathematical Society, 1984, p. 6</ref> but the set <math>\{\pi,M(1,1/\sqrt{2}),M'(1,1/\sqrt{2})\}</math> (where the prime denotes the [[derivative]] with respect to the second variable) is not algebraically independent over <math>\mathbb{Q}</math>. In fact,<ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} p. 45</ref><math display="block">\pi=2\sqrt{2}\frac{M^3(1,1/\sqrt{2})}{M'(1,1/\sqrt{2})}.</math>The [[geometric–harmonic mean]] GH can be calculated using analogous sequences of geometric and [[harmonic mean|harmonic]] means, and in fact {{math|1= GH(''x'', ''y'') = 1/''M''(1/''x'', 1/''y'') = ''xy''/''M''(''x'', ''y'')}}.<ref>{{cite journal
 | last = Newman | first = D. J.
 | doi = 10.2307/2007804
 | journal = Mathematics of Computation
 | pages = 207–210
 | title = A simplified version of the fast algorithms of Brent and Salamin
 | volume = 44
 | year = 1985| issue = 169
 | jstor = 2007804
 }}</ref>
The arithmetic–harmonic mean [[Geometric mean#Iterative means|is equivalent to the geometric mean]].

The arithmetic–geometric mean can be used to compute – among others – [[Logarithm#Arithmetic–geometric mean approximation|logarithms]],  [[Elliptic integral|complete and incomplete elliptic integrals of the first and second kind]],<ref>{{AS ref|17|598–599}}</ref> and [[Jacobi elliptic functions]].<ref>{{cite book |first=Louis V. |last=King |author-link=Louis Vessot King |url=https://archive.org/details/onthenumerical032686mbp |title=On the Direct Numerical Calculation of Elliptic Functions and Integrals |publisher=Cambridge University Press |year=1924 }}</ref>

==Proof of existence==
The [[inequality of arithmetic and geometric means]] implies that<math display=block>g_n \leq a_n</math>and thus<math display=block>g_{n + 1} = \sqrt{g_n \cdot a_n} \geq \sqrt{g_n \cdot g_n} = g_n</math>that is, the sequence {{math|''g<sub>n</sub>''}} is nondecreasing and bounded above by the larger of {{math|''x''}} and {{math|''y''}}. By the [[monotone convergence theorem]], the sequence is convergent, so there exists a {{math|''g''}} such that:<math display=block>\lim_{n\to \infty}g_n = g</math>However, we can also see that:<math display=block>a_n = \frac{g_{n + 1}^2}{g_n}</math>
and so:
<math display=block>\lim_{n\to \infty}a_n = \lim_{n\to \infty}\frac{g_{n + 1}^2}{g_{n}} = \frac{g^2}{g} = g</math>

[[Q.E.D.]]

==Proof of the integral-form expression==
This proof is given by Gauss.<ref name="Cox" />
Let

<math display=block>I(x,y) = \int_0^{\pi/2}\frac{d\theta}{\sqrt{x^2\cos^2\theta+y^2\sin^2\theta}} ,</math>

Changing the variable of integration to <math>\theta'</math>, where

<math display="block"> 
\begin{align}
\sin\theta &= \frac{2x\sin\theta'}{(x+y)+(x-y)\sin^2\theta'} \\
\Rightarrow d(\sin\theta) &= d\left(\frac{2x\sin\theta'}{(x+y)+(x-y)\sin^2\theta'}\right)\\
\Rightarrow \cos\theta\ d\theta &= 2x  \frac{(x+y)-(x-y)\sin^2\theta'}{((x+y)+(x-y)\sin^2\theta')^2}\ \cos\theta' d\theta' 
\end{align}
</math>

<math display="block">
\begin{align}
\cos\theta 
&= \frac{\sqrt{(x+y)^2-2(x^2+y^2)\sin^2\theta'+(x-y)^2\sin^4\theta'}}{(x+y)+(x-y)\sin^2\theta'}\\
&= \frac{\cos\theta'\sqrt{(x-y)^2\cos^2\theta'+4xy}}{(x+y)+(x-y)\sin^2\theta'}\\
&= \frac{\cos\theta'\sqrt{(x+y)^2\cos^2\theta'+4xy\sin^2\theta'}}{(x+y)+(x-y)\sin^2\theta'},
\end{align}
</math>

<math display="block">
\Rightarrow 
\cos\theta\ d\theta =\frac{\cos\theta'\sqrt{(x+y)^2\cos^2\theta'+4xy\sin^2\theta'}}{(x+y)+(x-y)\sin^2\theta'}\ d\theta =2x  \frac{(x+y)-(x-y)\sin^2\theta'}{((x+y)+(x-y)\sin^2\theta')^2}\ \cos\theta' d\theta' ,
</math>

<math display="block">  
\Rightarrow 
d\theta = \frac{x((x+y)-(x-y)\sin^2\theta')}{((x+y)+(x-y)\sin^2\theta')} \frac{2 d\theta'}{\sqrt{(x+y)^2\cos^2\theta'+4xy\sin^2\theta'}}  
 \ ,
</math>

<math display="block"> 
\begin{align}
\sqrt{x^2\cos^2\theta+y^2\sin^2\theta} 
&= \frac{\sqrt{{x^2 ((x+y)^2-2(x^2+y^2)\sin^2\theta'+(x-y)^2\sin^4\theta')+4x^2y^2\sin^2\theta'}}}{((x+y)+(x-y)\sin^2\theta')}\\
&= \frac{x ((x+y)-(x-y)\sin^2\theta')}{((x+y)+(x-y)\sin^2\theta')}
\end{align}
</math>

