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| {{mergeto|arithmetic–geometric mean|date=September 2012}}
| | Hello from Italy. I'm glad to came here. My first name is June. <br>I live in a small city called Poggio San Francesco in east Italy.<br>I was also born in Poggio San Francesco 24 years ago. Married in March 2009. I'm working at the college.<br><br>Here is my page - [http://www.calscottages.com/uggboots.asp UGG Boots USA] |
| In [[mathematics]], the '''AGM method''' (for [[arithmetic–geometric mean]]) makes it possible to construct fast [[algorithm]]s for calculation of [[exponential function|exponential]] and [[trigonometric functions]], and some [[mathematical constant]]s and in particular, to quickly [[computing π|compute <math>\pi</math>]].
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| ==Method==
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| [[Gauss]] noticed<ref>{{cite journal |author=B. C. Carlson |title=Algorithms involving arithmetic and geometric means |journal=Amer. Math. Monthly |volume=78 |year=1971 |pages=496–505 |doi=10.2307/2317754 |mr=0283246}}</ref><ref>{{cite journal |author=B. C. Carlson |title=An algorithm for computing logarithms and arctangents |journal=Math.Comp. |volume=26 |issue=118 |year=1972 |pages=543–549 |doi=10.2307/2005182 |mr=0307438}}</ref> that the sequences
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| : <math>
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| \begin{align}
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| a_0 & & b_0 \\
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| a_1 & = \frac{a_0+b_0}{2}, & b_1 & = \sqrt{a_0 b_0} \\
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| a_2 & = \frac{a_1+b_1}{2}, & b_2 & = \sqrt{a_1 b_1} \\
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| & {}\ \ \vdots & & {}\ \ \vdots \\
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| a_{N+1} & = \frac{a_N + b_N}{2}, & b_{N+1} & = \sqrt{a_N b_N}
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| \end{align}
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| </math>
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| as
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| : <math>N\to +\infty, \, </math>
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| have the same limit:
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| : <math>
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| \lim_{N\to\infty}a_N = \lim_{N\to\infty}b_N = M(a,b), \,
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| </math>
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| the [[arithmetic–geometric mean]].
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| It is possible to use this fact to construct [[fast algorithms]] for calculating [[elementary function|elementary]] [[transcendental function]]s and some classical constants, in particular, the constant [[pi|{{pi}}]].
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| ==Applications==
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| ===The number ''π''===
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| For example, according to the Gauss–[[Eugene Salamin (mathematician)|Salamin]] formula:<ref>{{cite journal |author=[[Eugene Salamin (mathematician)|E. Salamin]] |title=Computation of <math>\pi</math> using arithmetic-geometric mean |journal=Math. Comp. |volume=30 |issue=135 |year=1976 |pages=565–570 |doi=10.2307/2005327 |mr=0404124}}</ref>
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| : <math>
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| \pi = \frac{4 \left( M(1; \frac{1}{\sqrt{2}}) \right)^2} {\displaystyle 1 - \sum_{j=1}^\infty 2^{j+1} c_j^2}
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| ,
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| </math>
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| where
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| : <math>c_j = \frac 12\left(a_{j-1}-b_{j-1}\right).</math>
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| ===Complete elliptic integral ''K''(''α'')===
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| At the same time, if we take
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| : <math>
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| a_0 = 1, \quad b_0 = \cos\alpha,
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| </math>
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| then
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| : <math>
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| \lim_{N\to\infty}a_N = \frac{\pi}{2K(\alpha)},
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| </math>
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| where ''K''(''α'') is a complete [[elliptic integral]]
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| : <math> | |
| K(\alpha) = \int_0^{\pi/2}(1 - \alpha \sin^2\theta)^{-1/2} \, d\theta .
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| </math>
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| ===Other applications===
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| Using this property of the AGM and also the ascending transformations of Landen,<ref>{{cite journal |author=J. 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}}</ref> [[Richard Brent (scientist)|Richard Brent]]<ref>{{cite journal |author=[[Richard Brent (scientist)|R.P. Brent]] |title=Fast Multiple-Precision Evaluation of Elementary Functions |journal=J. Assoc. Comput. Mach. |volume=23 |issue=2 |year=1976 |pages=242–251 |doi=10.1145/321941.321944 |mr=0395314}}</ref> suggested the first AGM algorithms for fast evaluation of elementary transcendental functions (''e''<sup>''x''</sup>, cos ''x'', sin ''x''). Later many authors have been going on to study and use the AGM algorithms, see, for example, the book ''Pi and the AGM'' by [[Jonathan Borwein|Jonathan]] and [[Peter Borwein]].<ref>{{cite book |author1-link=Jonathan Borwein |first1=J.M. |last1=Borwein| author2-link=Peter Borwein |first2=P.B. |last2=Borwein |title=Pi and the AGM |publisher=Wiley |place=New York |year=1987 |isbn=0-471-83138-7 |mr=0877728}}</ref>
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| ==See also==
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| *[[Gauss–Legendre algorithm]]
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| ==References==
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| {{reflist|2}}
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| {{DEFAULTSORT:AGM method}}
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| [[Category:Computer arithmetic algorithms]]
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Hello from Italy. I'm glad to came here. My first name is June.
I live in a small city called Poggio San Francesco in east Italy.
I was also born in Poggio San Francesco 24 years ago. Married in March 2009. I'm working at the college.
Here is my page - UGG Boots USA