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			{{For\|the midpoint rule in numerical [[quadrature]]\|rectangle method}}

			[[File:Midpoint method illustration.png\|right\|thumb\|Illustration of the midpoint method assuming that <math>y_n</math> equals the exact value <math>y(t_n).</math> The midpoint method computes <math>y_{n+1}</math> so that the red chord is approximately parallel to the tangent line at the midpoint (the green line).]]
			In [[numerical analysis]], a branch of [[applied mathematics]], the '''midpoint method''' is a one-step method for [[Numerical ordinary differential equations\|numerically]] solving the [[ordinary differential equation\|differential equation]],
			:<math> y'(t) = f(t, y(t)), \quad y(t_0) = y_0 </math>
			and is given by the formula
			:<math> y_{n+1} = y_n + hf\left(t_n+\frac{h}{2},y_n+\frac{h}{2}f(t_n, y_n)\right), \qquad\qquad (1)</math>
			for <math>n=0, 1, 2, \dots</math> Here, <math>h</math> is the ''step size'' — a small positive number, <math>t_n=t_0 + n h,</math> and <math>y_n</math> is the computed approximate value of <math>y(t_n).</math> The midpoint method is also known as the '''modified Euler method'''.<ref>{{harvnb\|Süli\|Mayers\|2003\|p=328}}</ref>

	~~You can download~~ from the ~~according to hyperlink~~, ~~if you~~'~~re looking for clash~~ of ~~families zero-cost gems, elixir and gold rings. You~~'~~ll get the greatest secret text~~ to ~~get accessibility having to do with assets and endless stones by [http://browse~~.~~deviantart.com~~/~~?qh=&section=&~~global~~=1&q=downloading downloading] from adhering to links.~~<br><br>~~Trying the higher level: it is important when it comes with game~~, ~~but when you~~'~~re Clash of Clans~~, ~~you get a lot~~ more ~~subtle actions~~. ~~Despite making use~~ of ~~clash~~ of ~~clans hack tools, you can also acquire experience points~~ as ~~a result~~ of ~~matching on top of other players~~. ~~Increased metabolism purpose~~ of ~~earning Pvp is to enable even further enhancements~~ for ~~your indigneous group~~. ~~The restoration consists of better strive equipment~~, ~~properties~~, ~~troops while tribe people~~.<br><br>~~Temperance is essential in most things, and enjoying clash of clans cheats is no dissimilar~~. ~~Playing for hours on finish seriously is not good-for-you~~, ~~bodily or emotionally~~. ~~There are some games out there which know this and can include measures to remind in order to take rests~~. ~~Go ahead and take initiative yourself, although! Place an alarm as well as don't play~~ for ~~a lot more an hour right.~~<br><br>~~An outstanding~~ method ~~to please children with~~ a ~~gaming device~~ and ~~ensure they survive fit~~ is to it's ~~Wii. This gaming system needs real task to play. Your children won~~'t ~~be sitting for hours on termination playing clash of clans hack~~. ~~They need to be moving around as~~ the right ~~way to play the games in such a particular system.~~<br><br>~~Deliver the in-online game songs opportunity~~. ~~If you liked~~ this ~~post and you would such~~ as ~~to receive more details relating to clash of clans hack android [[http:~~//~~circuspartypanama~~.~~com have~~ a ~~peek at these guys~~]] ~~kindly see the web-site. If, nonetheless, you might exist annoyed by~~ using ~~it again soon after one moment approximately, don't be nervous to mute~~ the ~~tv set or personal computer and as a consequence play some audio of one's very own. You will find a far more pleasurable game playing experience this way and therefore are a whole lot unlikely~~ to ~~get a frustration from actively playing.~~<br><br>~~Your tutorial will guide you thru your first few raids~~, ~~constructions~~, ~~and upgrades~~, ~~yet unfortunately youre~~ left ~~to private wiles pretty quickly. Your buildings take real-time to construct and upgrade~~, ~~your army units historic recruit~~, and ~~your guide buildings take time create food and gold~~. ~~Like all of the truck bed cover~~'s ~~genre cousins~~, ~~Throne Push is meant to took part in multiple short bursts the sun sets~~. ~~This type of fanatical gaming definitely works more beneficial on mobile devices which can always with you and could send push notifications when timed tasks are launched~~. ~~Then again~~, ~~the success of so many hit Facebook games through the years indicates that people check Facebook often enough to short play sessions employment there too~~.~~<br><br>Plainly individuals who produced this Crack Clash of Loved ones are true fans linked with the sport themselves, in addition~~ to ~~this is exactly what ensures the potency of our alternative, because we needed to do this task ourselves~~.		The name of the method comes from the fact that in the formula above the function <math>f</math> is evaluated at <math>t=t_n+h/2,</math> which is the midpoint between <math>t_n</math> at which the value of ''y''(''t'') is known and <math>t_{n+1}</math> at which the value of ''y''(''t'') needs to be found.

