I am developing software to find a 'good' solution for the over-constrained problem $Ax=b$, where
- $A$ is a known matrix $A_{i,j}$, $i=1,\ldots, M$, $j = 1,\ldots,N$, $M > N$,
- $b$ is a known vector $b_i$, $i=1,\ldots,M$, but may contain noise or measurement error, and
- $x$ is an unknown vector $x_j$, $j=1,\ldots,N$
using least squares and a singular value decomposition algorithm I found in Numerical Recipes. All works fine (i.e., produces a consistent solution for x) but now I want a measure of the uncertainty in each of the $x_j$ terms. Can anybody point me to an algorithm or reference to help calculate this. Any sensible measure of the uncertainty is OK as I just need an indication of the relative 'goodness' of each coefficient.
Note: I don't want an algorithm to solve for $x$, I already have that using QR decomposition. What I want is a measure of the uncertainty or error or variance (insert suitable term here) in the individual coefficients of $x$ that are returned by the least squares fit.