Improve svd gaugefixing performance (#238)

* Improve svd gaugefixing performance

* Fixes

* Reduce allocs

* Restore the alloc

* Use abs2 for complex comparisons

Author: kshyatt

src/common/gauge.jl

@@ -10,7 +10,8 @@ is real and positive.
 
 # Helper functions
 _argmaxabs(x) = reduce(_largest, x; init = zero(eltype(x)))
-_largest(x, y) = abs(x) < abs(y) ? y : x
+_largest(x::Real, y::Real) = abs(x) < abs(y) ? y : x
+_largest(x::Complex, y::Complex) = abs2(x) < abs2(y) ? y : x
 
 function gaugefix!(::typeof(qr_householder!), Q, R, Rd)
     ax = Base.OneTo(length(Rd))
@@ -67,12 +68,10 @@ end
 
 function gaugefix!(::Union{typeof(svd_compact!), typeof(svd_trunc!)}, U, Vᴴ)
     @assert axes(U, 2) == axes(Vᴴ, 1)
-    for j in axes(U, 2)
-        u = view(U, :, j)
-        v = view(Vᴴ, j, :)
-        s = sign(_argmaxabs(u))
-        u .*= conj(s)
-        v .*= s
-    end
+    signs = reduce(_largest, U; dims = 1, init = zero(eltype(U)))
+    @. signs = sign(signs)
+    signs_t = transpose(signs)
+    @. U = U * conj(signs)
+    @. Vᴴ = signs_t * Vᴴ
     return (U, Vᴴ)
 end