Tolerance arguments for scipy.sparse.linalg.{lsmr,lsqr} do not match related functions #19704
Description
There were a bunch of long-outstanding changes regarding tol/atol/rtol in scipy.sparse.linalg.*, c.f. #15738, #18934, #18943 and #19702.
While preparing the latter (to try to make the imminent 1.12 more consistent), I fixed a couple outstanding functions, but noticed that lsmr and lsqr have a completely different setup for their tolerance arguments:
scipy/scipy/sparse/linalg/_isolve/lsmr.py
Lines 57 to 74 in c4ce0c4
What's worse, it uses atol -- a kwarg that's pretty universally used to mean absolute tolerance -- as a relative tolerance, which is IMO pretty bad as a footgun for users coming from any of the other scipy.sparse.linalg.* functions.
I don't have a good suggestion how to fix this (much quickly enough for 1.12!), but perhaps we can find a way to harmonize this a bit better?
FWIW, as of SciPy 1.14, the other functions in scipy.sparse.linalg.* should uniformly default to rtol=1.e-5 as relative tolerance and atol=0 as absolute tolerance (with any remaining tol= usages removed).
CC @tupui who touched this in 3d53ac2, and CC @mdhaber for the related #7219