Make svdvals(::Matrix{<:Complex}) type inferrable (#22443)

* Make svdvals(::Matrix{<:Complex}) type inferrable

Ensure that svdvals(zeros(Complex128,0,0)) returns a complex real matrix
to avoid type instability.  Also add some simplistic but explicit tests
for svdvals and svdfact, including ensuring this case is inferred.

* Add unitarity test for SVD U and Vt parts.

* Use \approxeq in tests instead of clobbering \approx

base/linalg/svd.jl

@@ -128,7 +128,7 @@ end
 Returns the singular values of `A`, saving space by overwriting the input.
 See also [`svdvals`](@ref).
 """
-svdvals!(A::StridedMatrix{T}) where {T<:BlasFloat} = findfirst(size(A), 0) > 0 ? zeros(T, 0) : LAPACK.gesdd!('N', A)[2]
+svdvals!(A::StridedMatrix{T}) where {T<:BlasFloat} = isempty(A) ? zeros(real(T), 0) : LAPACK.gesdd!('N', A)[2]
 svdvals(A::AbstractMatrix{<:BlasFloat}) = svdvals!(copy(A))
 
 """

test/linalg/svd.jl

@@ -4,6 +4,31 @@ using Base.Test
 
 using Base.LinAlg: BlasComplex, BlasFloat, BlasReal, QRPivoted
 
+@testset "Simple svdvals / svdfact tests" begin
+    ≊(x,y) = isapprox(x,y,rtol=1e-15)
+
+    m1 = [2 0; 0 0]
+    m2 = [2 -2; 1 1]/sqrt(2)
+    m2c = Complex.([2 -2; 1 1]/sqrt(2))
+    @test @inferred(svdvals(m1))  ≊ [2, 0]
+    @test @inferred(svdvals(m2))  ≊ [2, 1]
+    @test @inferred(svdvals(m2c)) ≊ [2, 1]
+
+    sf1 = svdfact(m1)
+    sf2 = svdfact(m2)
+    @test sf1.S ≊ [2, 0]
+    @test sf2.S ≊ [2, 1]
+    # U & Vt are unitary
+    @test sf1.U*sf1.U'   ≊ eye(2)
+    @test sf1.Vt*sf1.Vt' ≊ eye(2)
+    @test sf2.U*sf2.U'   ≊ eye(2)
+    @test sf2.Vt*sf2.Vt' ≊ eye(2)
+    # SVD not uniquely determined, so just test we can reconstruct the
+    # matrices from the factorization as expected.
+    @test sf1.U*Diagonal(sf1.S)*sf1.Vt' ≊ m1
+    @test sf2.U*Diagonal(sf2.S)*sf2.Vt' ≊ m2
+end
+
 n = 10
 
 # Split n into 2 parts for tests needing two matrices