Type piracy in defining `qr` for `Tridiagonal`

[jishnub]

BandedMatrices.jl/src/banded/bandedqr.jl

Line 4 in 4c4a5ad

qr(A::Tridiagonal{T}) where T = qr!(BandedMatrix{float(T)}(A, (1,2)))

This is an egregious example of type piracy. Do we really need this here? Loading BandedMatrices changes the type of the result.

julia> A = Tridiagonal([1:2;], [1:3;], [1:2;])
3×3 Tridiagonal{Int64, Vector{Int64}}:
 1  1  ⋅
 1  2  2
 ⋅  2  3

julia> qr(A)
LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}
Q factor: 3×3 LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}
R factor:
3×3 Matrix{Float64}:
 -1.41421  -2.12132  -1.41421
  0.0      -2.12132  -3.29983
  0.0       0.0      -0.333333

julia> using BandedMatrices

julia> qr(A)
QR{Float64, BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}, Vector{Float64}}
Q factor: 3×3 LinearAlgebra.QRPackedQ{Float64, BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}, Vector{Float64}}
R factor:
3×3 BandedMatrix{Float64} with bandwidths (1, 2):
 -1.41421  -2.12132  -1.41421
  0.0      -2.12132  -3.29983
   ⋅        0.0      -0.333333

[dlfivefifty]

I don't remember why this was added but note that the standard qr(::Tridiagonal) is basically useless....

We could change this to be a new BandedMatrices.qr and see if that breaks anything?