|x1-x2|<=?

[Matthew Knepley]

On 8/23/07, Alejandro Garzon <gtg100n at mail.gatech.edu> wrote:
> Hi, I have found two aproximations x1 and x2 to the solution of a linear system
> A*x=b by two different methods with the same relative residual "e". That is
> |A*x1 - b| < e*|b| and |A*x2 - b| < e*|b|. For debugging purposes I want to know
> if an upper bound for |x1 - x2| can be derived from the two inequalities above.
> I have gone this far in trying to find it:
>
>  From the triangle inequality
>
> |A*x1 - b -(A*x2 - b)| <= |A*x1 - b| + |A*x2 - b| = 2*e*|b|,
>
> eliminating the b's in the left hand side,
>
> |A*(x1-x2)| <= 2*e*|b|,
>
> Does anybody know if from here a condition of the form
>
> |x1-x2| <= ?
>
> can be derived?

This is exactly the condition number blowup of residual into error:

  |e| \le |A^{-1} r|
     |e| \le |A^{-1}| |r|

which looks like the condition number of A.

    Matt

> Thanks
> --
> Alejandro
>
>
>
>
>


-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which
their experiments lead.
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