My question is concerned about solving linear, over-determined systems of equations of the form $Ax = b$, where $A$ is a matrix of coefficients of dimension $m \times k$, $(m > k)$, $x$ is the vector of unknowns of dimension $k \times 1$ and, consequently, $b$ is a vector of constants of dimension $m \times 1$.
If I add now a certain amount of noise $n$ modelled by a uniform random variable to $b$ we have now a different vector of unknowns $b' = b + n$. After checking out matrix $A$ is full-rank and applying least-squares minimization or QR factorization to $A$ the solution I get for $x$ is still far from good: while several coefficients $x_i$ match the expected results pretty accurately, some others still differ significantly from what I should get -and this different of course becomes larger when the noise variance gets increased.
I was wondering whether any method to 'distribute' the uncertainty among all the coefficients $x_i$ exists (I have the feeling that I could 'sacrifice' the accuracy with which some values of $x_i$ are resolved to improve the estimation of the others). Glancing through the literature, I have read about ill-conditioned matrix and how the condition number of a matrix can affect its sensitivity when it is perturbed by external noise. In my case, the condition number of my matrix is in the order of $10^1$, which is not -as I have read- not that large to have that much problem, is it?
Many thanks in advance, and best regards.