Linear Algebra operations with matrices of abstract element types are promoted incorrectly

[jakebolewski]

Follow up to #285.

Ex.

julia> a = Real[1.0 big(realmax(Float64))+1
       1.0 big(1.0)]
2x2 Array{Real,2}:
 1.0  1.797693134862315708145274237317043567980705675258449965989174768031572607800285e+308
 1.0  1.000000000000000000000000000000000000000000000000000000000000000000000000000000

julia> eigfact(a)
Base.LinAlg.Eigen{Float64,Float64,Array{Float64,2},Array{Float64,1}}([1.3407807929942597e154,-1.3407807929942597e154],2x2 Array{Float64,2}:
 1.0           -1.0
 7.45834e-155   7.45834e-155)

To be correct, this would have to widen the element type in a manner similar to map.

cc: @andreasnoack

[dkarrasch]

One could resolve this by defining

eigtype(A::AbstractArray{T}) where T = isconcretetype(T) ? eigtype(T) : eigtype(mapreduce(a -> typeof(zero(a)/sqrt(abs2(one(a)))), promote_type, A))

while keeping

eigtype(T) = promote_type(Float32, typeof(zero(T)/sqrt(abs2(one(T)))))

and then call eigtype(A) throughout. For the specific example, though, this will make eigen fail because we don't have a generic eigendecomposition.

[StefanKarpinski]

Wow, that may be the longest time between one comment and another on this repo. Eight years!