butools.mc.DTMCSolve¶

butools.mc.DTMCSolve()¶

Matlab:	pi = DTMCSolve(Q)
Mathematica:	pi = DTMCSolve[Q]
Python/Numpy:	pi = DTMCSolve(Q)

Computes the stationary solution of a discrete time Markov chain.

Parameters:	P : matrix, shape (M,M) The transition probability matrix of the Markov chain
Returns:	pi : row vector, shape (1,M) The vector that satisfies \(\pi\, P = \pi, \sum_i \pi_i=1\)

Notes

The procedure raises an exception if butools.checkInput is set to true and CheckProbMatrix (P) fails.

Examples

For Matlab:

>>> Q = [0.1, 0.5, 0.4; 0.9, 0.1, 0; 0.3, 0.3, 0.4];
>>> ret = DTMCSolve(Q);
>>> disp(ret);
      0.40909      0.31818      0.27273
>>> disp(ret*Q-ret);
  -5.5511e-17   5.5511e-17   5.5511e-17

For Mathematica:

>>> Q = {{0.1, 0.5, 0.4},{0.9, 0.1, 0},{0.3, 0.3, 0.4}};
>>> ret = DTMCSolve[Q];
>>> Print[ret];
{0.4090909090909091, 0.3181818181818182, 0.2727272727272727}
>>> Print[ret.Q-ret];
{-5.551115123125783*^-17, 0., 5.551115123125783*^-17}

For Python/Numpy:

>>> Q = ml.matrix([[0.1, 0.5, 0.4],[0.9, 0.1, 0],[0.3, 0.3, 0.4]])
>>> ret = DTMCSolve(Q)
>>> print(ret)
[[ 0.40909  0.31818  0.27273]]
>>> print(ret*Q-ret)
[[ -5.55112e-17   0.00000e+00   5.55112e-17]]