Improve printing of LDLt factorization

[goggle]

This PR improves the REPL printing of LDLt factorizations as proposed in #24588.

using LinearAlgebra
S = SymTridiagonal([1., 2., 3.], [-1., 1.])

Prior to this PR it looked like this:

julia> F = ldlt(S)
LDLt{Float64,SymTridiagonal{Float64,Array{Float64,1}}}([1.0 -1.0 0.0; -1.0 1.0 1.0; 0.0 1.0 2.0])

With this PR it now looks like this:

LDLt{Float64,SymTridiagonal{Float64,Array{Float64,1}}}
L factor:
3×3 UnitLowerTriangular{Float64,SymTridiagonal{Float64,Array{Float64,1}}}:
  1.0   ⋅    ⋅ 
 -1.0  1.0   ⋅ 
  0.0  1.0  1.0
diagonal values:
3-element Array{Float64,1}:
 1.0
 1.0
 2.0

Instead of printing diagonal values, we could call it D factor and print a diagonal matrix. Let me know which method is preferred.

I have also added a new test file stdlib/LinearAlgebra/test/ldlt.jl to add a basic test for this.

[andreasnoack]

Thanks for adding this. I think it would be a good idea to have the extractors (getproperty) defined since useeers would probably expect to have a way to extract the factors when they are printed likee that. Would you be up for adding a getproperty method as well and use it in the show method?

[goggle]

@andreasnoack Yes sure, I will try to do this. If I have questions, I will ask here.

[goggle]

Ok, so I have added a getproperty method for LDLt.

using LinearAlgebra
S = SymTridiagonal([1., 2., 3.], [-1., 1.]);

The factorization looks like this:

julia> F = ldlt(S)
LDLt{Float64,SymTridiagonal{Float64,Array{Float64,1}}}
L factor:
3×3 UnitLowerTriangular{Float64,SymTridiagonal{Float64,Array{Float64,1}}}:
  1.0   ⋅    ⋅ 
 -1.0  1.0   ⋅ 
  0.0  1.0  1.0
D factor:
3×3 Diagonal{Float64,Array{Float64,1}}:
 1.0   ⋅    ⋅ 
  ⋅   1.0   ⋅ 
  ⋅    ⋅   2.0

I'm not sure if the Lt should also be shown on that output...

I'm also not sure about the types that should be returned. Currently it is like this:

Component	Type
F.L	UnitLowerTriangular
F.D	Diagonal
F.Lt	UnitUpperTriangular
F.d	Array{T, 1}

If that should be changed, let me know.

Note that something like

julia> F.L*F.D*F.Lt

does not work:

ERROR: ArgumentError: cannot set off-diagonal entry (2, 1)
Stacktrace:
 [1] copyto!(::SymTridiagonal{Float64,Array{Float64,1}}, ::Base.Broadcast.Broadcasted{LinearAlgebra.StructuredMatrixStyle{SymTridiagonal{Float64,Array{Float64,1}}},Tuple{Base.OneTo{Int64},Base.OneTo{Int64}},typeof(*),Tuple{SymTridiagonal{Float64,Array{Float64,1}},Array{Float64,2}}}) at /home/alex/Projects/github/goggle/julia/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/tridiag.jl:385
 [2] materialize! at ./broadcast.jl:817 [inlined]
 [3] rmul!(::SymTridiagonal{Float64,Array{Float64,1}}, ::Diagonal{Float64,Array{Float64,1}}) at /home/alex/Projects/github/goggle/julia/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/diagonal.jl:175
 [4] rmul!(::UnitLowerTriangular{Float64,SymTridiagonal{Float64,Array{Float64,1}}}, ::Diagonal{Float64,Array{Float64,1}}) at /home/alex/Projects/github/goggle/julia/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/diagonal.jl:187
 [5] * at /home/alex/Projects/github/goggle/julia/usr/share/julia/stdlib/v1.3/LinearAlgebra/src/diagonal.jl:165 [inlined]
 [6] *(::UnitLowerTriangular{Float64,SymTridiagonal{Float64,Array{Float64,1}}}, ::Diagonal{Float64,Array{Float64,1}}, ::UnitUpperTriangular{Float64,SymTridiagonal{Float64,Array{Float64,1}}}) at ./operators.jl:529
 [7] top-level scope at REPL[8]:1

I would have assumed that the multiplication gets dispatched to a more generic method if there is no specialized method... What's the reason for this? Is it some bug?

[andreasnoack]

Great work. The error is probably because we don't have specialized methods for UnitLowerTriangular{Float64,SymTridiagonal{Float64,Array{Float64,1}}}. Eventually, it would be good to have them but it can go into a separate PR.