When solving $A x = b$ on a computer, error is inherently incurred. Instead of the exact solution $x \text{,}$ an approximate solution $\hat x$ is computed, which instead solves $A \hat x = \hat b \text{.}$ The difference between $x$ and $\hat x$ satisfies
We can compute $\hat b = A \hat x$ and hence we can compute $\delta\!b = b - \hat b \text{.}$ We can then solve $A \delta\!x = \delta\!b \text{.}$ If this computation is completed without error, then $x = \hat x + \delta\!x$ and we are left with the exact solution. Obviously, there is error in $\delta\!x$ as well, and hence we have merely computed an improved approximate solution to $A x = b \text{.}$ This process can be completed. As long as solving with $A$ yields at least one digit of accuracy, this process can be used to improve the computed result, limited by the accuracy in the right-hand side $b$ and the condition number of $A \text{.}$