butools.reptrans.TransformToAcyclic¶

butools.reptrans.TransformToAcyclic()¶

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

Transforms an arbitrary matrix to a Markovian bi-diagonal matrix.

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 (N,N) Transient (bi-diagonal) generator matrix of the Markovian acyclic representation.

Notes

Calls the ‘TransformToMonocyclic’ procedure if all the eigenvalues are real, otherwise it raises an error if no Markovian acyclic generator has been found up to the given size.

Raises an error if no Markovian acyclic generator has been found up to the given size.

References

[R46]

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

Examples

For Matlab:

>>> A = [-0.8, 0.8, 0.; 0.1, -0.3, 0.1; 0.2, 0., -0.5];
>>> B = TransformToAcyclic(A);
>>> disp(B);
      -0.1203       0.1203            0
            0      -0.6158       0.6158
            0            0      -0.8639
>>> Cm = SimilarityMatrix(A, B);
>>> err = norm(A*Cm-Cm*B);
>>> disp(err);
   7.0942e-09

For Mathematica:

>>> A = {{-0.8, 0.8, 0.},{0.1, -0.3, 0.1},{0.2, 0., -0.5}};
>>> B = TransformToAcyclic[A];
>>> Print[B];
{{-0.12030366824989391, 0.12030366824989391, 0},
 {0, -0.6157989613063427, 0.6157989613063427},
 {0, 0, -0.8638973704437634}}
>>> Cm = SimilarityMatrix[A, B];
>>> err = Norm[A.Cm-Cm.B];
>>> Print[err];
3.2936898311300537*^-16

For Python/Numpy:

>>> A = ml.matrix([[-0.8, 0.8, 0.],[0.1, -0.3, 0.1],[0.2, 0., -0.5]])
>>> B = TransformToAcyclic(A)
>>> print(B)
[[-0.1203  0.1203  0.    ]
 [ 0.     -0.6158  0.6158]
 [ 0.      0.     -0.8639]]
>>> Cm = SimilarityMatrix(A, B)
>>> err = la.norm(A*Cm-Cm*B)
>>> print(err)
8.75449540243e-09