Document numerical error in rank (#48127)

Co-authored-by: Daniel Karrasch <@dkarrasch>
Co-authored-by: Fredrik Ekre <ekrefredrik@gmail.com>
Co-authored-by: Steven G. Johnson <stevenj@alum.mit.edu>

stdlib/LinearAlgebra/src/generic.jl

@@ -947,13 +947,22 @@ dot(x::AbstractVector, transA::Transpose{<:Real}, y::AbstractVector) = adjoint(d
     rank(A::AbstractMatrix; atol::Real=0, rtol::Real=atol>0 ? 0 : n*ϵ)
     rank(A::AbstractMatrix, rtol::Real)
 
-Compute the rank of a matrix by counting how many singular
-values of `A` have magnitude greater than `max(atol, rtol*σ₁)` where `σ₁` is
-`A`'s largest singular value. `atol` and `rtol` are the absolute and relative
+Compute the numerical rank of a matrix by counting how many outputs of
+`svdvals(A)` are greater than `max(atol, rtol*σ₁)` where `σ₁` is `A`'s largest
+calculated singular value. `atol` and `rtol` are the absolute and relative
 tolerances, respectively. The default relative tolerance is `n*ϵ`, where `n`
 is the size of the smallest dimension of `A`, and `ϵ` is the [`eps`](@ref) of
 the element type of `A`.
 
+!!! note
+    Numerical rank can be a sensitive and imprecise characterization of
+    ill-conditioned matrices with singular values that are close to the threshold
+    tolerance `max(atol, rtol*σ₁)`. In such cases, slight perturbations to the
+    singular-value computation or to the matrix can change the result of `rank`
+    by pushing one or more singular values across the threshold. These variations
+    can even occur due to changes in floating-point errors between different Julia
+    versions, architectures, compilers, or operating systems.
+
 !!! compat "Julia 1.1"
     The `atol` and `rtol` keyword arguments requires at least Julia 1.1.
     In Julia 1.0 `rtol` is available as a positional argument, but this
@@ -981,7 +990,7 @@ function rank(A::AbstractMatrix; atol::Real = 0.0, rtol::Real = (min(size(A)...)
     isempty(A) && return 0 # 0-dimensional case
     s = svdvals(A)
     tol = max(atol, rtol*s[1])
-    count(x -> x > tol, s)
+    count(>(tol), s)
 end
 rank(x::Union{Number,AbstractVector}) = iszero(x) ? 0 : 1