[petsc-users] Condition Number and GMRES iteration

[Victor Eijkhout]

On , 2020Feb7, at 12:31, Mark Adams <mfadams at lbl.gov<mailto: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.)

Victor.


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