What do you mean by "tolerance"?
A QR solution has no need for a tolerance, nor does linsolve, as they are not iterative methods.
Anyway, you cannot specify the desired accuracy of a solution in a linear system like this, solved using QR. Just wanting a more accurate solution is not relevant. The solution exists, and is unique, or there are infinitely many solutions, all of which are equally good (or poor), or there is no exact solution, and that which qr/linsolve returns is entirely valid.
Perhaps you have a singular system, or one that is nearly singular. In that case, a tolerance is still not really relevant. Yes, you could decide which singular values to discard, in case of a pinv style solution. But that is not a "tolerance" in the way tolerances are traditionally used. And if your system is singular, then again, tolerances have no real meaning.
As far as your matrix being non-square, that just means (assuming you have an over-determined system) that you will never be able to achieve an exact solution. (Well, not entirely true. It IS possible that an over-determined system has an exact solution. But that is terribly rare.) So if no exact solution exists, then specifying a tolerance is again a meaningless goal.
If your problem is non-square in the sense that it is under-determined, then again, a tolerance is irrelevant. A solution may exist, or not. If one solution does exist, then there will be infinitely many solutions, all equally good. A tolerance is meaningless here.
I think that perhaps you are being confused, thinking that the solution to a linear system is like that of a nonlinear system, where you will employ convergence tolerances, etc. Or, perhaps you think that simply by cranking down on a tolerance, you can always gain an arbitrarily good solution. (Many people seem to think that. They are flat out wrong.) Or perhaps you are thinking that a solution like that provided by linsolve/qr is like an iterative solver, like that from lsqr.
If this has not cleared things up, then I will suggest you need to explain CLEARLY what you want to do, and why you think that a tolerance is a meaningful thing to ask for here. As well, you need to explain why it is that you think what I have said above does not apply.
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