sparseEqSolver
Updated:
Tags: documentation, solver
Sintax
X = sparseEqSolver(A,B,C,E)
Description
Consider the matrix equation
\[\begin{cases} \left[\mathbf{A}\mathbf{X}\mathbf{B}-\mathbf{C}\right]_{i,j} = 0 & \;,\; \mathbf{E}_{i,j} \neq 0\\ \left[\mathbf{X}\right]_{i,j} = 0 & \;,\; \mathbf{E}_{i,j} = 0 \end{cases}\;,\]where $\mathbf{A}\in \mathbb{R}^{n\times n}$, $\mathbf{B}\in \mathbb{R}^{o\times o}$, $\mathbf{C}\in \mathbb{R}^{n\times o}$, and $\mathbf{E}\in \mathbb{R}^{n\times o}$, which represents a sparsity pattern, are known and $\mathbf{X} \in \mathbb{R}^{n\times o}$ is the unknown. This equation arises, for instance, in the computation of the decentralized finite-horizon Kalman filter gain and decentralized finite-horizon LQR gain.
The function
X = sparseEqSolver(A,B,C,E)returns the solution to the matrix equation above, assuming an unique solution exists. The algorithm used is detailed in [Section 3, 1].
Input arguments
A: matrix $\mathbf{A}\in \mathbb{R}^{n\times n}$B: matrix $\mathbf{B}\in \mathbb{R}^{o\times o}$C: matrix $\mathbf{C}\in \mathbb{R}^{n\times o}$E: sparsity pattern $\mathbf{E}\in \mathbb{R}^{n\times o}$
Output Arguments
X: solution $\mathbf{X}\in \mathbb{R}^{n\times o}$
Examples
Random equation
To open this example execute the following command in the MATLAB command window
open sparseEqSolverTutorial
Generate matrices $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, and $\mathbf{E}$ randomly
n = 5;
o = 4;
A = rand(n,n)
A =
0.9425 0.6404 0.8957 0.7099 0.8676
0.5626 0.0884 0.8046 0.9169 0.5056
0.7992 0.0585 0.4475 0.1403 0.1463
0.4330 0.0572 0.4644 0.8736 0.8064
0.0730 0.8537 0.0831 0.9491 0.6901
B = rand(o,o)
B =
0.0217 0.4170 0.1672 0.8852
0.4153 0.1643 0.3241 0.7918
0.7735 0.6083 0.3607 0.4422
0.3626 0.3697 0.4211 0.8731
C = rand(n,o)
C =
0.1569 0.5736 0.3799 0.6590
0.0243 0.9060 0.4589 0.5584
0.6823 0.8052 0.4666 0.7979
0.2590 0.6943 0.5368 0.4517
0.4906 0.7305 0.1866 0.5506
E = round(rand(n,o))
E =
1 0 1 1
1 1 0 0
1 1 1 0
1 0 1 1
0 0 1 1
Compute the solution to the random equation
X = sparseEqSolver(A,B,C,E)
X =
-2.5837 0 0.9561 1.5036
2.5151 -2.5040 0 0
4.4700 -3.0202 0.7491 0
-2.4387 0 -3.5041 6.5209
0 0 2.5341 -4.0368
which has the desired sparsity pattern. Finally, verify that it is the solution to the matrix equation
sum(sum(abs((A*X*B-C).*E)))
ans =
6.0958e-15