[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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