Multiplicatively closed set - Revision history

en>Trappist the monk: /* References */replace mr template with mr parameter in CS1 templates; using AWB

2014-09-25T12:31:31Z

References: replace mr template with mr parameter in CS1 templates; using AWB

← Older revision		Revision as of 14:31, 25 September 2014
Line 1:		Line 1:
	~~{{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>]].

	~~==Method==~~
	~~[[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~~

	~~: <math>~~
	~~\begin{align}~~
	~~a_0 & & b_0 \\~~
	~~a_1 & = \frac{a_0+b_0}{2}, & b_1 & = \sqrt{a_0 b_0} \\~~
	~~a_2 & = \frac{a_1+b_1}{2}, & b_2 & = \sqrt{a_1 b_1} \\~~
	~~& {}\ \ \vdots & & {}\ \ \vdots \\~~
	~~a_{N+1} & = \frac{a_N + b_N}{2}, & b_{N+1} & = \sqrt{a_N b_N}~~
	~~\end{align}~~
	~~</math>~~

	as
	~~: <math>N\to +\infty, \, </math>~~

	~~have the same limit:~~
	~~: <math>~~
	~~\lim_{N\to\infty}a_N = \lim_{N\to\infty}b_N = M(a,b), \,~~
	~~</math>~~

	~~the [[arithmetic–geometric mean]]~~.

	~~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}}]]~~.

	~~==Applications==~~
	~~===The number ''π'~~'~~===~~
	~~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>~~

	~~: <math>~~
	~~\pi = \frac{4 \left( M(1; \frac{1}{\sqrt{2}}) \right)^2} {\displaystyle 1 - \sum_{j=1}^\infty 2^{j+1} c_j^2}~~
	,
	~~</math>~~

	~~where~~

	~~: <math>c_j = \frac 12\left(a_{j-1}-b_{j~~-~~1}\right).</math>~~

	~~===Complete elliptic integral ''K''(''α'')===~~
	~~At the same time, if we take~~

	~~: <math>~~
	~~a_0 = 1, \quad b_0 = \cos\alpha,~~
	~~</math>~~

	~~then~~

	~~: <math>~~
	~~\lim_{N\to\infty}a_N = \frac{\pi}{2K(\alpha)},~~
	~~</math>~~

	~~where ''K''(''α'') is a complete [~~[~~elliptic integral]]~~

	: ~~<math>~~
	~~K(\alpha) = \int_0^{\pi/2}(1 - \alpha \sin^2\theta)^{-1/2} \, d\theta .~~
	</~~math>~~

	~~===Other applications===~~
	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>~~

	~~==See also==~~
	*[[Gauss–Legendre algorithm]]

	~~==References==~~
	~~{{reflist\|2}}~~

	~~{{DEFAULTSORT:AGM method}}~~
	~~[[Category:Computer arithmetic algorithms]~~]

en>Ebony Jackson: It is standard not to exclude 0: see Talk page. Added references. Krull's lemma was removed because it is contained in the statement about unions of prime ideals.

2013-11-24T03:36:06Z

It is standard not to exclude 0: see Talk page. Added references. Krull's lemma was removed because it is contained in the statement about unions of prime ideals.

← Older revision		Revision as of 05:36, 24 November 2013
Line 1:		Line 1:
			{{mergeto\|arithmetic–geometric mean\|date=September 2012}}
			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>]].

