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

[Mark F. Adams]

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

> 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.

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.

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