			The local error at each step of the midpoint method is of order <math>O\left(h^3\right)</math>, giving a global error of order <math>O\left(h^2\right)</math>. Thus, while more computationally intensive than [[Euler's method]], the midpoint method generally gives more accurate results.

			The method is an example of a class of higher-order methods known as [[Runge-Kutta methods]].

			==Derivation of the midpoint method==
			[[File:Numerical integration illustration, h=1.png\|right\|thumb\|Illustration of numerical integration for the equation <math>y'=y, y(0)=1.</math> Blue: the [[Euler method]], green: the midpoint method, red: the exact solution, <math>y=e^t.</math> The step size is <math>h=1.0.</math>]]
			[[File:Numerical integration illustration, h=0.25.png\|right\|thumb\|The same illustration for <math>h=0.25.</math> It is seen that the midpoint method converges faster than the Euler method.]]

			The midpoint method is a refinement of the Euler's method
			:<math> y_{n+1} = y_n + hf(t_n,y_n),\, </math>
			and is derived in a similar manner.
			The key to deriving Euler's method is the approximate equality
			:<math> y(t+h) \approx y(t) + hf(t,y(t)) \qquad\qquad (2)</math>
			which is obtained from the slope formula
			:<math> y'(t) \approx \frac{y(t+h) - y(t)}{h} \qquad\qquad (3)</math>
			and keeping in mind that <math> y' = f(t, y).</math>

			For the midpoint method, one replaces (3) with the more accurate
			:<math> y'\left(t+\frac{h}{2}\right) \approx \frac{y(t+h) - y(t)}{h} </math>
			when instead of (2) we find
			:<math> y(t+h) \approx y(t) + hf\left(t+\frac{h}{2},y\left(t+\frac{h}{2}\right)\right). \qquad\qquad (4)</math>

			One cannot use this equation to find <math> y(t+h)</math> as one does not know <math>y</math> at <math>t+h/2.</math> The solution is then to use a [[Taylor series]] expansion exactly as if using the [[Euler method]] to solve for <math>y(t+h/2)</math>:
			:<math>y\left(t + \frac{h}{2}\right) \approx y(t) + \frac{h}{2}y'(t)=y(t) + \frac{h}{2}f(t, y(t)),</math>
			which, when plugged in (4), gives us
			:<math>y(t + h) \approx y(t) + hf\left(t + \frac{h}{2}, y(t) + \frac{h}{2}f(t, y(t))\right)</math>
			and the midpoint method (1).

			==See also==
			* [[Rectangle method]]
			* [[Heun's method]]
			* [[Leapfrog integration]] and [[Verlet integration]]

			==Notes==
			{{reflist}}

			==References==

			* {{cite book
			\|author=Griffiths,D. V.; Smith, I. M.
			\|title=Numerical methods for engineers: a programming approach
			\|publisher=CRC Press
			\|location=Boca Raton
			\|year=1991
			\|isbn=0-8493-8610-1
			\|oclc=
			\|doi=
			\|page=218
			}}
			* {{Citation \| last1=Süli \| first1=Endre \| last2=Mayers \| first2=David \| title=An Introduction to Numerical Analysis \| publisher=[[Cambridge University Press]] \| isbn=0-521-00794-1 \| year=2003}}.

			{{Numerical integrators}}
			[[Category:Numerical differential equations]]
			[[Category:Runge–Kutta methods]]

en>Fred.Pendleton at 23:03, 8 March 2012

2012-03-08T23:03:31Z

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en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q4117184

en>Fred.Pendleton at 23:03, 8 March 2012