Add linalg.solve_triangular (#63568)
Summary:
Pull Request resolved: https://github.com/pytorch/pytorch/pull/63568
This PR adds the first solver with structure to `linalg`. This solver
has an API compatible with that of `linalg.solve` preparing these for a
possible future merge of the APIs. The new API:
- Just returns the solution, rather than the solution and a copy of `A`
- Removes the confusing `transpose` argument and replaces it by a
correct handling of conj and strides within the call
- Adds a `left=True` kwarg. This can be achieved via transposes of the
inputs and the result, but it's exposed for convenience.
This PR also implements a dataflow that minimises the number of copies
needed before calling LAPACK / MAGMA / cuBLAS and takes advantage of the
conjugate and neg bits.
This algorithm is implemented for `solve_triangular` (which, for this, is
the most complex of all the solvers due to the `upper` parameters).
Once more solvers are added, we will factor out this calling algorithm,
so that all of them can take advantage of it.
Given the complexity of this algorithm, we implement some thorough
testing. We also added tests for all the backends, which was not done
before.
We also add forward AD support for `linalg.solve_triangular` and improve the
docs of `linalg.solve_triangular`. We also fix a few issues with those of
`torch.triangular_solve`.
Resolves https://github.com/pytorch/pytorch/issues/54258
Resolves https://github.com/pytorch/pytorch/issues/56327
Resolves https://github.com/pytorch/pytorch/issues/45734
cc jianyuh nikitaved pearu mruberry walterddr IvanYashchuk xwang233 Lezcano
Test Plan: Imported from OSS
Reviewed By: jbschlosser
Differential Revision: D32588230
Pulled By: mruberry
fbshipit-source-id: 69e484849deb9ad7bb992cc97905df29c8915910
