butools.mam.GM1FundamentalMatrix¶

butools.mam.GM1FundamentalMatrix()¶

Matlab:	R = GM1FundamentalMatrix(A, precision, maxNumIt, method)
Mathematica:	R = GM1FundamentalMatrix[A, precision, maxNumIt, method]
Python/Numpy:	R = GM1FundamentalMatrix(A, precision, maxNumIt, method)

Returns matrix R corresponding to the G/M/1 type Markov chain given by matrices A.

Matrix R is the minimal non-negative solution of the following matrix equation:

\[R = A_0 + R A_1 + R^2 A_2 + R^3 A_3 + \dots.\]

The implementation is based on [R15], please cite it if you use this method.

Parameters:

A : length(M) list of matrices of shape (N,N) Matrix blocks of the G/M/1 type generator in the regular part, from 0 to M-1. precision : double, optional Matrix R is 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”, “RR”, “NI”, “FI”, “IS”}, optional The method used to solve the matrix-quadratic equation (CR: cyclic reduction, RR: Ramaswami reduction, NI: Newton iteration, FI: functional iteration, IS: invariant subspace method). The default is “CR”.

Returns:

R : matrix, shape (N,N) The R matrix of the G/M/1 type Markov chain.

References

[R15]

(1, 2) Bini, D. A., Meini, B., Steffé, S., Van Houdt, B. (2006, October). Structured Markov chains solver: software tools. In Proceeding from the 2006 workshop on Tools for solving structured Markov chains (p. 14). ACM.

Examples

For Matlab:

>>> A0 = [0.1, 0.; 0., 0.1];
>>> A1 = [0., 0.2; 0., 0.2];
>>> A2 = [0., 0.1; 0., 0.];
>>> A3 = [0.3, 0.2; 0.3, 0.2];
>>> A4 = [0., 0.1; 0.2, 0.];
>>> A = {A0, A1, A2, A3, A4};
>>> R = GM1FundamentalMatrix(A);
>>> disp(R);
      0.10065     0.026961
   0.00065531      0.12569

For Mathematica:

>>> A0 = {{0.1, 0.},{0., 0.1}};
>>> A1 = {{0., 0.2},{0., 0.2}};
>>> A2 = {{0., 0.1},{0., 0.}};
>>> A3 = {{0.3, 0.2},{0.3, 0.2}};
>>> A4 = {{0., 0.1},{0.2, 0.}};
>>> A = {A0, A1, A2, A3, A4};
>>> R = GM1FundamentalMatrix[A];
"The evaluation of the iteration required "64" roots\n"
"The evaluation of the iteration required "32" roots\n"
"The evaluation of the iteration required "16" roots\n"
"The evaluation of the iteration required "8" roots\n"
"The evaluation of the iteration required "8" roots\n"
"Final Residual Error for G: "5.551115123125783*^-17
>>> Print[R];
{{0.10065149910973312, 0.026960920607274754},
 {0.0006553100576153258, 0.12568710472819553}}

For Python/Numpy:

>>> A0 = ml.matrix([[0.1, 0.],[0., 0.1]])
>>> A1 = ml.matrix([[0., 0.2],[0., 0.2]])
>>> A2 = ml.matrix([[0., 0.1],[0., 0.]])
>>> A3 = ml.matrix([[0.3, 0.2],[0.3, 0.2]])
>>> A4 = ml.matrix([[0., 0.1],[0.2, 0.]])
>>> A = [A0, A1, A2, A3, A4]
>>> R = GM1FundamentalMatrix(A)
The Shifted PWCR evaluation of Iteration  1  required  64  roots
The Shifted PWCR evaluation of Iteration  2  required  32  roots
The Shifted PWCR evaluation of Iteration  3  required  16  roots
The Shifted PWCR evaluation of Iteration  4  required  8  roots
The Shifted PWCR evaluation of Iteration  5  required  8  roots
Final Residual Error for G:  5.20417042793e-17
>>> print(R)
[[ 0.10065  0.02696]
 [ 0.00066  0.12569]]