I have a a system of equations
$$ (B^TB + C)x = B^Tb $$
Here, $B$ is a very large, dense matrix and $C$ is a very sparse symmetric matrix. Computing $B^TB$ is infeasible (although computing $BB^T$ would be sometimes feasible).
I'm running into issues finding $x$ with an iterative solver without preconditioning. I would like to find a factorization of $(B^TB + C)$ without computing the resulting matrix first. Is this possible?