Editing Arithmetic–geometric mean (section)

==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 }}</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>