# [petsc-dev] KSP convergence test based on error on residual

Jed Brown jedbrown at mcs.anl.gov
Sun Jul 17 21:31:51 CDT 2011

```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.
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