On Sun, Oct 30, 2011 at 10:52 AM, Matthew Knepley <span dir="ltr"><<a href="mailto:knepley@gmail.com">knepley@gmail.com</a>></span> wrote:<div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
<div class="gmail_quote"><div>More commentary: There are lots of papers about estimating these norms (1-norms too), and</div><div>nothing works well. There are no good ways to generically approximate the matrix norm. For</div>
<div>certain very special classes of matrix, you can do it, but these are also the matrices for which</div><div>you have a specialize very fast solver, like the Laplacian, so you rarely care.</div><div> </div></div></blockquote>
</div><br><div>There is a nice paper by John D. Dixon, "Estimating Extremal Eigenvalues and Condition Numbers of Matrices", <a href="http://www.jstor.org/pss/2157241">http://www.jstor.org/pss/2157241</a>, 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.</div>
<div><br></div><div>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.</div><div><br></div><div>Jack</div>