# Solve ill-posed linear system without transposing matrices?

I am attempting to use an iterative solver to solve $p$ in

$$Jp = -r$$

where $J$ is an $m\times m$ matrix. Unfortunately, this is an ill-posed problem and possibly infinite versions of $p$ exist. I cannot compute the least-squares version

$$J^TJp=J^Tr$$

because I only can access the product of $Jp$ and not $J^Tp$ (a limitation of an automatic differentiation approach I am taking).

• Is there a way I can compute $p$ in $Jp = -r$ according to some criterion (e.g. least-squares solution) without transposing $J$?
• If I use an iterative solver (e.g. GMRES, BiCGStab, etc...) with an initial guess of $p=0$, what solution will the solver converge to? The minimum-norm?
• how big is $m$? – sssssssssssss Sep 19 '17 at 19:20
• is $J$ sparse? if so, are you familiar with the MINRES family of solvers? if you're looking for the min-length solution, then MINRES-QLP may be a good option for your problem. – GoHokies Sep 19 '17 at 19:52
• as a side-comment, why is automatic differentiation limiting your options when it comes to $J^T$? If $J$ is the Jacobian of some $m$-variable (and at least locally smooth) function $f$, then reverse-mode AD gives you $J^T p$ for any direction $p$. – GoHokies Sep 19 '17 at 20:30
• The MINRES methods require $A$ to be symmetric (in which case all methods are effectively transpose free...) – Brian Borchers Sep 20 '17 at 2:27
• @GoHokies Reverse-mode AD requires excessive memory in my application. Beyond the point of being reasonably used. – bfletch Sep 20 '17 at 16:33