Re: General 3-state stochastic matrix (again)
- To: mathgroup at smc.vnet.net
- Subject: [mg58266] Re: General 3-state stochastic matrix (again)
- From: Virgil Stokes <virgil.stokes at neuro.ki.se>
- Date: Sat, 25 Jun 2005 01:56:24 -0400 (EDT)
- References: <200506230929.FAA16127@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"> <html> <head> <meta content="text/html;charset=ISO-8859-1" http-equiv="Content-Type"> <title></title> </head> <body bgcolor="#ffffff" text="#000000"> Steven M. Christensen wrote: <blockquote cite="mid200506230929.FAA16127 at smc.vnet.net" type="cite"> <pre wrap=""> Your email has come to me with a lot of non-ascii symbols. Please resend this in pure ascii so we can understand what it is. Moderator I have tried to find the limit (as n, the power of the matrix, goes to infinity) for the general 3-state stochastic matrix using the following code: Clear[\116\0612, \116\0613, \116\0621, \116\0623, \116\0631, \116\0632] T = {{1 - \116\0612 - \116\0613, \116\0612, \116\0613}, {\116\0621, 1 - \116\0621 - \116\0623, \116\0623}, {\116\0631, \116\0632, 1 - \116\0631 - \116\0632}}; MatrixForm[T] xx = MatrixPower[T, n]; MatrixForm[%] TimeUsed[] zz = Limit[xx, n -> \142\010\036, Assumptions -> 0 < \116\0612 < 1 && 0 < \116\0613 < 1 && 0 < \116\0612 + \116\0613 < 1 && 0 < \116\0621 < 1 && 0 < \116\0623 < 1 && 0 < \116\0621 + \116\0623 < 1 && 0 < \116\0631 < 1 && 0 < \116\0632 < 1 && 0 < \116\0631 + \116\0632 < 1] // FullSimplify; TimeUsed[] MatrixForm[zz] However, it does not find a symbolic solution. I would appreciate it greatly if someone else could look at this and see if they are able to get a symbolic solution. Warning! this can take considerable CPU time. Note, for a general 2-state stochastic matrix, the above approach works fine. --Thanks, V. Stokes </pre> </blockquote> You might try this (an extension to your notebook):<br> <br> <img src="cid:part1.09070602.08010506 at neuro.ki.se" alt=""><br> <br> --V. 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