			==Method==
			[[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

	~~What then are some of~~ the ~~important issues you can do~~ to ~~create~~ a ~~much better profile? Here are three specific profile suggestions you can use to improve your profile right now.~~<br>~~<br>If you are pretty new at~~ the ~~sport~~ and ~~certain individuals appear to be more successful than you~~, ~~notice them! View and listen to how they interact with~~ the ~~other people~~. ~~These people are most likely just more assured than you are~~, ~~and this is where~~ [~~http~~:/~~/www~~.~~all-for-ls~~.de/~~index~~/~~users/view/id/133219 http~~:~~//geekdates.net] increases your confidence, simply because you can see what they are performing that functions.~~<br><br><br><br>~~Ask concerns to continue~~ the ~~conversation. In the exact~~ same ~~manner~~, ~~don't just reply to the questions requested. That would be boring. online dating is not a mundane query-answer procedure. It is definitely much more than that. Talk as~~ if ~~you were in the exact same room.~~<br><br>~~You should also bear in mind the type of site you are signed up with, as they vary. For example~~, a ~~lady posting a photograph ad on a lingerie website should post a photo of herself in lingerie.~~<br><br>~~This is among~~ the ~~very best technique to permit you to obtain personal financial institution account~~ and ~~that is~~ the ~~mother~~ of ~~all presents from a Sugar Daddy. Ask about precisely how he manages his profit and how he started to be so successful. Also~~, ~~make certain he understands how you are scared which he may depart you~~. ~~By having your own bank-account that he fills up you'll learn how to manage~~ a ~~greater price and will have weekly deposits~~ for ~~types investing. Tell him you listened to~~ some ~~thing to kind it out becoming a tax create off in order to help his business price. Your sugar daddy will think it is so adorable that you are trying to discover about his company but will signal~~ the ~~checks to you instantly to assist you deposit them to your financial institution account.~~<br><br>~~The fourth issue with totally free nerd dating websites is that there is a chance~~ of ~~identification theft~~. ~~If you believe in a dater and begin passing out individual information like your address then they can use it to steal your identity info~~. ~~If you satisfied a individual in the bar and give them your address or give them accessibility to your wallet/purse at any time , they can take your identification also~~. ~~People who steal identities can be superb con artists~~. ~~A [http://www.Tumblr~~.~~com/tagged~~/~~matchmaker matchmaker] could also get fooled by a individual who is utilizing a fake identity but it is a lot simpler to capture somebody when you look into past addresses and work~~.<~~br><br~~>Saturday is ladies's buying working day or their day to celebration with their female friends. So when a woman accepted your invitation for a date during Saturday afternoon or Saturday night, there is a great chance that she likes you. She likes you if she is prepared not to attend a VIP party with her friends on the ~~newest and most popular club just to have a day with you. A woman is usually accessible~~ for ~~the guy she likes. Moreover, she is also putting you on the top~~ of ~~her list of priorities if she truly, really likes you.~~<br><br>~~The truth is~~, many ~~different dating singles are being much more and much more apt to discover the love of their life~~ on ~~the World Broad Internet. This provides people the capability~~ to ~~bring anyone~~ and ~~everybody with accessibility to the internet to their dens with out much much more than a couple clicks of~~ the ~~mouse.<br><br>Jumping into a relationship~~, ~~numerous women want the entire shebang correct from the initial day. Roses~~, ~~wine and dine, intimate lines~~, ~~grand gestures.~~ the ~~whole thing. This can put enormous pressure on~~ the ~~guy, especially the guy who might not have a all-natural knack for romance. Reduce him some slack~~ and ~~display him that you just want to have a good time with him~~.<br>~~<br>Some paid out for sites offer totally free trials~~. ~~This is perfect as it gives you a chance to see what their website is like and whether or not you want to adhere with it~~. ~~When registering for a free demo check to see how to unsubscribe if you require to~~. ~~If you are not satisfied you might find that you are being billed which is concealed~~.		: <math>
			\begin{align}
			a_0 & & b_0 \\
			a_1 & = \frac{a_0+b_0}{2}, & b_1 & = \sqrt{a_0 b_0} \\
			a_2 & = \frac{a_1+b_1}{2}, & b_2 & = \sqrt{a_1 b_1} \\
			& {}\ \ \vdots & & {}\ \ \vdots \\
			a_{N+1} & = \frac{a_N + b_N}{2}, & b_{N+1} & = \sqrt{a_N b_N}
			\end{align}
			</math>

			as
			: <math>N\to +\infty, \, </math>

			have the same limit:
			: <math>
			\lim_{N\to\infty}a_N = \lim_{N\to\infty}b_N = M(a,b), \,
			</math>

			the [[arithmetic–geometric mean]].

			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}}]].

			==Applications==
			===The number ''π''===
			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>

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

			where

			: <math>c_j = \frac 12\left(a_{j-1}-b_{j-1}\right).</math>

			===Complete elliptic integral ''K''(''α'')===
			At the same time, if we take

			: <math>
			a_0 = 1, \quad b_0 = \cos\alpha,
			</math>

			then

			: <math>
			\lim_{N\to\infty}a_N = \frac{\pi}{2K(\alpha)},
			</math>

			where ''K''(''α'') is a complete [[elliptic integral]]

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

			===Other applications===
			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>

			==See also==
			*[[Gauss–Legendre algorithm]]

			==References==
			{{reflist\|2}}

			{{DEFAULTSORT:AGM method}}
			[[Category:Computer arithmetic algorithms]]

en>Zundark: fix a link

2012-03-27T11:51:16Z

fix a link

New page

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