inv is broken for Bidiagonal

[jakebolewski]

This fails in factorize:

Ex

julia> BD
5x5 Base.LinAlg.Bidiagonal{Float64}:
 -2.33818   0.0160447  0.0        0.0        0.0
  0.0      -1.86915    0.106191   0.0        0.0
  0.0       0.0        1.65496   -0.849353   0.0
  0.0       0.0        0.0        0.246264  -0.00214547
  0.0       0.0        0.0        0.0       -1.19128

julia> inv(BD)
ERROR: MethodError: `factorize` has no method matching factorize(::Base.LinAlg.Bidiagonal{Float64})
 in inv at linalg/generic.jl:237

[tkelman]

Would be resolved by JuliaLang/julia#8240

[andreasnoack]

@tkelman I don't think JuliaLang/julia#8240 is enough here. Bidiagonal matrices in LAPACK's band layout doesn't cover our present Bidiagonal layout so we would't be able to use a LAPACK routine here.

However, an inv(Bidiagonal) but the routine wouldn't be hard to write. But again, I'm not really sure we want inv(Bidiagonal). It really depends on the interpretation of the special matrices.

[tkelman]

Using lapack for bidiagonal (or tridiagonal) is pretty pointless and should not be held sacred here. I'd rather have a generic system than go out of our way to support an "optimized" path that doesn't actually matter for performance.

[andreasnoack]

I think that such a system would be great, but I also think that it is bit more than what JuliaLang/julia#8240 proposes.

[jiahao]

Note that the inverse of Bidiagonal will be in general Triangular.

[tkelman]

Oh right, duh, no need to convert to Tridiagonal. Just define factorize(A::Bidiagonal) = A and this should work.

[tkelman]

Also not implemented: multiplying Bidiagonal and a dense matrix