Redesign of triangular matrix types

[andreasnoack]

Instead of using parameters to specify if a triangular matrix is upper, lower and/or has unit diagonal I have decided to write out the four possibilities as types as suggested some time ago by @StefanKarpinski. The four types are UpperTriangular, LowerTriangular, UpperTriangularUnit, LowerTriangularUnit, and right now I have only exported the first two as the last two are probably less used.

The main reason for this change is that the use of parameters makes it cumbersome to work with the types in a way the compiler understands because the type parameters aren't types (:L, :U, true, false). The new Val type would solve this, but I think it is ugly to write Triangular(A, Val{:L}) compared to simply LowerTriangular and as we don't plan to extend the value beyond :L, :U, true and false, the type parameters give us nothing.

The problems with writing the old Triangular types in a way the compiler understands caused \ to have inferred return type Any for the performance critical case where both left and right hand side has the same BlasFloat elements as reported in JuliaLang/LinearAlgebra.jl#152. This PR fixes JuliaLang/LinearAlgebra.jl#152 and supersedes #9196.

cc: @jiahao

[tkelman]

I think I'd prefer UnitTriangular over TriangularUnit

[andreasnoack]

Yes. That is probably better.

[ViralBShah]

That does sound better.

[jiahao]

This will break a lot of packages.

Would be very helpful to create an entry in Compat.jl for backward compatibility.

[IainNZ]

Here are the ones that PackageEval found:
davidavdav/GaussianMixtures.jl#6
JuliaStats/MultivariateStats.jl#8
Rpedigree/Pedigrees.jl#1
brian-j-smith/Mamba.jl#22

[jiahao]

Formatting error in doc change fixed in e98b817

[davidavdav]

Yep---I'd be all for support in Compat.jl for backward compatibility.

[davidavdav]

@andreasnoack Would it make sense to export AbstractTriangular, this would make compatibility somewhat easier? I think I can resolve the references to Triangular in NamedArrays by replacing them by AbstractTriangular.

Now I have my own compat.jl

if VERSION < v"0.4.0-dev"
    typealias AbstractTriangular Triangular
else
    import Base.LinAlg.AbstractTriangular
end