octave> markov2(0.1,0.15,0.1) # aperiodic
ans =

  0.90000  0.15000
  0.10000  0.85000



octave> markov2(0.9999,0.9,0.9999)
ans =

   1.0000e-04   9.0000e-01
   9.9990e-01   1.0000e-01



octave> P=[0.0001,0.9999;0.9999,0.0001];
octave> [X,L]=eig(P);
octave> X # eigenvectors
X =

   0.66900  -0.70711
   0.74326   0.70711
octave> L # eigenvalues
L =

   1.00000   0.00000
   0.00000  -0.89990
octave> X(:,1)/sum(X(:,1)) # make it a distribution
ans =

  0.47371
  0.52629
octave>
octave> markov2(1.0,1.0,0.0) # periodic
ans =

  0  1
  1  0



octave> markov3(0.55,0.4,0.1,0.1,0.25,0.3,0.8,0.1);
ans =

  0.55000  0.40000  0.10000
  0.10000  0.25000  0.30000
  0.35000  0.35000  0.60000


octave> markov3(0.55,0.4,0.1,0.1,0.25,0.3,0.1,0.4); # same P, different x0
ans =

  0.55000  0.40000  0.10000
  0.10000  0.25000  0.30000
  0.35000  0.35000  0.60000


octave> markov3(0.05,0.05,0.9,0.9,0.05,0.05,0.33,0.1);
ans =

  0.05000  0.05000  0.90000
  0.90000  0.05000  0.05000
  0.05000  0.90000  0.05000



octave> markov3(0.0,0.0,1.0,1.0,0.0,0.0,1.0,0.0);
ans =

  0  0  1
  1  0  0
  0  1  0