Multibrot set - Revision history

en>Jimw338: /* Positive powers */ added table formatting - should really be a 3x3 table instead of 3 separate 3x1's

2014-12-05T05:06:34Z

Positive powers: added table formatting - should really be a 3x3 table instead of 3 separate 3x1's

← Older revision		Revision as of 07:06, 5 December 2014
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	~~The~~ '~~''Bogacki–Shampine method''' is~~ a ~~method for the~~ [~~[numerical ordinary differential equations\|numerical solution of ordinary differential equations]], that was proposed by Przemyslaw Bogacki and Lawrence F~~. ~~Shampine in 1989 {{harv\|Bogacki\|Shampine\|1989}}~~. ~~The Bogacki–Shampine method is a [[Runge–Kutta method~~]~~] of order three with four stages with the First Same As Last (FSAL) property, so that it uses approximately three function evaluations per step~~. ~~It has an embedded second-order method which can be used to implement~~ [[~~adaptive step size]~~]. ~~The Bogacki–Shampine method is implemented in the~~ <~~code~~>~~ode23~~<~~/code~~> ~~function in [[MATLAB]~~] ~~{{harv\|Shampine\|Reichelt\|1997}}.~~		Hi there! :) My name is Jerome, I'm a student studying Computing and Information Science from Bitti, Italy.<br>xunjie フォーマルな場面で着用するも、
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	~~Low-order methods are more suitable than higher-order methods like the~~ [[Dormand–Prince method]] of order five, if only a crude approximation to the solution is required. Bogacki and Shampine argue that their method outperforms other third-order methods with an embedded method of order two.		Also visit my blog [http://belpars.by/Files/Config/tp/new/best/toms.html トムズシューズキッズ]

	~~The Butcher tableau for the Bogacki–Shampine method is:~~
	~~{\| cellpadding=3px cellspacing=0px~~
	~~\|width="20px"\| \|\| style="border-right:1px solid;" \| 0~~
	\|-
	~~\|\|\| style="border-right:1px solid;" \| 1/2 \|\| 1/2~~
	\|-
	~~\|\|\| style="border-right:1px solid;" \| 3/4 \|\| 0 \|\| 3/4~~
	\|-
	~~\|\|\| style="border-right:1px solid; border-bottom:1px solid;" \| 1 \|\| style="border-bottom:1px solid;" \| 2/9 \|\| style="border-bottom~~:~~1px solid;" \| 1~~/~~3 \|\| style="border-bottom:1px solid;" \| 4~~/~~9 \|\| style="border-bottom:1px solid;" \|~~
	\|-
	~~\|\|\| style="border-right:1px solid;" \| \|\| 2/9 \|\| 1/3 \|\| 4/9 \|\| 0~~
	\|-
	~~\|\|\| style="border-right:1px solid;" \| \|\| 7/24 \|\| 1/4 \|\| 1/3 \|\| 1/8~~
	\|}

	~~Following the standard notation, the differential equation to be solved is <math>y'=f(t,y)</math>~~. ~~Furthermore, <math>y_n</math> denotes the numerical solution at time <math>t_n</math> and <math>h_n</math> is the step size, defined~~ by ~~<math>h_n = t_{n+1}-t_n<~~/~~math>. Then, one step of the Bogacki–Shampine method is given by:~~
	~~:<math> \begin{align}~~
	~~k_1 &= f(t_n, y_n) \\~~
	~~k_2 &= f(t_n + \tfrac12 h_n, y_n + \tfrac12 h k_1) \\~~
	~~k_3 &= f(t_n + \tfrac34 h_n, y_n + \tfrac34 h k_2) \\~~
	~~y_{n+1} &= y_n + \tfrac29 h k_1 + \tfrac13 h k_2 + \tfrac49 h k_3 \\~~
	~~k_4 &= f(t_n + h_n, y_{n+1}) \\~~
	~~z_{n+1} &= y_n + \tfrac7{24} h k_1 + \tfrac14 h k_2 + \tfrac13 h k_3 + \tfrac18 h k_4.~~
	~~\end{align} </math>~~

	~~Here, <math>z_{n+1}</math> is a second-order approximation to the exact solution. The method for calculating <math>y_{n+1}<~~/~~math> is due to {{harvtxt\|Ralston\|1965}}. On the other hand, <math>y_{n+1}<~~/~~math> is a third-order approximation, so the difference between <math>y_{n+1}<~~/math> and <math>z_{n+1}</math> can be used to [[adaptive stepsize\|adapt the step size]]. The FSAL—first same as last—property is that the stage value <math>k_4</math> in one step equals <math>k_1</math> in the next step; thus, only three function evaluations are needed per step.

	~~== References ==~~
	* {{Citation \| last1=Bogacki \| first1=Przemyslaw \| last2=Shampine \| first2=Lawrence F. \| title=A 3(2) pair of Runge–Kutta formulas \| doi=10.1016/~~0893-9659(89)90079-7 \| year=1989 \| journal=Applied Mathematics Letters \| issn=0893-9659 \| volume=2 \| issue=4 \| pages=321–325}}.~~
	* {{Citation \| last1=Ralston \| first1=Anthony \| title=A First Course in Numerical Analysis \| publisher=[[McGraw-Hill]] \| location=New York \| year=1965}}.
	* {{Citation \| last1=Shampine \| first1=Lawrence F. \| last2=Reichelt \| first2=Mark W. \| title=The Matlab ODE Suite \| doi=10.1137/~~S1064827594276424 \| year=1997 \| journal=[[SIAM Journal on Scientific Computing]] \| issn=1064-8275 \| volume=18 \| issue=1 \| pages=1–22}}~~.

