For what it's worth, here's what I remember from my Numerical Analysis course.
The thing we were taught to use to figure out whether the matrix is ill conditioned is the condition number of a matrix (k(A) = norm(A) * norm(A^-1)). Here's a nice explanation of it [1]. Suppose you want to solve Ax = b. How much worse results will you get using A if you're not really solving Ax = b but A(x + delta) = b + epsilon (x is still a solution for Ax = b). So, by perturbing the b vector by epsilon, how much worse is delta going to be? There's a short proof [1, page 4] but the inequality you get is: norm(delta) / norm(x) <= k(A) * norm(epsilon) / norm(b) The rule of thumb is that if m = log10(k(A)), you lose m digits of accuracy. So, equivalently, if m' = log2(k(A)) you lose m' bits of accuracy. Since floats are 32bits, you can decide that say, at most 2 bits may be lost, therefore any k(A) > 4 is not acceptable. Anyway there are lots of possible norms and you need to look at ways of actually interpreting the condition number but from what I learned this is probably the thing you want to be looking at. Good luck! [1] http://www.math.ufl.edu/~kees/ConditionNumber.pdf [2] http://www.rejonesconsulting.com/CS210_lect07.pdf On Thu, Apr 4, 2013 at 5:26 PM, Sean Owen <sro...@gmail.com> wrote: > I think that's what I'm saying, yes. Small rows X shouldn't become > large rows of A -- and similarly small changes in X shouldn't mean > large changes in A. Not quite the same thing but both are relevant. I > see that this is just the ratio of largest and smallest singular > values. Is there established procedure for evaluating the > ill-conditioned-ness of matrices -- like a principled choice of > threshold above which you say it's ill-conditioned, based on k, etc.? > > On Thu, Apr 4, 2013 at 3:19 PM, Koobas <koo...@gmail.com> wrote: > > So, the problem is that the kxk matrix is ill-conditioned, or is there > more > > to it? > > >
