[petsc-users] Condition Number and GMRES iteration

[Fande Kong]

On Fri, Feb 7, 2020 at 11:43 AM Victor Eijkhout <eijkhout at tacc.utexas.edu>
wrote:

>
>
> On , 2020Feb7, at 12:31, Mark Adams <mfadams at lbl.gov> wrote:
>
> BTW, one of my earliest talks, in grad school before I had any real
> results, was called "condition number does not matter”
>
>
> After you learn that the condition number gives an _upper_bound_ on the
> number of iterations, you learn that if a few eigenvalues are separated
> from a cluster of other eigenvalues, your number of iterations is 1 for
> each separated one, and then a bound based on the remaining cluster.
>
> (Condition number predicts a number of iterations based on Chebychev
> polynomials. Since the CG polynomials are optimal, they are at least as
> good as Chebychev. Hence the number of iterations is at most what you got
> from Chebychev, which is the condition number bound.)
>

I like this explanation for normal matrices. Thanks so much, Victor,


Fande,


>
> Victor.
>
>
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20200207/954a042a/attachment.html>