Re: Minimizing matrix norm

In article <e8e6g6$uu$1@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>, Peter
Spellucci <spellucci@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

In article <030720061342294845%bruck@xxxxxxxxxxxx>,
Ronald Bruck <bruck@xxxxxxxxxxxx> writes:
>In fact, I also tried something else: instead of using the Frobenius
>norm, I used the [square root of] the sum of the squares of the
>OFF-DIAGONAL elements.
>
>This isn't a norm, of course, but it is still easily minimized. My
>rationale was sort of a Gershgorin approach: since the spectrum is
>contained in a union of disks centered at the diagonal elements, this
>tends to concentrate the spectrum near the diagonal.
>
>For my immediate problem, it USUALLY (about 97% of the time) resulted
>in a smaller value than the Frobenius value (which is good, from my
>point of view). But it was only about 10% less, which wasn't enough of
>a reduction to make it worthwhile.
>
>Is anybody familiar with other applications of this idea? Minimizing
>the off-diagonal contributions?
>
>--
>Ron Bruck
>
>
>
>In article <030720061317371148%bruck@xxxxxxxxxxxx>, Ronald Bruck
><bruck@xxxxxxxxxxxx> wrote:
>
>> Yes, I've done this too. In fact, since my matrices have integer
>> entries, the solutions are rational (as is the square of the Frobenius
>> norm), and I implemented a solver in exact rational arithmetic (using
>> the Gnu Multi-Precision software package).
>>
>> But the bounds I get are just too loose, so I'm trying the [much
>> harder] operator norm.

the Jacobi rotation method minimizes the sum of squares of off diagonal
elements
in order to get the eigenvalues of a hermitian matrix and this has been
generalized to general matrices by eberlein.

Ah, Peter, always helpful! I've wondered why Jacobi works, but never
had the time to look into it.

Well, the idea was too simple not to have been used before.

Many thanks, --
Ron Bruck
.

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