On Sun, Jul 17, 2011 at 21:23, Mark F. Adams <mark.adams at columbia.edu>wrote:
> Humm, the only linear algebra proof that I know gives bounds on the error > of the form > > | error |_2 <= Condition-number * | residual |_2, > This looks like relative error. > > for SPD matrices of course. This is pessimistic but I'm not sure how you > could get a bound on error with only the lowest eigen value ... > Suppose you have | A x - b | < c Then there is some y such that A (x + y) - b = 0 and for which |A y| < c Suppose s is the smallest singular value of A, thus 1/s is the largest singular value of A^{-1}. Then |y| = | A^{-1} A y | <= (1/s) |A y| < c/s. So you can bound the absolute error in the solution if you know the residual and the smallest singular value. -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://lists.mcs.anl.gov/pipermail/petsc-dev/attachments/20110717/315b718e/attachment.html>