Suppose that solutions are computed to the linear system Ax = b where the entries of A have about 6 digits of accuracy, b has about 4 digits of accuracy. Given that the condition number of A is about $10^3$
How to estimate the number of digits of accuracy can be expected in the solution x.
Very good question. It could be phrased as How do we compute significant digits in our least squares results? An example is worked here: What parameters can be used to tell a least squares fit is “well fit”?
In this case, the least squares solution vector is computed as $$ \left[ \begin{array}{c} a_{0} \\ a_{1} \\ \end{array} \right]_{LS} = \left[ \begin{array}{c} 4.8 \pm 4.9\\ 9.41 \pm 0.86 \\ \end{array} \right] $$ The uncertainties are built from the diagonal terms of the curvature matrix $$ \alpha = \left( \mathbf{A}^{*} \mathbf{A} \right)^{-1}. $$
