On Thu, Apr 16, 2009 at 11:34 AM, Chetan Jhurani <chetan at ices.utexas.edu>wrote:
> > From: Yixun Liu > > > > Hi, > > For Ax=b, A is mxn, m>n. I use CG to resolve it and find the solution > > makes no sense. I guess rank(A) < min(m,n). How to resolve this > > singular system? Use SVD? > > Only a square matrix can be singular. No, a singular matrix has a kernel. A non-square matrix can be singular. > > If reinterpreting as a least-squares problem, SVD would be slower. > > If rank(A) = n, see > <http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse#The_QR_method> QR will work for a matrix of rank < n. In this case, a null space basis fills out U. Matt > If A is dense, use LAPACK for QR, otherwise sparse QR factorization > should be faster. http://www.cise.ufl.edu/research/sparse/CSparse/ > > If A is not full rank (rank(A) < n), it is more complicated. The > pseudoinverse does not have a simple formula, although it is still > computable for getting the minimum norm solution. The book by Ake > Bjorck would be useful, as Matt already suggested. > > Chetan > > > Best, > > > > Yixun > > > > -- What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead. -- Norbert Wiener -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20090416/996094a6/attachment.htm>
