Random vector on a sphere - Revision history

Carl McBride: Clarified a reference

2010-10-21T12:55:07Z

Clarified a reference

← Older revision		Revision as of 14:55, 21 October 2010
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	:<math>\left(2V_1 \sqrt{1-S},~ 2V_2 \sqrt{1-S},~ 1-2S\right)</math>		:<math>\left(2V_1 \sqrt{1-S},~ 2V_2 \sqrt{1-S},~ 1-2S\right)</math>
	==Fortran 90 implementation==		==Fortran 90 implementation==
	This Fortran 90 implementation is adapted from Refs. <ref>[http://molsim.chem.uva.nl/frenkel_smit Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications", Second Edition (2002)] Algorithm 42 (p. 578)</ref> and <ref>[http://www.oup.com/uk/catalogue/?ci=9780198556459 M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", Oxford University Press (1989)] p. 349 </ref>. The function '''ran()''' calls a [[Random_numbers \| randon number generator]]:		This Fortran 90 implementation is adapted from Refs. <ref>[http://molsim.chem.uva.nl/frenkel_smit Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications", Second Edition (2002)] Algorithm 42 (p. 578)</ref> and <ref>[http://www.oup.com/uk/catalogue/?ci=9780198556459 M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", Oxford University Press (1989)] Appendix G.5 (p. 349) </ref>. The function '''ran()''' calls a [[Random_numbers \| randon number generator]]:
	<small><pre>		<small><pre>
	! The following is taken from Allen & Tildesley, p. 349		! The following is taken from Allen & Tildesley, p. 349

Carl McBride: Added an external link

2010-10-21T12:47:36Z

Added an external link

← Older revision		Revision as of 14:47, 21 October 2010
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	==References==		==References==
	<references/>		<references/>
			==External links==
			*[http://mathworld.wolfram.com/SpherePointPicking.html Sphere Point Picking] from Wolfram MathWorld
	{{Source}}		{{Source}}
	[[category: random numbers]]		[[category: random numbers]]
	[[category: computer simulation techniques]]		[[category: computer simulation techniques]]

Carl McBride: Slight tidy

2010-10-21T11:34:53Z

Slight tidy

← Older revision		Revision as of 13:34, 21 October 2010
Line 8:		Line 8:
	:<math>\left(2V_1 \sqrt{1-S},~ 2V_2 \sqrt{1-S},~ 1-2S\right)</math>		:<math>\left(2V_1 \sqrt{1-S},~ 2V_2 \sqrt{1-S},~ 1-2S\right)</math>
	==Fortran 90 implementation==		==Fortran 90 implementation==
	This Fortran 90 implementation is adapted from ~~Ref~~. <ref>Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications" p. ~~410 Academic~~ Press (~~1996~~)</ref> The function '''ran()''' calls a [[Random_numbers \| randon number generator]]:		This Fortran 90 implementation is adapted from Refs. <ref>[http://molsim.chem.uva.nl/frenkel_smit Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications", Second Edition (2002)] Algorithm 42 (p. 578)</ref> and <ref>[http://www.oup.com/uk/catalogue/?ci=9780198556459 M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", Oxford University Press (1989)] p. 349 </ref>. The function '''ran()''' calls a [[Random_numbers \| randon number generator]]:
	<small><pre>		<small><pre>
	! The following is taken from Allen & Tildesley, p. 349		! The following is taken from Allen & Tildesley, p. 349
Line 41:		Line 41:
	==References==		==References==
	<references/>		<references/>
	~~'''Related reading'''~~
	* M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", p. 349 Clarendon Press (1989)
	{{Source}}		{{Source}}
	[[category: random numbers]]		[[category: random numbers]]
	[[category: computer simulation techniques]]		[[category: computer simulation techniques]]

