When considering matrix inversion it is worth while worrying about the condition number of the matrix you wish to invert. Matrices that are poorly conditioned can often create inaccurate results. This I understand.
However, no where have I been able to see a guide of what the rough order of the condition number should be before we should start using preconditioning techniques. I imagine it varies between 32 and 62 bit processors, floats vs singles etc.
It is not complicated. We assume that we know the decomposition $A=LU$ where the entries of $A\in M_n$ are known with $p$ significant digits. We calculate, with an iterative algorithm, an approximation $K\approx 10^q$ of $cond(A)$. cf.http://www.alglib.net/matrixops/rcond.php
The complexity is only in $O(n^2)$ but the true value of $cond(A)$ can be $\approx 10 K\approx 10^{q+1}$. Then, with LU, we obtain $A^{-1}$ with theoretically $p-q$ significant digits but, in fact, with $p-q-1$ digits.
EDIT: answer to copper.hat. Yes I was unclear. That I'd want to say is that if $p-q$ is small or negative, then we must use a preconditioner matrix $P$, otherwise it is clearly useless. Moreover the preconditioned matrix $P^{-1}A$ must satisfy $cond(P^{-1}A)<<cond(A)$ and the calculation of $cond(P^{-1}A)$ must be done.