Need help with relative and absolute errors?

Lets assume I have $Ax=b$ equation, where $A$ is $2$x$2$ matrix.

1) I want to find an A, x, and b such that relative error in x is small but absolute error in x is large
2) Also want to find A, x, and b such that absolute error is small and relative error in x is large

I know the following formulas:
$ABS_{error}=\|\hat{x}-x\|^2$
$REL_{error}=ABS_{error}/\|x\|$

I was thinking to do reverse engineering to find $A$ and $b$ based on $x$ and $\hat{x}$, but then i did not know how should i find $\hat{x}$? Which method should I use?

Help please

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1 Answer

Absolute error is proportional to $||A^{-1}||$ and relative error is proportional to $||A^{-1}|| \cdot ||A||$.

For example, $$H =\pmatrix{4 & -6 \\ -6 & 12}$$

$||H^{-1}||_2 \approx 1.3$ and $||H^{-1}||_2 \cdot ||H||_2 \approx 19.3$, where $||\cdot||_2$ is the matrix 2-norm or frobenius norm. So in this example, the relative error is large compared to the absolute error.

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