TITLE: Conjugate Gradient for a "slightly" singular system.
QUESTION [6 upvotes]: Suppose I have a symmetric $N \times N$ matrix A which has a one-dimensional Nullspace $N$.  A is positive definite on  $N^\bot$.  In my case $N$ is the space of constant vectors (i.e. generated by the all-one vector).
I have to solve the problem $Ax = b$, with $b \in R(A)$ which has infinitely many solutions.  I am looking for the minimum norm solution. The matrix $A$ is very large and sparse, direct methods aren't really an option.  The rank-deficient least squares algorithms I have seen also appear to be prohibitive.
I was solving a non-singular version of this problem with Conjugate Gradient.  Is there anyway I can modify the algorithm to solve this particular problem?
EDIT: Boiling the problem down to the bone the question is if $A$ is positive semi-definite with exactly one 0 eigenvalue, does CG work?
Thanks.

REPLY [4 votes]: Plain CG, without the regularization, will converge to the pseudo-inverse solution provided you start the iteration with $y=0$.  The regularized (preconditioned) solution will be different and the operation count will depend on the sparsity, so it's good to maintain this if you can. I guess partly it must depend on whether you can tolerate a solution that has a nonzero component in the null space.