implements a rank(::SVD) method and adds unit tests. fixes #1126 (#1127)

* added rank method for SVD object alongwith unit tests for the method. issue #1126

* simplified DOCSTRING for rank(::SVD)

* simplified rank(::SVD) implementation

Author: ajinkya-k

src/svd.jl

@@ -245,6 +245,24 @@ svdvals(A::AbstractVector{T}) where {T} = [convert(eigtype(T), norm(A))]
 svdvals(x::Number) = abs(x)
 svdvals(S::SVD{<:Any,T}) where {T} = (S.S)::Vector{T}
 
+"""
+    rank(S::SVD{<:Any, T}; atol::Real=0, rtol::Real=min(n,m)*ϵ) where {T}
+
+Compute the numerical rank of the SVD object `S` by counting how many singular values are greater 
+than `max(atol, rtol*σ₁)` where `σ₁` is the largest calculated singular value.
+This is equivalent to the default [`rank(::AbstractMatrix)`](@ref) method except that it re-uses an existing SVD factorization.
+`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 UΣV'
+and `ϵ` is the [`eps`](@ref) of the element type of `S`.
+
+!!! compat "Julia 1.12"
+    The `rank(::SVD)` method requires at least Julia 1.12.
+"""
+function rank(S::SVD; atol::Real = 0.0, rtol::Real = (min(size(S)...)*eps(real(float(eltype(S))))))
+    tol = max(atol, rtol*S.S[1])
+    count(>(tol), S.S)
+end
+
 ### SVD least squares ###
 function ldiv!(A::SVD{T}, B::AbstractVecOrMat) where T
     m, n = size(A)

test/svd.jl

@@ -294,4 +294,16 @@ end
     @test F.S ≈ F32.S
 end
 
+@testset "rank svd" begin
+    # Test that the rank of an svd is computed correctly
+    @test rank(svd([1.0 0.0; 0.0 1.0])) == 2
+    @test rank(svd([1.0 0.0; 0.0 0.9]), rtol=0.95) == 1
+    @test rank(svd([1.0 0.0; 0.0 0.9]), atol=0.95) == 1
+    @test rank(svd([1.0 0.0; 0.0 1.0]), rtol=1.01) == 0
+    @test rank(svd([1.0 0.0; 0.0 1.0]), atol=1.01) == 0
+
+    @test rank(svd([1.0 2.0; 2.0 4.0])) == 1
+    @test rank(svd([1.0 2.0 3.0; 4.0 5.0 6.0 ; 7.0 8.0 9.0])) == 2
+end
+
 end # module TestSVD