I m trying to analyse the stability against perturbations for a specific system of linear equations $Ax=b$.
For this, i use the standard condition number $||A||_{\infty}||A^{-1}||_{\infty}$.
Here is an exemple for an instance of my problem :
$A = \begin{pmatrix} 1 & -1 & 0 & 0 \\ -0.5 & 1 & -0.3 & -0.2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$
$b =\begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \\ \end{pmatrix}$
If $n$ is the size of the square matrix, matrix $A$ is such :
- $A(i,i) = 1$ for all $i = 1 \cdots n$ (all on-diagonal elements are equals to 1);
- $A(i,j) \in [-1,0]$ for all $i \neq j$ (all off-diagonal elements are negative and are between 0 and -1 inclusively);
- $\sum_{j=1,j \neq i}^{n} |A(i,j)| \in \{0,1\}$, for all $i = 1 \cdots n$ (for a given row, all aboslute value of off-diagonal elements must sum up to 0 or 1);
- $\exists i | \sum_{j=1,j \neq i}^{n} |A(i,j)| = 1$, in other cases the problem would be trivial.
- $\exists i | \sum_{j=1,j \neq i}^{n} |A(i,j)| = 0$,
- $b(i) \in \{0,1\} $,
- if $b(i) = 1 \Rightarrow \sum_{j=1,j \neq i}^{n} |A(i,j)| = 0 $,
- $A$ is non-singular ($A^{-1}$ exist and the system $Ax=b$ admit one solution)
Obviously, my matrices are diagonaly dominant but not strictly which means i could not use the varga bound. My matrices are z-matrix and maybe m-matrix with all the nice properties such that the inverse elements are all positives.
It's easy to see that $||A||_{\infty}=2$, i wanna determine un upper bound for $||A^{-1}||_{\infty}$.
For the exemple given, the inverse matrix is : $A^{-1} = \begin{pmatrix} 2 & 2 & 0.6 & 0.4 \\ 1 & 2 & 0.6 & 0.4 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$
And $||A^{-1}||_{\infty}=5$.
Any help will be very helpfull.
