[petsc-users] Condition Number and GMRES iteration

[Matthew Knepley]

On Fri, Feb 7, 2020 at 1:43 PM 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.
>

This is _only_ for normal matrices. Not true for general matrices.

   Matt


> (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.)
>
> Victor.
>
>
>

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

https://www.cse.buffalo.edu/~knepley/ <http://www.cse.buffalo.edu/~knepley/>
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