`trace_test` fails for witness sets with real linear subspace

[oskarhenriksson]

The trace test fails for witness sets where the linear subspace L is a LinearSubspace{Float64}:

julia> L = LinearSubspace([1.0 0.0], [2.0]);

julia> W = witness_set([x^2+y^2-1], L)
Witness set for dimension 1 of degree 2

julia> trace_test(W)
ERROR: InexactError: Float64(2.7923774848451757 + 0.04248034231480903im)
Stacktrace:
  [1] Real
    @ ./complex.jl:44 [inlined]
  [2] convert
    @ ./number.jl:7 [inlined]
  [3] setindex!
    @ ./array.jl:985 [inlined]
  [4] macro expansion
    @ ./broadcast.jl:995 [inlined]
  [5] macro expansion
    @ ./simdloop.jl:77 [inlined]
  [6] copyto!
    @ ./broadcast.jl:994 [inlined]
  [7] copyto!
    @ ./broadcast.jl:947 [inlined]
  [8] materialize!
    @ ./broadcast.jl:905 [inlined]
  [9] materialize!
    @ ./broadcast.jl:902 [inlined]
 [10] translate!(L::LinearSubspace{Float64}, δb::Vector{ComplexF64}, ::Coordinates{:Extrinsic})
    @ HomotopyContinuation ~/.julia/packages/HomotopyContinuation/s3sko/src/linear_subspaces.jl:686
 [11] translate (repeats 2 times)
    @ ~/.julia/packages/HomotopyContinuation/s3sko/src/linear_subspaces.jl:682 [inlined]
 [12] trace_test(W₀::WitnessSet{MixedSystem{…}, LinearSubspace{…}, PathResult}; options::@Kwargs{})
    @ HomotopyContinuation ~/.julia/packages/HomotopyContinuation/s3sko/src/witness_set.jl:597
 [13] trace_test(W₀::WitnessSet{MixedSystem{(0xa6d77f86f464958b, 1)}, LinearSubspace{Float64}, PathResult})
    @ HomotopyContinuation ~/.julia/packages/HomotopyContinuation/s3sko/src/witness_set.jl:582
 [14] top-level scope
    @ REPL[194]:1
Some type information was truncated. Use `show(err)` to see complete types.

The problem seems to arise when trace_test runs translate and attempts to shift the vector L.extrensic.b (which is of type Vector{Float64}) by vector δb (which is of type Vector{ComplexF64}) here:

HomotopyContinuation.jl/src/linear_subspaces.jl

Lines 684 to 691 in 9ef422d

	function translate!(L::LinearSubspace, δb, ::Coordinates{:Extrinsic} = Extrinsic)
	ext = extrinsic(L)
	ext.b .+= δb
	int = intrinsic(L)
	LA.mul!(int.b, ext.A', δb, true, true)
	stiefel_coordinates_intrinsic!(int.Y, int.A, int.b)
	L
	end

[PBrdng]

Thanks. This is an easy fix: promote everything to complex.

[oskarhenriksson]

Do you think the promotion should happen in trace_test or translate! (or elsewhere)?

[PBrdng]

See #728