This yields 
<math display="block">  \frac{d\theta}{\sqrt{x^2\cos^2\theta+y^2\sin^2\theta}} = \frac{2 d\theta'}{\sqrt{(x+y)^2\cos^2\theta'+4xy\sin^2\theta'}} = \frac{d\theta'}{\sqrt{((\frac{x+y}{2})^2\cos^2\theta'+(\sqrt{xy})^2\sin^2\theta'}},
</math>

gives

<math display="block">\begin{align}
I(x,y) &= \int_0^{\pi/2}\frac{d\theta'}{\sqrt{((\frac{x+y}{2})^2\cos^2\theta'+(\sqrt{xy})^2\sin^2\theta'}}\\
       &= I\bigl(\tfrac{x+y}{2},\sqrt{xy}\bigr) .
\end{align}</math>

Thus, we have

<math display=block>
\begin{align}
I(x,y) &= I(a_1, g_1) = I(a_2, g_2) = \cdots\\
  &= I\bigl(M(x,y),M(x,y)\bigr) = \pi/\bigr(2M(x,y)\bigl) .
\end{align}
</math>
The last equality comes from observing that <math>I(z,z) = \pi/(2z)</math>.

Finally, we obtain the desired result

<math display=block>M(x,y) = \pi/\bigl(2 I(x,y) \bigr) .</math>

==Applications==

===The number ''π''===
According to the [[Gauss–Legendre algorithm]],<ref>{{cite journal |first=Eugene |last=Salamin |author-link=Eugene Salamin (mathematician) |title=Computation of π using arithmetic–geometric mean |journal=[[Mathematics of Computation]] |url=https://link.springer.com/chapter/10.1007/978-3-319-32377-0_1 |volume=30 |issue=135 |year=1976 <!-- |month=July -->|pages=565–570 |doi=10.2307/2005327 |jstor=2005327 |mr=0404124 |url-access=subscription }}</ref>

<math display=block>\pi = \frac{4\,M(1,1/\sqrt{2})^2} {1 - \displaystyle\sum_{j=1}^\infty 2^{j+1} c_j^2} ,</math>

where

<math display=block>c_j = \frac{1}{2}\left(a_{j-1}-g_{j-1}\right) ,</math>

with <math>a_0=1</math> and <math>g_0=1/\sqrt{2}</math>, which can be computed without loss of precision using

<math display=block>c_j = \frac{c_{j-1}^2}{4a_j} .</math>

===Complete elliptic integral ''K''(sin''α'')===
Taking <math>a_0 = 1</math> and <math>g_0 = \cos\alpha</math> yields the AGM

<math display=block>M(1,\cos\alpha) = \frac{\pi}{2K(\sin \alpha)} ,</math>

where {{math|''K''(''k'')}} is a complete [[elliptic integral|elliptic integral of the first kind]]:

<math display=block>K(k) = \int_0^{\pi/2}(1 - k^2 \sin^2\theta)^{-1/2} \, d\theta.</math>

That is to say that this [[quarter period]] may be efficiently computed through the AGM,
<math display=block>K(k) =  \frac{\pi}{2M(1,\sqrt{1-k^2})} .</math>

===Other applications===
Using this property of the AGM along with the ascending transformations of [[John Landen]],<ref>{{cite journal |first=John |last=Landen |title=An investigation of a general theorem for finding the length of any arc of any conic hyperbola, by means of two elliptic arcs, with some other new and useful theorems deduced therefrom |journal=[[Philosophical Transactions of the Royal Society]] |volume=65 |year=1775 |pages=283–289 |doi=10.1098/rstl.1775.0028|s2cid=186208828 }}</ref> [[Richard P. Brent]]<ref>{{cite journal |first=Richard P. |last=Brent |title=Fast Multiple-Precision Evaluation of Elementary Functions |journal=[[Journal of the ACM]] |volume=23 |issue=2 |year=1976 |pages=242–251 |doi=10.1145/321941.321944 |mr=0395314 |citeseerx=10.1.1.98.4721 |s2cid=6761843 |url=https://link.springer.com/chapter/10.1007/978-3-319-32377-0_2 }}</ref> suggested the first AGM algorithms for the fast evaluation of elementary [[transcendental function]]s ({{math|''e''<sup>''x''</sup>}}, {{math|cos&nbsp;''x''}}, {{math|sin&nbsp;''x''}}). Subsequently, many authors went on to study the use of the AGM algorithms.<ref>{{cite book |author1-link=Jonathan Borwein |first1=Jonathan M. |last1=Borwein |author2-link=Peter Borwein |first2=Peter B. |last2=Borwein |title=Pi and the AGM |publisher=Wiley |place=New York |year=1987 |isbn=0-471-83138-7 |mr=0877728 }}</ref>

==See also==
* [[Landen's transformation]]
* [[Gauss–Legendre algorithm]]
* [[Generalized mean]]

==References==

===Notes===
{{reflist|group=note}}

===Citations===
{{reflist}}

===Sources===
{{refbegin}}
* {{cite journal |author1-last=Daróczy |author1-first=Zoltán |author2-last=Páles |author2-first=Zsolt |year=2002 |title=Gauss-composition of means and the solution of the Matkowski–Suto problem |journal=[[Publicationes Mathematicae Debrecen]] |volume=61 |issue=1–2 |pages=157–218 |doi=10.5486/PMD.2002.2713 }}
* {{SpringerEOM |title=Arithmetic–geometric mean process|mode=cs1}}
* {{mathworld |urlname=Arithmetic-GeometricMean}}
{{refend}}
{{Statistics|descriptive}}
{{DEFAULTSORT:Arithmetic-Geometric Mean}}
[[Category:Means]]
[[Category:Special functions]]
[[Category:Elliptic functions]]
[[Category:Articles containing proofs]]