butools.mam.FluidFundamentalMatrices¶

butools.mam.FluidFundamentalMatrices()¶

Matlab:	M = FluidFundamentalMatrices(Fpp, Fpm, Fmp, Fmm, matrices, precision, maxNumIt, method)
Mathematica:	M = FluidFundamentalMatrices[Fpp, Fpm, Fmp, Fmm, matrices, precision, maxNumIt, method]
Python/Numpy:	M = FluidFundamentalMatrices(Fpp, Fpm, Fmp, Fmm, matrices, precision, maxNumIt, method)

Returns the fundamental matrices corresponding to the given canonical Markov fluid model. Matrices Psi, K and U are returned depending on the “matrices” parameter. The canonical Markov fluid model is defined by the matrix blocks of the generator of the background Markov chain partitioned according to the sign of the associated fluid rates (i.e., there are “+” and “-” states).

Parameters:

Fpp : matrix, shape (Np,Np) The matrix of transition rates between states having positive fluid rates Fpm : matrix, shape (Np,Nm) The matrix of transition rates where the source state has a positive, the destination has a negative fluid rate associated. Fpm : matrix, shape (Nm,Np) The matrix of transition rates where the source state has a negative, the destination has a positive fluid rate associated. Fpp : matrix, shape (Nm,Nm) The matrix of transition rates between states having negative fluid rates matrices : string Specifies which matrices are required. ‘P’ means that only matrix Psi is needed. ‘UK’ means that matrices U and K are needed. Any combinations of ‘P’, ‘K’ and ‘U’ are allowed, in any order. precision : double, optional The matrices are computed iteratively up to this precision. The default value is 1e-14 maxNumIt : int, optional The maximal number of iterations. The default value is 50. method : {“CR”, “ADDA”, “SDA”}, optional The method used to solve the algebraic Riccati equation (CR: cyclic reduction, ADDA: alternating- directional doubling algorithm, SDA: structured doubling algorithm). The default is “CR”.

Returns:

M : list of matrices The list of calculated matrices in the order as requested in the ‘matrices’ parameter.

Notes

Thanks to Benny Van Houdt for the implementation of the Riccati solvers.

Examples

For Matlab:

>>> Fpp = [-5., 1.; 2., -3.];
>>> Fpm = [2., 1., 1.; 1., 0., 0.];
>>> Fmm = [-8., 4., 1.; 2., -12., 3.; 2., 0., -2.];
>>> Fmp = [3., 0.; 2., 5.; 0., 0.];
>>> [Psi, K, U] = FluidFundamentalMatrices(Fpp, Fpm, Fmp, Fmm, 'PKU');
Final Residual Error for Psi:    1.1657e-15
>>> disp(Psi);
      0.33722      0.16517      0.49761
       0.3318      0.12995      0.53825
>>> disp(K);
       -3.658       1.8258
       3.2553      -2.3502
>>> disp(U);
      -6.9883       4.4955       2.4928
       4.3334       -11.02       6.6865
            2            0           -2

For Mathematica:

>>> Fpp = {{-5., 1.},{2., -3.}};
>>> Fpm = {{2., 1., 1.},{1., 0., 0.}};
>>> Fmm = {{-8., 4., 1.},{2., -12., 3.},{2., 0., -2.}};
>>> Fmp = {{3., 0.},{2., 5.},{0., 0.}};
>>> {Psi, K, U} = FluidFundamentalMatrices[Fpp, Fpm, Fmp, Fmm, "PKU"];
"Final Residual Error for Psi: "1.1657341758564144*^-15
>>> Print[Psi];
{{0.33722394414970486, 0.16516588217551262, 0.4976101736747833},
 {0.3317962853815385, 0.12995245394948857, 0.5382512606689742}}
>>> Print[K];
{{-3.65799640319986, 1.8258294108775632},
 {3.255293764043593, -2.350237730252557}}
>>> Print[U];
{{-6.988328167550885, 4.4954976465265375, 2.4928305210243495},
 {4.333429315207102, -11.019905965901533, 6.686476650694438},
 {2., 0., -2.}}

For Python/Numpy:

>>> Fpp = ml.matrix([[-5., 1.],[2., -3.]])
>>> Fpm = ml.matrix([[2., 1., 1.],[1., 0., 0.]])
>>> Fmm = ml.matrix([[-8., 4., 1.],[2., -12., 3.],[2., 0., -2.]])
>>> Fmp = ml.matrix([[3., 0.],[2., 5.],[0., 0.]])
>>> Psi, K, U = FluidFundamentalMatrices(Fpp, Fpm, Fmp, Fmm, "PKU")
Final Residual Error for G:  1.7208456881689926e-15
>>> print(Psi)
[[ 0.33722  0.16517  0.49761]
 [ 0.3318   0.12995  0.53825]]
>>> print(K)
[[-3.658    1.82583]
 [ 3.25529 -2.35024]]
>>> print(U)
[[ -6.98833   4.4955    2.49283]
 [  4.33343 -11.01991   6.68648]
 [  2.        0.       -2.     ]]