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.<div>
<br></div><div>Jack<br><br><div class="gmail_quote">On Sun, Oct 30, 2011 at 11:33 AM, behzad baghapour <span dir="ltr"><<a href="mailto:behzad.baghapour@gmail.com">behzad.baghapour@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
A good paper, I will work on it.<br>Thanks a lot dear Jack.<div><div></div><div class="h5"><br><br><div class="gmail_quote">On Sun, Oct 30, 2011 at 7:54 PM, Jack Poulson <span dir="ltr"><<a href="mailto:jack.poulson@gmail.com" target="_blank">jack.poulson@gmail.com</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div>On Sun, Oct 30, 2011 at 10:52 AM, Matthew Knepley <span dir="ltr"><<a href="mailto:knepley@gmail.com" target="_blank">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><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" target="_blank">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><font color="#888888"><div>
Jack</div>
</font></blockquote></div><br><br clear="all"><br></div></div><div><div></div><div class="h5">-- <br>==================================<br>Behzad Baghapour<br>Ph.D. Candidate, Mechecanical Engineering<br>University of Tehran, Tehran, Iran<br>
<a href="https://sites.google.com/site/behzadbaghapour" target="_blank">https://sites.google.com/site/behzadbaghapour</a><br>
Fax: 0098-21-88020741<br>==================================<br><br>
</div></div></blockquote></div><br></div>