Add docstring to the rank(::QRPivoted) method

With the release of Julia 1.12, LinearAlgebra.jl is adding dispatches for rank to SVD and QRPivoted objects, allowing the re-use of existing matrix factorizations where they have already been computed. The SVD method already has an appropriate docstring; this PR adds a near-identical one to the QRPivoted method as well, with some additional context for how it differs from the default SVD-based computation.

Author: Luis-Varona

src/qr.jl

@@ -535,6 +535,28 @@ function ldiv!(A::QRCompactWY{T}, B::AbstractMatrix{T}) where {T}
     return B
 end
 
+"""
+    rank(A::QRPivoted{<:Any, T}; atol::Real=0, rtol::Real=min(n,m)*ϵ) where {T}
+
+Compute the numerical rank of the QR factorization `A` by counting how many diagonal entries of
+`A.factors` are greater than `max(atol, rtol*Δ₁)` where `Δ₁` is the largest calculated such entry.
+This is similar to the [`rank(::AbstractMatrix)`](@ref) method insofar as it counts the number of
+(numerically) nonzero coefficients from a matrix factorization, although the default method uses an
+SVD instead of a QR factorization. Like [`rank(::SVD)`](@ref), this method also re-uses an existing
+matrix factorization.
+
+Using a QR factorization to compute rank should typically produce the same result as using SVD,
+although it may be more prone to overestimating the rank in pathological cases where the matrix is
+ill-conditioned. It is also worth noting that it is generally faster to compute a QR factorization
+than it is to compute an SVD, so this method may be preferred when performance is a concern.
+
+`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`.
+
+!!! compat "Julia 1.12"
+    The `rank(::QRPivoted)` method requires at least Julia 1.12.
+"""
 function rank(A::QRPivoted; atol::Real=0, rtol::Real=min(size(A)...) * eps(real(float(eltype(A)))) * iszero(atol))
     m = min(size(A)...)
     m == 0 && return 0