add issingular

[ChrisRackauckas]

@YingboMa do you have ideas for other dispatches that would be good? Should there be a fallback?

[YingboMa]

It would be good to have UpperTriangular, LowerTriangular, Symmetric, and Hermitian, too. For the Hermitian type we can use isposdef, and Symmetric can use bunchkaufman.

Also, could you add some tests?

[ChrisRackauckas]

Can you take it from here? It seems you have a bunch of ideas so that might be quicker.

[YingboMa]

Done. @ChrisRackauckas could you take a look?

[dkarrasch]

One could add here

issingular(B::Bidiagonal) = any(iszero, A.dv)
issingular(S::SymTridiagonal) = diaganyzero(iszero, ldlt(S).data)
issingular(T::Tridiagonal) = !issuccess(lu(A, check=false))

And perhaps as a fallback (which fulfills the promise of the docstring)

issingular(A::AbstractMatrix) = issingular(Matrix(A))
issingular(A::AbstractSparseMatrix) = !issuccess(lu(A, check=false))

[ChrisRackauckas]

Note I reverted the README change @YingboMa but it's expected that issingular is the best way to check for it: if it's ever faster to use rank, then the dispatch should be to use rank.