Separate parent matrix type from Cholesky parameter

[jishnub]

This adds an extra type parameter to PDMat corresponding to the parent matrix type. This lets the PDMat constructor succeed even if the parent type doesn't match those of the Cholesky factors. E.g. the following works after this:

julia> S = SymTridiagonal(fill(4,4), fill(1,3))
4×4 SymTridiagonal{Int64, Vector{Int64}}:
 4  1  ⋅  ⋅
 1  4  1  ⋅
 ⋅  1  4  1
 ⋅  ⋅  1  4

julia> PDMat(S)
4×4 PDMat{Float64, Bidiagonal{Float64, Vector{Float64}}, SymTridiagonal{Float64, Vector{Float64}}}:
 4.0  1.0  0.0  0.0
 1.0  4.0  1.0  0.0
 0.0  1.0  4.0  1.0
 0.0  0.0  1.0  4.0

where the Cholesky factors are Bidiagonal, but the parent is a SymTridiagonal.

[andreasnoack]

Thanks. I'm wondering if this susggests that https://github.com/JuliaLang/julia/blob/197295c84aab217bffde01a4339f1d0e8e95fb98/stdlib/LinearAlgebra/src/tridiag.jl#L957 should use SymTridiagonal instead of Bidiagonal.

[devmotion]

Is this issue fixed by #188? I think if we make a breaking change to the type structure, we should also address the other problems covered by #188 at the same time.

[devmotion]

I just checked and the example is indeed fixed by #188. I added the test in this PR to #188.

[jishnub]

Thanks, I'll close this in favor of that PR