Reorganize and cleanup to match Julia v1 conventions

[Omar-Elrefaei]

Implement StridedArray interface for compatibility with BLAS/LAPACK operations.

This allows all QuasiTriangular matrisies to work with LinearAlgebra.BLAS operations or our ones in ExtendedMul (in fact both do a ccall to libblas)

I think this practically makes obsolete all A_mul_B functions except the ones that specify extra arguments (ma, nc, etc...) for submatrix operations.

Reference: https://docs.julialang.org/en/v1/manual/interfaces/#man-interface-strided-arrays

[MichelJuillard]

It is more complicated than that. We could only use BLAS/LAPACK operations if the storage of QuasiUpperTriangular matrix was quasi upper triangular. It is not the way that Triangular matrices work in Julia

using LinearAlgebra
julia> UT = UpperTriangular(rand(3,3))
3×3 UpperTriangular{Float64, Matrix{Float64}}:
 0.249811  0.439284  0.218579
  ⋅                0.378951  0.888622
  ⋅                ⋅                 0.467821

julia> UT.data
3×3 Matrix{Float64}:
 0.249811  0.439284  0.218579
 0.689945  0.378951  0.888622
 0.540507  0.720874  0.467821

Only specialized algorithms that ignore the lower triangle can produce correct results. BLAS/LAPACK operations on UT.data would return wrong results.

In addition for QuasiUpperTriangular, we have to make sure that there is no contiguous nonzero elements on the first sub-diagonal.

[Omar-Elrefaei]

Decisions log

• Tests: mul!(C, Q, B) ≈ mul!(C, T, B) put our Quasi version on the left hand side to assert that we are not mutating B mistakingly (as some of the original functions where).
• Deleted optimized mul implementations that used BLAS for a more generic version for now.
  • Bring back from git when needed. (Specialize methods on things like StridedArray, BlasFloat)
• Remove the in-place functions A_mul_B!(A, B) that mutated A or B. There should be lmul! or rmul!. To be implemented later if needed. I actually leveraged their implementation to write the 3-arg mul!(C, A, B) in this pr.

Conventions

• Formatting with JuliaFormatter SciMl style
• Capitalize matrices A as opposed to vectors a
• Use adjA to represent A'. As used in stdlib and to be able to capitalize the A. (as opposed to aAdj)

Learning notes:

• If QuasiTriangular extends LinearAlgebra then QuasiTriangular.mul! and LinearAlgebra.mul! dispatch to the same method table no matter the type of inputs.
• Since QuasiUpperTriangular is a subtype of AbstractMatrix, any functions not defined "safely" to the ones defined in LinearAlgebra.
  • This is true since getindex is overridden to ignore any values below the subdiagonal, and setindex is overridden to disallow assigning values below the subdiagonal.