On Thu, Apr 16, 2009 at 11:34 AM, Chetan Jhurani <span dir="ltr"><<a href="mailto:chetan@ices.utexas.edu">chetan@ices.utexas.edu</a>></span> wrote:<br><div class="gmail_quote"><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
> From: Yixun Liu<br>
><br>
> Hi,<br>
> For Ax=b, A is mxn, m>n. I use CG to resolve it and find the solution<br>
> makes no sense. I guess rank(A) < min(m,n). How to resolve this<br>
> singular system? Use SVD?<br>
<br>
Only a square matrix can be singular.</blockquote><div><br>No, a singular matrix has a kernel. A non-square matrix can be singular.<br> </div><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
<br>
If reinterpreting as a least-squares problem, SVD would be slower.<br>
<br>
If rank(A) = n, see<br>
<<a href="http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse#The_QR_method" target="_blank">http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse#The_QR_method</a>></blockquote><div><br>QR will work for a matrix of rank < n. In this case, a null space basis fills out U.<br>
<br> Matt<br><br></div><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;"><br>
If A is dense, use LAPACK for QR, otherwise sparse QR factorization<br>
should be faster. <a href="http://www.cise.ufl.edu/research/sparse/CSparse/" target="_blank">http://www.cise.ufl.edu/research/sparse/CSparse/</a><br>
<br>
If A is not full rank (rank(A) < n), it is more complicated. The<br>
pseudoinverse does not have a simple formula, although it is still<br>
computable for getting the minimum norm solution. The book by Ake<br>
Bjorck would be useful, as Matt already suggested.<br>
<br>
Chetan<br>
<br>
> Best,<br>
><br>
> Yixun<br>
><br>
<br>
</blockquote></div><br><br clear="all"><br>-- <br>What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>-- Norbert Wiener<br>