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

[Jack Poulson]

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
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