butools.ph.CanonicalFromPH2¶

butools.ph.CanonicalFromPH2()¶

Matlab:	[beta, B] = CanonicalFromPH2(alpha, A, prec)
Mathematica:	{beta, B} = CanonicalFromPH2[alpha, A, prec]
Python/Numpy:	beta, B = CanonicalFromPH2(alpha, A, prec)

Returns the canonical form of an order-2 phase-type distribution.

Parameters:	alpha : matrix, shape (1,2) Initial vector of the phase-type distribution A : matrix, shape (2,2) Transient generator of the phase-type distribution prec : double, optional Numerical precision, default value is 1e-14
Returns:	beta : matrix, shape (1,2) The initial probability vector of the canonical form B : matrix, shape (2,2) Transient generator matrix of the canonical form

Notes

This procedure calculates 3 moments of the input and calls ‘PH2From3Moments’.

Examples

For Matlab:

>>> a = [0.12,0.88];
>>> A = [-1.28, 0; 3.94, -3.94];
>>> [b, B] = CanonicalFromPH2(a, A);
>>> disp(b);
      0.96102     0.038985
>>> disp(B);
        -1.28         1.28
            0        -3.94
>>> Cm = SimilarityMatrix(A, B);
>>> err1 = norm(A*Cm-Cm*B);
>>> err2 = norm(a*Cm-b);
>>> disp(max(err1, err2));
    6.669e-15

For Mathematica:

>>> a = {0.12,0.88};
>>> A = {{-1.28, 0},{3.94, -3.94}};
>>> {b, B} = CanonicalFromPH2[a, A];
>>> Print[b];
{0.9610152284263966, 0.03898477157360336}
>>> Print[B];
{{-1.2800000000000014, 1.2800000000000014},
 {0, -3.9399999999999946}}
>>> Cm = SimilarityMatrix[A, B];
>>> err1 = Norm[A.Cm-Cm.B];
>>> err2 = Norm[a.Cm-b];
>>> Print[Max[err1, err2]];
1.881192080999035*^-15

For Python/Numpy:

>>> a = ml.matrix([[0.12,0.88]])
>>> A = ml.matrix([[-1.28, 0],[3.94, -3.94]])
>>> b, B = CanonicalFromPH2(a, A)
>>> print(b)
[[ 0.96102  0.03898]]
>>> print(B)
[[-1.28  1.28]
 [ 0.   -3.94]]
>>> Cm = SimilarityMatrix(A, B)
>>> err1 = la.norm(A*Cm-Cm*B)
>>> err2 = la.norm(a*Cm-b)
>>> print(np.max(err1, err2))
4.86095148111e-15