Carl McBride: Slight tidy

2010-10-21T10:54:06Z

Slight tidy

← Older revision		Revision as of 12:54, 21 October 2010
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	~~{{Source}}~~
	The ability to generate a randomly orientated vector is very useful in [[Monte Carlo]] simulations of anisotropic [[models]]		The ability to generate a randomly orientated vector is very useful in [[Monte Carlo]] simulations of anisotropic [[models]]
	or molecular systems.		or molecular systems.
	==Marsaglia algorithm==		==Marsaglia algorithm==
	This is the algorithm proposed by George Marsaglia (~~Ref. 1~~):		This is the algorithm proposed by George Marsaglia <ref name="Marsaglia">[http://dx.doi.org/10.1214/aoms/1177692644 George Marsaglia "Choosing a Point from the Surface of a Sphere", The Annals of Mathematical Statistics '''43''' pp. 645-646 (1972)]</ref>
	*Independently generate V<sub>1</sub> and V<sub>2</sub>, taken from a uniform distribution on (-1,1) such that		*Independently generate V<sub>1</sub> and V<sub>2</sub>, taken from a uniform distribution on (-1,1) such that
	:<math>S=(V_1^2+V_2^2) < 1</math>		:<math>S=(V_1^2+V_2^2) < 1</math>
	*The random vector is then (~~Ref. 1~~ Eq. 4):		*The random vector is then (Eq. 4 in <ref name="Marsaglia"> </ref> ):
	:<math>\left(2V_1 \sqrt{1-S},~ 2V_2 \sqrt{1-S},~ 1-2S\right)</math>		:<math>\left(2V_1 \sqrt{1-S},~ 2V_2 \sqrt{1-S},~ 1-2S\right)</math>
	==Fortran 90 implementation==		==Fortran 90 implementation==
	This Fortran 90 implementation is adapted from Ref. 2. The function '''ran()''' calls a [[Random_numbers \| randon number generator]]:		This Fortran 90 implementation is adapted from Ref. <ref>Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications" p. 410 Academic Press (1996)</ref> The function '''ran()''' calls a [[Random_numbers \| randon number generator]]:
	<small><pre>		<small><pre>
	! The following is taken from Allen & Tildesley, p. 349		! The following is taken from Allen & Tildesley, p. 349
Line 41:		Line 40:

	==References==		==References==
	~~#[http:~~/~~/links.jstor.org/sici?sici=0003-4851%28197204%2943%3A2%3C645%3ACAPFTS%3E2.0.CO%3B2-%23 George Marsaglia "Choosing a Point from the Surface of a Sphere", The Annals of Mathematical Statistics~~ '''43''' ~~pp. 645-646 (1972)]~~		<references/>
	~~#Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications" p. 410 Academic Press (1996)~~		'''Related reading'''
	# M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", p. 349 Clarendon Press (1989)		* M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", p. 349 Clarendon Press (1989)
			{{Source}}
	[[category: random numbers]]		[[category: random numbers]]
	[[category: computer simulation techniques]]		[[category: computer simulation techniques]]

Carl McBride at 13:15, 30 October 2007

2007-10-30T13:15:01Z

← Older revision		Revision as of 15:15, 30 October 2007
Line 1:		Line 1:
			{{Source}}
	The ability to generate a randomly orientated vector is very useful in [[Monte Carlo]] simulations of anisotropic [[models]]		The ability to generate a randomly orientated vector is very useful in [[Monte Carlo]] simulations of anisotropic [[models]]
	or molecular systems.		or molecular systems.

Carl McBride: Typo.

2007-10-30T12:48:25Z

Typo.

← Older revision		Revision as of 14:48, 30 October 2007
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	==Marsaglia algorithm==		==Marsaglia algorithm==
	This is the algorithm proposed by George Marsaglia (Ref. 1):		This is the algorithm proposed by George Marsaglia (Ref. 1):
	*Independently generate V<sub>1</sub> and V<sub>2</sub>, taken from a ~~unifor~~ distribution on (-1,1) such that		*Independently generate V<sub>1</sub> and V<sub>2</sub>, taken from a uniform distribution on (-1,1) such that
	:<math>S=(V_1^2+V_2^2) < 1</math>		:<math>S=(V_1^2+V_2^2) < 1</math>
	*The random vector is then (Ref. 1 Eq. 4):		*The random vector is then (Ref. 1 Eq. 4):

