Re: SingularValueDecomposition

*To*: mathgroup at smc.vnet.net*Subject*: [mg119728] Re: SingularValueDecomposition*From*: Dana DeLouis <dana.del at gmail.com>*Date*: Sun, 19 Jun 2011 07:24:48 -0400 (EDT)

... In[19]:= MatrixPower[N[m],100][[2]]//Chop ... Out[19]= {0,0,0,0,0,0,0,0,0,0,0,0,0.809663,0.190337} Hi. Sorry I can't help, but as a side note, this converged in 7 loops vs 100. FixedPoint[Chop[#.#]&,m][[2]] {0,0,0,0,0,0,0,0,0,0,0,0,0.809663,0.190337} = = = = = = = = = = Dana DeLouis On Jun 18, 5:45 pm, John Snyder <jsny... at wi.rr.com> wrote: > I read Jon McLoone's recent post on the WolframBlog concerning the > solution of the drunken sailor's walk problem by using a Markov chain > transition probability matrix. He mentions that it may also be possible > to solve the problem using the SingularValueDecomposition function, but > he does not illustrate this. I am trying to figure out how this could be > done. > > Here is a simple "toy" example. Assume that I have the following Markov > chain transition probability matrix m where each row sums to 1: > > m={{1/4,1/4,0,1/4,0,0,0,0,0,0,0,0,1/4,0},{1/4,1/4,1/4,0,1/4,0,0,0,0,0,0,0,0,0},{0,1/4,1/4,0,0,1/4,0,0,0,0,0,0,1/4,0},{0,0,0,1/4,1/4,0,1/4,0,0,0,0,0,1/4,0},{0,0,0,1/4,1/4,1/4,0,1/4,0,0,0,0,0,0},{0,0,0,0,1/4,1/4,0,0,1/4,0,0,0,1/4,0},{0,0,0,0,0,0,1/4,1/4,0,1/4,0,0,1/4,0},{0,0,0,0,0,0,1/4,1/4,1/4,0,1/4,0,0,0},{0,0,0,0,0,0,0,1/4,1/4,0,0,1/4,1/4,0},{0,0,0,0,0,0,0,0,0,1/4,1/4,0,1/4,1/4},{0,0,0,0,0,0,0,0,0,1/4,1/4,1/4,0,1/4},{0,0,0,0,0,0,0,0,0,0,1/4,1/4,1/4,1/4},{0,0,0,0,0,0,0,0,0,0,0,0,1,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,1}}; > > Assuming that I start in the position 2 (column 2 out of 3, in the first > of 4 rows) I want to find the so-called "fixed point", the ultimate > state density function, as the number of steps goes to infinity. I know > that I can do this numerically using MatrixPower as follows (here is 100 > steps which appears to be more than enough in this case): > > In[19]:= MatrixPower[N[m],100][[2]]//Chop > Out[19]= {0,0,0,0,0,0,0,0,0,0,0,0,0.809663,0.190337} > > I believe that it is also possible to get this same result by using the > SingularValueDecomposition function, but I cannot figure out how to get > this to work. Can someone please show me how to use > SingularValueDecomposition to get the same answer to this question? I > know there are other ways to solve this, but I am really interested in > using SingularValueDecomposition in this case. Thanks.