some LAPACK glitches

[bsxfan]

• This was already mentioned in closed issue Provide logabsdet(::Matrix)? LinearAlgebra.jl#13: a failed cholesky factorization tries to throw PosDefException, but finds it is undefined.

• Similar problems occur with DimensionMismatch, e.g.:

  julia> lufact(randn(3,3)) \ randn(2)
  ERROR: in getrs!: DimensionMismatch not defined
  

• Method signatures for \ involving Cholesky and LU differ and thus cause difficulties. In factorization.jl, we have:

  \{T<:BlasFloat}(C::Cholesky{T}, B::StridedVecOrMat{T}) = ...
  

  but

  function (\)(A::LU, B::StridedVecOrMat)    
    ...
  

  Now try:

  julia> cholfact(eye(3)) \ [1,2,3]
  ERROR: no method \(Cholesky{Float64},Array{Int32,1})
  

  but

  julia> lufact(eye(3)) \ [1,2,3]
  ERROR: no method getrs!(Char,Array{Float64,2},Array{Int32,1},Array{Int32,1})
  

  This difference is harmless until you try to extend the applicability of \. For Cholesky, one can do e.g.:

  (\){T<:BlasFloat,S<:Union(Integer,Rational)}(C::Cholesky{T},B::Vector{S}) =
  C\convert(Vector{T},B)
  

But if you try similar for LU, you will just get a stackoverflow, because the signature (\)(A::LU, B::StridedVecOrMat) is too general and will not be invoked after conversion.

[ViralBShah]

Cc: @jiahao @andreasnoackjensen

The exception stuff is probably due to the recent refactoring of the exceptions code.

[ViralBShah]

Can this be closed?

[bsxfan]

OK, I've tried my examples above on the new build. They work now, except for a very minor issue. The error type definitions are now reachable, but some of the error messages in LAPACK are missing, have bad grammar or are misleading. Examples:

julia> lufact(randn(3,3)) \ randn(2)
ERROR: DimensionMismatch("Matrix must be square")

The LHS is square as it should be, but does not match the RHS

julia> cholfact((X->X*X')(randn(3,4)))\randn(2)
ERROR: DimensionMismatch("Left and right hand side does not fit")

maybe this should rather be "Left and right hand sides do not fit"?

julia> qrfact(randn(3,3))\randn(2)
ERROR: DimensionMismatch("")

These are hardly bugs and probably won't bother anyone.