Carl McBride: Added description of Marsaglia algorithm

2007-10-30T12:43:12Z

Added description of Marsaglia algorithm

← Older revision		Revision as of 14:43, 30 October 2007
Line 1:		Line 1:
	Fortran 90 implementation from Ref. 2. ran() ~~is some~~ [[Random_numbers \| randon number generator]]:		The ability to generate a randomly orientated vector is very useful in [[Monte Carlo]] simulations of anisotropic [[models]]
			or molecular systems.
			==Marsaglia algorithm==
			This is the algorithm proposed by George Marsaglia (Ref. 1):
			*Independently generate V<sub>1</sub> and V<sub>2</sub>, taken from a unifor distribution on (-1,1) such that
			:<math>S=(V_1^2+V_2^2) < 1</math>
			*The random vector is then (Ref. 1 Eq. 4):
			:<math>\left(2V_1 \sqrt{1-S},~ 2V_2 \sqrt{1-S},~ 1-2S\right)</math>
			==Fortran 90 implementation==
			This Fortran 90 implementation is adapted from Ref. 2. The function '''ran()''' calls a [[Random_numbers \| randon number generator]]:
	<small><pre>		<small><pre>
	! The following is taken from Allen & Tildesley, p. 349		! The following is taken from Allen & Tildesley, p. 349
Line 32:		Line 41:
	==References==		==References==
	#[http://links.jstor.org/sici?sici=0003-4851%28197204%2943%3A2%3C645%3ACAPFTS%3E2.0.CO%3B2-%23 George Marsaglia "Choosing a Point from the Surface of a Sphere", The Annals of Mathematical Statistics '''43''' pp. 645-646 (1972)]		#[http://links.jstor.org/sici?sici=0003-4851%28197204%2943%3A2%3C645%3ACAPFTS%3E2.0.CO%3B2-%23 George Marsaglia "Choosing a Point from the Surface of a Sphere", The Annals of Mathematical Statistics '''43''' pp. 645-646 (1972)]
	#Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications" p. 410 Academic Press (1996)~~, algorithm based on:~~		#Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications" p. 410 Academic Press (1996)
	# M.P. Allen ~~(Author)~~ and D.J. Tildesley "Computer Simulation of Liquids" p. 349 Clarendon Press (1989)		# M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", p. 349 Clarendon Press (1989)

	[[category: random numbers]]		[[category: random numbers]]
	[[category: computer simulation techniques]]		[[category: computer simulation techniques]]

Dduque at 09:10, 30 October 2007

2007-10-30T09:10:26Z

@@ Line 1: / Line 1: @@
 ==References==
 #[http://links.jstor.org/sici?sici=0003-4851%28197204%2943%3A2%3C645%3ACAPFTS%3E2.0.CO%3B2-%23 George Marsaglia "Choosing a Point from the Surface of a Sphere", The Annals of Mathematical Statistics '''43''' pp. 645-646 (1972)]
 [[category: random numbers]]
 [[category: computer simulation techniques]]

Carl McBride: New page: ==References== #[http://links.jstor.org/sici?sici=0003-4851%28197204%2943%3A2%3C645%3ACAPFTS%3E2.0.CO%3B2-%23 George Marsaglia "Choosing a Point from the Surface of a Sphere", The Annals o...

2007-10-30T08:05:39Z

New page: ==References== #[http://links.jstor.org/sici?sici=0003-4851%28197204%2943%3A2%3C645%3ACAPFTS%3E2.0.CO%3B2-%23 George Marsaglia "Choosing a Point from the Surface of a Sphere", The Annals o...

New page

==References==
#[http://links.jstor.org/sici?sici=0003-4851%28197204%2943%3A2%3C645%3ACAPFTS%3E2.0.CO%3B2-%23 George Marsaglia "Choosing a Point from the Surface of a Sphere", The Annals of Mathematical Statistics '''43''' pp. 645-646 (1972)]
[[category: random numbers]]
[[category: computer simulation techniques]]