On Sun, Jul 17, 2011 at 21:23, Mark F. Adams <[email protected]>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.