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MathGroup Archive 2005

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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

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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 -&gt; \142\010\036, Assumptions -&gt; 0 &lt; \116\0612 &lt; 1 &amp;&amp;
0 &lt; \116\0613 &lt;  1 &amp;&amp; 0 &lt; \116\0612 + \116\0613 &lt; 1 &amp;&amp; 0 &lt; \116\0621 &lt; 1 &amp;&amp;
       0 &lt; \116\0623 &lt; 1 &amp;&amp; 0 &lt; \116\0621 + \116\0623 &lt;  1 &amp;&amp; 0 &lt; \116\0631 &lt; 1
&amp;&amp; 0 &lt; \116\0632 &lt; 1 &amp;&amp; 0 &lt;
\116\0631 + \116\0632 &lt; 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. Stokes<br>
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