[petsc-users] How to calculate Induced Norm of Matrix?

[Jack Poulson]

Not a problem; though for some reason I repeatedly wrote "condition number"
when I meant "two norm". Dixon's paper certainly provides a method for
computing an estimate to the condition number, but the latter also requires
the ability to apply the inverse of your operator and the inverse of its
adjoint.

Jack

On Sun, Oct 30, 2011 at 11:33 AM, behzad baghapour <
behzad.baghapour at gmail.com> wrote:

> A good paper, I will work on it.
> Thanks a lot dear Jack.
>
>
> On Sun, Oct 30, 2011 at 7:54 PM, Jack Poulson <jack.poulson at gmail.com>wrote:
>
>> On Sun, Oct 30, 2011 at 10:52 AM, Matthew Knepley <knepley at gmail.com>wrote:
>>
>>> More commentary: There are lots of papers about estimating these norms
>>> (1-norms too), and
>>> nothing works well. There are no good ways to generically approximate
>>> the matrix norm. For
>>> certain very special classes of matrix, you can do it, but these are
>>> also the matrices for which
>>> you have a specialize very fast solver, like the Laplacian, so you
>>> rarely care.
>>>
>>>
>>
>> There is a nice paper by John D. Dixon, "Estimating Extremal Eigenvalues
>> and Condition Numbers of Matrices", http://www.jstor.org/pss/2157241,
>> which provides an extremely robust method for getting rough estimates of
>> the condition number, and it only requires the ability to apply your
>> operator and its adjoint. A typical usage would be to compute an estimate
>> of the condition number K, such that the true condition number is within a
>> factor of 2 of K with a probability of 1-10^-6.
>>
>> The 1-norm is actually pretty trivial to compute if you have access to
>> your matrix entries; it is the maximum vector one norm of the columns of
>> the matrix.
>>
>> Jack
>>
>
>
>
> --
> ==================================
> Behzad Baghapour
> Ph.D. Candidate, Mechecanical Engineering
> University of Tehran, Tehran, Iran
> https://sites.google.com/site/behzadbaghapour
> Fax: 0098-21-88020741
> ==================================
>
>
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