Find the relative error $\dfrac{\| \Delta x \|_1}{\| x \|_1}$ of the linear system $A\hat{x}=b+\Delta b$, where $\hat{x}=x+\Delta x$ and $x$ is the exact solution of the system $Ax=b$
The $x^*$ solution of $Ax=b$ is given by the maximum argument of $\dfrac{\| Ax \|_1}{\| x\|_1}$ and $\Delta b$ itself is the maximum argument of $\dfrac{\| A^{-1}x\|_1}{\| x\|_1}$. The matrices $A$ and $A^{-1}$ are given by:
$ A = \begin{bmatrix} 1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ -1 & -1 & 1 & 0 \\ -1 & -1 & -1 & 1 \end{bmatrix} = (a_{ij})$
$ A^{-1} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 2 & 1 & 1 & 0 \\ 4 & 2 & 1 & 1 \end{bmatrix} =(c_{ij})$
I know that the following relation is valid: $\dfrac{\| \Delta x\|_1}{\| x \|_1}=\mathcal{K}_1(A)\dfrac{\| \Delta b\|_1}{\| b\|_1}$. $\| A\|_1$ and $\| A^{-1}\|_1$ can be easily calculated. I don't know how to compute $\| b \|_1$ and $\| \Delta b \|_1$
By guessing, taking $e_{1}^{(4)}$, the canonical vector of $\mathbb{R}^4$, is easy to see that it is the argument that maximizes the 1-norm, the remaining steps are straightforward.