A_<op>_B Inconsistencies

[mihirparadkar]

The following code demonstrates some major inconsistencies with the A_*_B functions.

#Create a sparse positive semidefinite matrix
A = sparse(5*eye(5) - ones(5,5))
#Make it positive definite and guarantee its symmetry
AI = Symmetric(A + eye(A))

#I can factorize this just fine:
cholfact(AI, shift=1.)

#This works
A_ldiv_Bt(cholfact(AI, shift=1.), randn(2,5))

#So does this
cholfact(AI, shift=1.) \ randn(5,2)

#This doesn't (MethodError)
At_ldiv_B(cholfact(AI, shift=1.), randn(5,2))

#This also doesn't, same MethodError
A_ldiv_B!(cholfact(AI, shift=1.), randn(5,2))

I didn't try all of the possible combinations, but I don't know how to figure out which ones are implemented and which aren't.
Is this documented somewhere?

[andreasnoack]

There are some limitations in the library that we use for sparse Cholesky which mean that A_ldiv_B! cannot be implemented. However, the documentation for cholfact doesn't mention A_x_B methods. It positively mentions functions that are supported and therefore solving is supposed to work for all possible combinations of op(F)\op(B) where op can be either nothing or '. Therefore, we should define a fallback method for At_ldiv_B that calls Ac_ldiv_B in the real case and something like At_ldiv_B(F,b)=conj.(F'\conj.(b)) otherwise.

We could also update the documentation for A_ldiv_B! to mention that sparse matrices are not supported. Other suggestions for documentation improvements are also very welcome and it would be great if you could open pull request for them.

[fredrikekre]

Would it make sense to implement A_ldiv_B! for sparse using lufact, since A_ldiv_B!(lufact(A), b) already works? And in addition, throw a descriptive error for A_ldiv_B!(cholfact(A), b)?

[ViralBShah]

Since all these APIs are greatly changed, I'm closing this. Current issues with sparse linear algebra should be filed at https://github.com/JuliaSparse/SuiteSparse.jl/issues