On Jul 17, 2011, at 10:31 PM, Jed Brown wrote:

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.

Yes, I was confused last night, JFK will do that :) Mark

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.