Add SVD function specification

spec/API_specification/linear_algebra_functions.md

@@ -275,9 +275,35 @@ TODO
 TODO
 
 (function-svd)=
-### svd()
+### svd(x, /, *, compute_uv=True, full_matrices=True)
 
-TODO
+Computes the singular value decomposition `A = USV` of a matrix (or stack of matrices) `x`.
+
+#### Parameters
+
+-   **x**: _<array>_
+
+    -   input array having shape `(..., M, N)` and whose innermost two dimensions form matrices on which to perform singular value decomposition. Must have a data type of either `float32` or `float64`.
+
+-   **compute_uv**: _bool_
+
+    -   If `True`, compute the left and right singular vectors and return as `u` and `v`, respectively. Default: `True`.
+
+-   **full_matrices**: _bool_
+
+    -   If `True`, compute full-sized `u` and `v`, such that `u` has shape `(..., M, M)` and `v` has shape `(..., N, N)`. If `False`, compute on the leading `K` singular vectors, such that `u` has shape `(..., M, K)` and `v` has shape `(..., N, K)` and where `K = min(M, N)`. This option must be ignored if `compute_uv` is `False`. Default: `True`.
+
+#### Returns
+
+-   **out**: _Union\[ <array>, Tuple\[ <array>, ... ] ]_
+
+    -   if `compute_uv` is `False`, an array having shape `(..., K)` and containing the vector(s) containing the singular values. For each vector, the values must be sorted in descending order according to magnitude, such that `s[..., 0]` is the largest value, `s[..., 1]` is the second largest value, et cetera. The first `x.ndim-2` dimensions must have the same size as those of the input `x`.
+
+    -   if `compute_uv` is `True`, a namedtuple `(u, s, v)` whose
+
+        -   first element must be an array whose shape depends on the value of `full_matrices` and contain unitary array(s) (i.e., the left singular vectors). The left singular vectors must be stored as columns. If `full_matrices` is `True`, the array must have shape `(..., M, M)`. If `full_matrices` is `False`, the array must have shape `(..., M, K)`, where `K = min(M, N)`. The first `x.ndim-2` dimensions must have the same size as those of the input `x`.
+        -   second element must be an array having shape `(..., K)` and contain the vector(s) containing the singular values. For each vector, the values must be sorted in descending order according to magnitude, such that `s[..., 0]` is the largest value, `s[..., 1]` is the second largest value, et cetera. The first `x.ndim-2` dimensions must have the same size as those of the input `x`.
+        -   third element must be an array whose shape depends on the value of `full_matrices` and contain unitary array(s) (i.e., the right singular vectors). The right singular vectors must be stored as rows (i.e., the array is the adjoint). If `full_matrices` is `True`, the array must have shape `(..., N, N)`. If `full_matrices` is `False`, the array must have shape `(..., K, N)` where `K = min(M, N)`. The first `x.ndim-2` dimensions must have the same size as those of the input `x`. 
 
 (function-trace)=
 ### trace(x, /, *, axis1=0, axis2=1, offset=0)