That article is quite right regarding the take-home message to try to avoid inverting a matrix when other, more direct methods for solving the particular proble

That article is quite right regarding the take-home message to try to avoid inverting a matrix when other, more direct methods for solving the particular problem at hand exist. On page 2 of http://www.ti3.tuhh.de/paper/rump/Ru08a.pdf http://www.ti3.tuhh.de/paper/rump/Ru08a.pdf there is an innocent-looking matrix for which the numerically computed inverse is off from the true inverse by 47 orders of magnitude. However, one of the reasons he's given is not correct: Druinsky and Toledo have shown (http://arxiv.org/abs/1201.6035 http://arxiv.org/abs/1201.6035) that -- despite the very widespread belief to the contrary -- solving a linear system be calculating the inverse can be as accurate (though not nearly as efficient) as solving it directly. They even speculate as to how this myth was able to sustain it for so long. Namely, there is a certain error bound for the solution as calculated by employing the inverse which is quite simple to deduce and which predicts a pretty large error. However, this bound is too conservative. There is a better bound, which ensures that the error is of the same kind as for the direct method. However, this bound is much harder to derive.