Allow Banded Jacobian Matrix

[YingboMa]

It will be really nice to add banded Jacobian matrix support in NLsolve.jl.

[ChrisRackauckas]

I think this would be mostly solved by

#83

since the user can just set it to use a banded linear solver. That doesn't fix the memory usage though, so it would still be nice to allow the user to give a type which is the suggested Jacobian type. This would be a nice way IMO to handle sparsity and other things like Triangular Jacobians too.

[YingboMa]

I suggest that there should be a functionality like nlsolve(f!, u0, jac_upper=N, jac_lower=M), so user doesn't need to calculate the Jacobian matrix explicitly and factorize the banded Jacobian from there.

[pkofod]

Why would that be superior to simply supplying Jacobians of any type?

[ChrisRackauckas]

It wouldn't.

[pkofod]

This should work now. If it doesn't, it's a bug and we'll reopen. The syntax is to supply the OnceDifferentiable instance directly

OnceDifferentiable(f!, j!, fj! x, F, J)

where F is a typical storage for the residuals and J is the banded matrix