The project "Infinity System of Matrix Equations" was developed for solving the
following system of matrix equations
[4*B-A^2, X]=0,
(1) [[4*B-A^2, A], X]=0,
[[[4*B-A^2, A], A], X]=0,
[[[[4*B-A^2, A], A], A], X]=0,
[[[[[4*B-A^2, A], A], A], A], X]=0,
...................................
and so on to infinity,
...................................
where A and B are given matrices and matrix X is the unknown; [A, B]=A*B-B*A.
I have encountered this system when using stability preserving structural
transformations for stability investigation of systems of second-order
ordinary differential equations (for more detail see http://arxiv.org/abs/1204.1814).
It is easy to see that system (1) always possesses an infinite number
(multi-parametric family) of solutions. Thus, to find all the solutions of the system
we cannot take advantage of iterative methods, we have to use exact methods instead.
In the project I have used the Gaussian elimination method.
Evidently, to find the exact solution to a given system of linear equations
we have to confine ourselves to use rational arithmetic (no rounding errors).
In the project I took advantage of using the rational type mpq_t from The GNU
Multiple Precision Arithmetic Library (see http://gmplib.org/).
For the sake of convenience, I have implemented a C++ class for managing matrices
based on the type mpq_t (see class Matrix in Matrix.h and Matrix.cpp).
The project was developed under MS Visual Studio 2005.
How to use?
1) Extract the contents of the archive "infinity_system_of_matrix_equations.rar"
to some directory;
2) Open file "inf_matrix_system.vcproj" with MS Visual Studio;
3) Build the project;
4) Place files "inf_matrix_system.exe" (which was built in the previous step),
"A.txt" and "B.txt" (the latter two files you can find in the archive) to the
same directory;
5) Run inf_matrix_system.exe;
6) Open the file "Ans.txt", which will be created during the execution of
"inf_matrix_system.exe". This file contains the solution to the system (1) with the
matrices A and B loaded from files "A.txt" and "B.txt" respectively.
In the file A.txt from the archive you can find the description of matrix A:
4, 4,
4119/2635, 679/2635, -1162/2635, 1111/2635,
-437/2635, 4113/2635, 7576/2635, 4902/2635,
15/527, -105/527, 1614/527, 290/527,
1392/2635, -258/2635, -1786/2635, 2143/2635,
This description means that the matrix A has dimension 4x4 and looks like the following:
4119 679 -1162 1111
---- ---- ----- ----
2635 2635 2635 2635
-437 4113 7576 4902
---- ---- ---- ----
2635 2635 2635 2635
A=
15 -105 1614 290
--- ---- ---- ---
527 527 527 527
1392 -258 -1786 2143
---- ---- ----- ----
2635 2635 2635 2635
Similarly to this, the matrix B from the file B.txt can be imagened in the following form:
7499 -2192 86 -534
---- ----- --- ----
2635 2635 155 527
-15417 56697 -2041 3087
------ ----- ----- ----
2635 5270 310 527
B=
-725 2335 161 1255
---- ---- --- ----
527 1054 62 527
7027 -12557 811 948
---- ------ --- ---
2635 5270 310 527
In the file Ans.txt you can find the following:
4, 4,
1651/1536*C1+C2, -53/256*C1, 1/768*C1, -25/96*C1,
-4/3*C1, 2113/512*C1+C2, -121/96*C1, 5125/1536*C1,
25/768*C1, 925/1536*C1, 215/96*C1+C2, 2075/1536*C1,
29/48*C1, -451/384*C1, C1, C2,
This means that the solution X of system (1) has the form
1651*C1 -53*C1 1*C1 -25*C1
------- + C2 ------ ----- ------
1536 256 768 96
-4*C1 2113*C1 -121*C1 5125*C1
----- ------- + C2 ------- --------
3 512 96 1536
X=
25*C1 925*C1 215*C1 2075*C1
----- ------ ------ + C2 -------
768 1536 96 1536
29*C1, -451*C1
----- ------- C1 C2
48 384
Here C1, C2 are the arbitrary parameters.
