butools.dph.CanonicalFromDPH2¶

butools.dph.CanonicalFromDPH2()¶

Matlab:	[beta, B] = CanonicalFromDPH2(alpha, A, prec)
Mathematica:	{beta, B} = CanonicalFromDPH2[alpha, A, prec]
Python/Numpy:	beta, B = CanonicalFromDPH2(alpha, A, prec)

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

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

Examples

For Matlab:

>>> a = [0,1.0];
>>> A = [0.23, 0.22; 0.41, 0.48];
>>> [b, B] = CanonicalFromDPH2(a, A);
>>> disp(b);
      0.88663      0.11337
>>> disp(B);
      0.68031      0.31969
            0     0.029692
>>> ev = eig(A);
>>> disp(ev);
     0.029692
      0.68031
>>> flag = CheckDPHRepresentation(b, B);
>>> disp(flag);
     1
>>> Cm = SimilarityMatrix(A, B);
>>> err1 = norm(A*Cm-Cm*B);
>>> err2 = norm(a*Cm-b);
>>> a = [1.0,0];
>>> A = [0, 0.61; 0.56, 0.44];
>>> [b, B] = CanonicalFromDPH2(a, A);
>>> disp(b);
  -5.0834e-16            1
>>> disp(B);
         0.44         0.56
         0.61            0
>>> ev = eig(A);
>>> disp(ev);
      -0.4045
       0.8445
>>> flag = CheckDPHRepresentation(b, B);
>>> disp(flag);
     1
>>> Cm = SimilarityMatrix(A, B);
>>> err1 = norm(A*Cm-Cm*B);
>>> err2 = norm(a*Cm-b);
>>> disp(max(err1, err2));
   4.4871e-16

For Mathematica:

>>> a = {0,1.0};
>>> A = {{0.23, 0.22},{0.41, 0.48}};
>>> {b, B} = CanonicalFromDPH2[a, A];
>>> Print[b];
{0.8866338818412278, 0.1133661181587723}
>>> Print[B];
{{0.6803075467922624, 0.31969245320773765},
 {0, 0.029692453207737557}}
>>> ev = Eigenvalues[A];
>>> Print[ev];
{0.6803075467922624, 0.029692453207737557}
>>> flag = CheckDPHRepresentation[b, B];
>>> Print[flag];
True
>>> Cm = SimilarityMatrix[A, B];
>>> err1 = Norm[A.Cm-Cm.B];
>>> err2 = Norm[a.Cm-b];
>>> a = {1.0,0};
>>> A = {{0, 0.61},{0.56, 0.44}};
>>> {b, B} = CanonicalFromDPH2[a, A];
>>> Print[b];
{-5.083438757441196*^-16, 1.0000000000000002}
>>> Print[B];
{{0.4400000000000001, 0.5599999999999998},
 {0.6100000000000002, 0}}
>>> ev = Eigenvalues[A];
>>> Print[ev];
{0.8444997998398399, -0.4044997998398398}
>>> flag = CheckDPHRepresentation[b, B];
>>> Print[flag];
True
>>> Cm = SimilarityMatrix[A, B];
>>> err1 = Norm[A.Cm-Cm.B];
>>> err2 = Norm[a.Cm-b];
>>> Print[Max[err1, err2]];
1.059520934796808*^-15

For Python/Numpy:

>>> a = ml.matrix([[0,1.0]])
>>> A = ml.matrix([[0.23, 0.22],[0.41, 0.48]])
>>> b, B = CanonicalFromDPH2(a, A)
>>> print(b)
[[ 0.88663  0.11337]]
>>> print(B)
[[ 0.68031  0.31969]
 [ 0.       0.02969]]
>>> ev = la.eigvals(A)
>>> print(ev)
[ 0.02969+0.j  0.68031+0.j]
>>> flag = CheckDPHRepresentation(b, B)
>>> print(flag)
True
>>> Cm = SimilarityMatrix(A, B)
>>> err1 = la.norm(A*Cm-Cm*B)
>>> err2 = la.norm(a*Cm-b)
>>> a = ml.matrix([[1.0,0]])
>>> A = ml.matrix([[0, 0.61],[0.56, 0.44]])
>>> b, B = CanonicalFromDPH2(a, A)
>>> print(b)
[[ -5.08344e-16   1.00000e+00]]
>>> print(B)
[[ 0.44  0.56]
 [ 0.61  0.  ]]
>>> ev = la.eigvals(A)
>>> print(ev)
[-0.4045+0.j  0.8445+0.j]
>>> flag = CheckDPHRepresentation(b, B)
>>> print(flag)
True
>>> Cm = SimilarityMatrix(A, B)
>>> err1 = la.norm(A*Cm-Cm*B)
>>> err2 = la.norm(a*Cm-b)
>>> print(np.max(err1, err2))
3.90921887111e-16