	~~{{DEFAULTSORT:Bogacki-Shampine method}}~~
	~~[[Category:Runge–Kutta methods]~~]

en>Josep m batlle: /* Positive powers */

2014-01-19T22:40:06Z

Positive powers

← Older revision		Revision as of 00:40, 20 January 2014
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	~~Art Manager or Manager Jeromy from Chicoutimi~~, ~~enjoys~~ to ~~spend time going~~ to ~~movies, sex toys~~ and ~~poker~~. ~~Gets encouragement through travel and just spent~~ 2 ~~months~~ at ~~Fortress~~ of ~~Suomenlinna~~.<br><br>~~Visit my webpage~~ ... ~~[http:~~//~~wetter~~-~~foto~~.de/~~profil~~-~~362307~~-~~daniel3678pes0.html sex offenders~~]		The '''Bogacki–Shampine method''' is a method for the [[numerical ordinary differential equations\|numerical solution of ordinary differential equations]], that was proposed by Przemyslaw Bogacki and Lawrence F. Shampine in 1989 {{harv\|Bogacki\|Shampine\|1989}}. The Bogacki–Shampine method is a [[Runge–Kutta method]] of order three with four stages with the First Same As Last (FSAL) property, so that it uses approximately three function evaluations per step. It has an embedded second-order method which can be used to implement [[adaptive step size]]. The Bogacki–Shampine method is implemented in the <code>ode23</code> function in [[MATLAB]] {{harv\|Shampine\|Reichelt\|1997}}.

			Low-order methods are more suitable than higher-order methods like the [[Dormand–Prince method]] of order five, if only a crude approximation to the solution is required. Bogacki and Shampine argue that their method outperforms other third-order methods with an embedded method of order two.

			The Butcher tableau for the Bogacki–Shampine method is:
			{\| cellpadding=3px cellspacing=0px
			\|width="20px"\| \|\| style="border-right:1px solid;" \| 0
			\|-
			\|\|\| style="border-right:1px solid;" \| 1/2 \|\| 1/2
			\|-
			\|\|\| style="border-right:1px solid;" \| 3/4 \|\| 0 \|\| 3/4
			\|-
			\|\|\| style="border-right:1px solid; border-bottom:1px solid;" \| 1 \|\| style="border-bottom:1px solid;" \| 2/9 \|\| style="border-bottom:1px solid;" \| 1/3 \|\| style="border-bottom:1px solid;" \| 4/9 \|\| style="border-bottom:1px solid;" \|
			\|-
			\|\|\| style="border-right:1px solid;" \| \|\| 2/9 \|\| 1/3 \|\| 4/9 \|\| 0
			\|-
			\|\|\| style="border-right:1px solid;" \| \|\| 7/24 \|\| 1/4 \|\| 1/3 \|\| 1/8
			\|}

			Following the standard notation, the differential equation to be solved is <math>y'=f(t,y)</math>. Furthermore, <math>y_n</math> denotes the numerical solution at time <math>t_n</math> and <math>h_n</math> is the step size, defined by <math>h_n = t_{n+1}-t_n</math>. Then, one step of the Bogacki–Shampine method is given by:
			:<math> \begin{align}
			k_1 &= f(t_n, y_n) \\
			k_2 &= f(t_n + \tfrac12 h_n, y_n + \tfrac12 h k_1) \\
			k_3 &= f(t_n + \tfrac34 h_n, y_n + \tfrac34 h k_2) \\
			y_{n+1} &= y_n + \tfrac29 h k_1 + \tfrac13 h k_2 + \tfrac49 h k_3 \\
			k_4 &= f(t_n + h_n, y_{n+1}) \\
			z_{n+1} &= y_n + \tfrac7{24} h k_1 + \tfrac14 h k_2 + \tfrac13 h k_3 + \tfrac18 h k_4.
			\end{align} </math>

			Here, <math>z_{n+1}</math> is a second-order approximation to the exact solution. The method for calculating <math>y_{n+1}</math> is due to {{harvtxt\|Ralston\|1965}}. On the other hand, <math>y_{n+1}</math> is a third-order approximation, so the difference between <math>y_{n+1}</math> and <math>z_{n+1}</math> can be used to [[adaptive stepsize\|adapt the step size]]. The FSAL—first same as last—property is that the stage value <math>k_4</math> in one step equals <math>k_1</math> in the next step; thus, only three function evaluations are needed per step.

			== References ==
			* {{Citation \| last1=Bogacki \| first1=Przemyslaw \| last2=Shampine \| first2=Lawrence F. \| title=A 3(2) pair of Runge–Kutta formulas \| doi=10.1016/0893-9659(89)90079-7 \| year=1989 \| journal=Applied Mathematics Letters \| issn=0893-9659 \| volume=2 \| issue=4 \| pages=321–325}}.
			* {{Citation \| last1=Ralston \| first1=Anthony \| title=A First Course in Numerical Analysis \| publisher=[[McGraw-Hill]] \| location=New York \| year=1965}}.
			* {{Citation \| last1=Shampine \| first1=Lawrence F. \| last2=Reichelt \| first2=Mark W. \| title=The Matlab ODE Suite \| doi=10.1137/S1064827594276424 \| year=1997 \| journal=[[SIAM Journal on Scientific Computing]] \| issn=1064-8275 \| volume=18 \| issue=1 \| pages=1–22}}.

			{{DEFAULTSORT:Bogacki-Shampine method}}
			[[Category:Runge–Kutta methods]]

en>Lateg: Reverted 1 edit by 122.151.34.144 (talk): Test edit? (TW)

2012-04-13T23:58:29Z

Reverted 1 edit by 122.151.34.144 (talk): Test edit? (TW)

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