You want the smallest nonzero singular value, right? Even then, what if there are multiple equal (or nearly equal) values?

@Nick, your comment doesn't have anything to do with the original question or my answer. Perhaps you should ask a new question.

You'd do well to explain exactly what integral you actually want to evaluate. It seems unlikely that the strategy you're suggesting would be optimal. Is it just $\int_{-\infty}^{c} e^{-x^2 }\mbox{erf}(x+a)dx$?

Do you need to compute the Gaussian probability density or the comulative desnity (integral of the pdf from $-\infty$ to $x$) The former is trivial using an $\exp$ library function. The second is actually somewhat of a challenge.

@DimitriosPagonakis the document you've referenced is simply wrong. To see this, Try multiplying a random 2 by 3 matrix (say A=[1 2 3; 4 5 6]) times a 3 by 3 diagonal matrix (say D=[1 0 0; 0 2 0; 0 0 3]) and see what happens. This scales the columns of A by the factors on the diagonal of D.

I'm curious- what approach did you decide on in the end?

If you're still having trouble getting this through your library, contact me by email.

Note that you don't actually have to store multiple copies of $A$ and $A^{T}$. All that LSQR requires is that you be able to compute the matrix vector products $Ax$ and $A^{T}y$. One copy of the $A$ matrix is sufficient for this.

In the special case of LP, if an optimal BFS is non-degenerate then the result holds. I think this is equivalent to "perturbations preserve the optimal partitions."