lu_backward: more numerically stable and with complex support. (#53994)
Summary:
As per title.
Numerical stability increased by replacing inverses with solutions to systems of linear triangular equations.
Unblocks computing `torch.det` for FULL-rank inputs of complex dtypes via the LU decomposition once https://github.com/pytorch/pytorch/pull/48125/files is merged:
```
LU, pivots = input.lu()
P, L, U = torch.lu_unpack(LU, pivots)
det_input = P.det() * torch.prod(U.diagonal(0, -1, -2), dim=-1) # P is not differentiable, so we are fine even if it is complex.
```
Unfortunately, since `lu_backward` is implemented as `autograd.Function`, we cannot support both autograd and scripting at the moment.
The solution would be to move all the lu-related methods to ATen, see https://github.com/pytorch/pytorch/issues/53364.
Resolves https://github.com/pytorch/pytorch/issues/52891
TODOs:
* extend lu_backward for tall/wide matrices of full rank.
* move lu-related functionality to ATen and make it differentiable.
* handle rank-deficient inputs.
Pull Request resolved: https://github.com/pytorch/pytorch/pull/53994
Reviewed By: pbelevich
Differential Revision: D27188529
Pulled By: anjali411
fbshipit-source-id: 8e053b240413dbf074904dce01cd564583d1f064
