butools.dmap.CanonicalFromDMAP2¶

butools.dmap.CanonicalFromDMAP2()¶

Matlab:	[G0, G1] = CanonicalFromDMAP2(D0, D1, prec)
Mathematica:	{G0, G1} = CanonicalFromDMAP2[D0, D1, prec]
Python/Numpy:	G0, G1 = CanonicalFromDMAP2(D0, D1, prec)

Returns the canonical form of an order-2 discrete Markovian arrival process.

Parameters:	D0 : matrix, shape (2,2) The D0 matrix of the DMAP(2) D1 : matrix, shape (2,2) The D1 matrix of the DMAP(2) prec : double, optional Numerical precision to check the input, default value is 1e-14
Returns:	G0 : matrix, shape (1,2) The D0 matrix of the canonical DMAP(2) G1 : matrix, shape (2,2) The D1 matrix of the canonical DMAP(2)

Examples

For Matlab:

>>> D0 = [0.46, 0.28; 0.35, 0.23];
>>> D1 = [0.08, 0.18; 0.14, 0.28];
>>> [H0, H1] = CanonicalFromDMAP2(D0, D1);
>>> disp(H0);
       0.6785      0.31704
            0     0.011496
>>> disp(H1);
            0     0.004455
       0.6285         0.36
>>> Cm = SimilarityMatrix(H0, D0);
>>> err1 = norm(H0*Cm-Cm*D0);
>>> err2 = norm(H1*Cm-Cm*D1);
>>> disp(max(err1, err2));
   8.9731e-14
>>> D0 = [0.26, 0.28; 0.35, 0.23];
>>> D1 = [0.28, 0.18; 0.14, 0.28];
>>> [H0, H1] = CanonicalFromDMAP2(D0, D1);
>>> disp(H0);
         0.49      0.38875
     0.098265            0
>>> disp(H1);
      0.12125            0
      0.46299      0.43875
>>> Cm = SimilarityMatrix(H0, D0);
>>> err1 = norm(H0*Cm-Cm*D0);
>>> err2 = norm(H1*Cm-Cm*D1);
>>> disp(max(err1, err2));
   1.2787e-14
>>> D0 = [0.14, 0.34; 0.35, 0.23];
>>> D1 = [0.22, 0.3; 0.28, 0.14];
>>> [H0, H1] = CanonicalFromDMAP2(D0, D1);
>>> disp(H0);
         0.37      0.51734
      0.16778            0
>>> disp(H1);
            0      0.11266
      0.47222         0.36
>>> Cm = SimilarityMatrix(H0, D0);
>>> err1 = norm(H0*Cm-Cm*D0);
>>> err2 = norm(H1*Cm-Cm*D1);
>>> disp(max(err1, err2));
   1.2567e-15

For Mathematica:

>>> D0 = {{0.46, 0.28},{0.35, 0.23}};
>>> D1 = {{0.08, 0.18},{0.14, 0.28}};
>>> {H0, H1} = CanonicalFromDMAP2[D0, D1];
>>> Print[H0];
{{0.6785041229130464, 0.31704085460114795},
 {0, 0.011495877086969752}}
>>> Print[H1];
{{0, 0.004455022485805612},
 {0.6285041229130266, 0.3600000000000036}}
>>> Cm = SimilarityMatrix[H0, D0];
>>> err1 = Norm[H0.Cm-Cm.D0];
>>> err2 = Norm[H1.Cm-Cm.D1];
>>> Print[Max[err1, err2]];
8.953600966260704*^-14
>>> D0 = {{0.26, 0.28},{0.35, 0.23}};
>>> D1 = {{0.28, 0.18},{0.14, 0.28}};
>>> {H0, H1} = CanonicalFromDMAP2[D0, D1];
>>> Print[H0];
{{0.4900000000000001, 0.38874507866387564},
 {0.09826490956822952, 0}}
>>> Print[H1];
{{0.12125492133612426, 0},
 {0.462990011767894, 0.4387450786638765}}
>>> Cm = SimilarityMatrix[H0, D0];
>>> err1 = Norm[H0.Cm-Cm.D0];
>>> err2 = Norm[H1.Cm-Cm.D1];
>>> Print[Max[err1, err2]];
2.6634419263318992*^-14
>>> D0 = {{0.14, 0.34},{0.35, 0.23}};
>>> D1 = {{0.22, 0.3},{0.28, 0.14}};
>>> {H0, H1} = CanonicalFromDMAP2[D0, D1];
>>> Print[H0];
{{0.37, 0.5173403532281082},
 {0.16778122846668353, 0}}
>>> Print[H1];
{{0, 0.11265964677189179},
 {0.4722187715333161, 0.3600000000000004}}
>>> Cm = SimilarityMatrix[H0, D0];
>>> err1 = Norm[H0.Cm-Cm.D0];
>>> err2 = Norm[H1.Cm-Cm.D1];
>>> Print[Max[err1, err2]];
5.0785547211252284*^-15

For Python/Numpy:

>>> D0 = ml.matrix([[0.46, 0.28],[0.35, 0.23]])
>>> D1 = ml.matrix([[0.08, 0.18],[0.14, 0.28]])
>>> H0, H1 = CanonicalFromDMAP2(D0, D1)
>>> print(H0)
[[ 0.6785   0.31704]
 [ 0.       0.0115 ]]
>>> print(H1)
[[ 0.       0.00446]
 [ 0.6285   0.36   ]]
>>> Cm = SimilarityMatrix(H0, D0)
>>> err1 = la.norm(H0*Cm-Cm*D0)
>>> err2 = la.norm(H1*Cm-Cm*D1)
>>> print(np.max(err1, err2))
8.9717085547e-14
>>> D0 = ml.matrix([[0.26, 0.28],[0.35, 0.23]])
>>> D1 = ml.matrix([[0.28, 0.18],[0.14, 0.28]])
>>> H0, H1 = CanonicalFromDMAP2(D0, D1)
>>> print(H0)
[[ 0.49     0.38875]
 [ 0.09826  0.     ]]
>>> print(H1)
[[ 0.12125  0.     ]
 [ 0.46299  0.43875]]
>>> Cm = SimilarityMatrix(H0, D0)
>>> err1 = la.norm(H0*Cm-Cm*D0)
>>> err2 = la.norm(H1*Cm-Cm*D1)
>>> print(np.max(err1, err2))
1.74838096756e-15
>>> D0 = ml.matrix([[0.14, 0.34],[0.35, 0.23]])
>>> D1 = ml.matrix([[0.22, 0.3],[0.28, 0.14]])
>>> H0, H1 = CanonicalFromDMAP2(D0, D1)
>>> print(H0)
[[ 0.37     0.51734]
 [ 0.16778  0.     ]]
>>> print(H1)
[[ 0.       0.11266]
 [ 0.47222  0.36   ]]
>>> Cm = SimilarityMatrix(H0, D0)
>>> err1 = la.norm(H0*Cm-Cm*D0)
>>> err2 = la.norm(H1*Cm-Cm*D1)
>>> print(np.max(err1, err2))
1.92296268638e-16