Add missing constructors for Bidiagonal, Symmetric, and Hermitian

[mcognetta]

Bidiagonal, Symmetric, and Hermitian are currently all missing the X(A::Matrix, uplo::Char) constructor but have the X(A::Matrix, uplo::Symbol) constructor. This PR adds the Char constructor.

Two additional methods Symmetric(A::AbstractArray) = Symmetric(A, :U) and Hermitian(A::AbstractMatrix) = Hermitian(A, :U) were also added to avoid the method overwriting warning (due to the previous implementation having default arguments) and to preserve the historical default argument.

NOTE: This PR should be submitted after #42466, which adds better uplo checks to the constructors.

---

HEAD:

julia> Bidiagonal(rand(3,3), :U)
3×3 Bidiagonal{Float64,Array{Float64,1}}:
 0.971135  0.423251   ⋅ 
  ⋅        0.528393  0.696667
  ⋅         ⋅        0.343728

julia> Bidiagonal(rand(3,3), 'U')
ERROR: MethodError: no method matching Bidiagonal(::Array{Float64,2}, ::Char)
Closest candidates are:
  Bidiagonal(::AbstractArray{T,2} where T, ::Symbol) at /home/mc/github/julia/usr/share/julia/stdlib/v1.5/LinearAlgebra/src/bidiag.jl:101
Stacktrace:
 [1] top-level scope at REPL[3]:1

This PR:

julia> Bidiagonal(rand(3,3), :U)
3×3 Bidiagonal{Float64, Vector{Float64}}:
 0.87165  0.860882   ⋅ 
  ⋅       0.539619  0.278167
  ⋅        ⋅        0.4687

julia> Bidiagonal(rand(3,3), 'U')
3×3 Bidiagonal{Float64, Vector{Float64}}:
 0.499547  0.862715   ⋅ 
  ⋅        0.556194  0.751918
  ⋅         ⋅        0.706695

and likewise for Symmetric, and Hermitian.

[mcognetta]

Failures look unrelated.

[N5N3]

1. If Char constructor is introduced to Hermitian, you'd better to add it to hermitian to keep consistency.
2. Symbol constructor checks :L, :U via char_uplo, so I think we should do similar check in Char constructor.
3. As for Symmetric(A::AbstractArray) = Symmetric(A, :U), since

julia> f(a,b::Int = 1) = 1
f (generic function with 2 methods)
julia> f(a,b::Float64) = 2
f (generic function with 3 methods)
julia> f(1,1), f(1), f(1,1.0)
(1, 1, 2)

I think you only need to add the new constructor?
4. I also wonder why you say it's missing. Can you give us an example that Char constructor is better than Symbol constructor?

[mcognetta]

1. If Char constructor is introduced to Hermitian, you'd better to add it to hermitian to keep consistency.

Done. Actually, this and symmetric aren't exported. Perhaps they should be.

2. Symbol constructor checks :L, :U via char_uplo, so I think we should do similar check in Char constructor.

This is done in #42466. I will add a note not to submit this one until that is merged.

3. As for Symmetric(A::AbstractArray) = Symmetric(A, :U), since

julia> f(a,b::Int = 1) = 1
f (generic function with 2 methods)
julia> f(a,b::Float64) = 2
f (generic function with 3 methods)
julia> f(1,1), f(1), f(1,1.0)
(1, 1, 2)

I think you only need to add the new constructor?

Doing it this way causes a Method Overwriting warning during compilation (doesn't happen in the REPL I guess, but it happens when running make for example.). This discusses it (but, at the time, the warning would still appear in REPL. the same code does not produce the warning in REPL at HEAD):
https://discourse.julialang.org/t/method-definition-overwritten-warning-for-functions-with-default-arguments/1710

4. I also wonder why you say it's missing. Can you give us an example that Char constructor is better than Symbol constructor?

It's missing in the sense that some of the constructors allow Symbol and Char but some don't. I don't think Char is better, but adding those constructors makes the API more consistent.

---

Thanks for the comments.

[mcognetta]

Note that this should be submitted after #42466, so that the correct uplo checks are included in the base constructors.

[dkarrasch]

You can rebase onto current master now.

[dkarrasch]

Is this a new issue or has it always been there but we ignored the warning?

[mcognetta]

It is new if we add another method with a different default parameter type. The warning was not there before because there was only one Hermatian function with a default type parameter.

[dkarrasch]

I'm not sure, but I think not exporting symmetric and hermitian was done on purpose. Unless we have a good reason, I wouldn't export them, because you can do what you want by calling the wrapper constructors with capitalized initial letters.

[andreasnoack]

It's on purpose that we don't have Char constructors. The idea was that Char was just the internal representation whereas Symbol was the public API.

[mcognetta]

It's on purpose that we don't have Char constructors. The idea was that Char was just the internal representation whereas Symbol was the public API.

Oh, I see. In that case, should I close this PR?