[doc] Fix linalg.slogdet doc consistency issues (#51353)

Summary:
Pull Request resolved: #51353

Re-created from #51300 because of issues with ghstack.

This PR is part of a larger effort to ensure torch.linalg documentation is consistent (see #50287).

Updated torch.linalg.slogdet to include notes about cross-device synchronization, backend routines used and fix signature missing out argument.

Test Plan: Imported from OSS

Reviewed By: H-Huang

Differential Revision: D26148678

Pulled By: heitorschueroff

fbshipit-source-id: 40f6340226ecb72e4ec347c5606012f31f5877fb

torch/linalg/__init__.py

@@ -149,23 +149,29 @@
 """)
 
 slogdet = _add_docstr(_linalg.linalg_slogdet, r"""
-linalg.slogdet(input) -> (Tensor, Tensor)
+linalg.slogdet(input, *, out=None) -> (Tensor, Tensor)
 
 Calculates the sign and natural logarithm of the absolute value of a square matrix's determinant,
-or of the absolute values of the determinants of a batch of square matrices :attr`input`.
+or of the absolute values of the determinants of a batch of square matrices :attr:`input`.
 The determinant can be computed with ``sign * exp(logabsdet)``.
 
 Supports input of float, double, cfloat and cdouble datatypes.
 
 .. note:: When given inputs on a CUDA device, this function synchronizes that device with the CPU.
 
+.. note:: The determinant is computed using LU factorization. LAPACK's `getrf` is used for CPU inputs,
+          and MAGMA's `getrf` is used for CUDA inputs.
+
 .. note:: For matrices that have zero determinant, this returns ``(0, -inf)``.
           If :attr:`input` is batched then the entries in the result tensors corresponding to matrices with
           the zero determinant have sign 0 and the natural logarithm of the absolute value of the determinant -inf.
 
-Arguments:
+Args:
     input (Tensor): the input matrix of size :math:`(n, n)` or the batch of matrices of size :math:`(*, n, n)`
-                    where `*` is one or more batch dimensions.
+                    where :math:`*` is one or more batch dimensions.
+
+Keyword args:
+    out (tuple, optional): tuple of two tensors to write the output to.
 
 Returns:
     A namedtuple (sign, logabsdet) containing the sign of the determinant and the natural logarithm