butools.reptrans.TransformToMonocyclic¶

butools.reptrans.TransformToMonocyclic()¶

Matlab:	B = TransformToMonocyclic(A, maxSize, precision)
Mathematica:	B = TransformToMonocyclic[A, maxSize, precision]
Python/Numpy:	B = TransformToMonocyclic(A, maxSize, precision)

Transforms an arbitrary matrix to a Markovian monocyclic matrix (see [R47]).

Parameters:	A : matrix, shape (N,N) Matrix parameter of the initial representation maxSize : int, optional The maximal order of the resulting Markovian representation. The default value is 100 precision : double, optional Matrix entries smaller than the precision are considered to be zeros. The default value is 1e-14
Returns:	B : matrix, shape (M,M) Transient generator matrix of the Markovian monocyclic representation. Note that M>N if there are complex eigenvalues.

Notes

Raises an exception if no Markovian monocyclic generator has been found up to the given size.

References

[R47]

(1, 2) Mocanu, S., Commault, C.: “Sparse representations of phase-type distributions,” Stoch. Models 15, 759-778 (1999)

Examples

For Matlab:

>>> A = [-1, 0, 0; 0, -3, 2; 0, -2, -3];
>>> B = TransformToMonocyclic(A);
>>> disp(B);
           -1            1            0            0            0
            0           -3            3            0            0
            0            0           -3            3            0
            0            0            0           -3            3
            0      0.59259            0            0           -3
>>> Cm = SimilarityMatrix(A, B);
>>> err = norm(A*Cm-Cm*B);
>>> disp(err);
   1.4306e-14

For Mathematica:

>>> A = {{-1, 0, 0},{0, -3, 2},{0, -2, -3}};
>>> B = TransformToMonocyclic[A];
>>> Print[B];
{{-1, 1, 0, 0, 0},
 {0, -3, 3, 0, 0},
 {0, 0, -3, 3, 0},
 {0, 0, 0, -3, 3},
 {0, 16/27, 0, 0, -3}}
>>> Cm = SimilarityMatrix[A, B];
>>> err = Norm[A.Cm-Cm.B];
>>> Print[err];
0

For Python/Numpy:

>>> A = ml.matrix([[-1, 0, 0],[0, -3, 2],[0, -2, -3]])
>>> B = TransformToMonocyclic(A)
>>> print(B)
[[-1.       1.       0.       0.       0.     ]
 [ 0.      -3.       3.       0.       0.     ]
 [ 0.       0.      -3.       3.       0.     ]
 [ 0.       0.       0.      -3.       3.     ]
 [ 0.       0.59259  0.       0.      -3.     ]]
>>> Cm = SimilarityMatrix(A, B)
>>> err = la.norm(A*Cm-Cm*B)
>>> print(err)
1.29225863835e-14