Specializing Broyden's method for sparse Jacobians

[ChrisRackauckas]

https://www.ams.org/journals/mcom/1970-24-109/S0025-5718-1970-0258276-9/

This might be putting it in the realm of NonlinearSolve instead, to keep the SimpleNonlinearSolve.jl version simple?

[isaacsas]

There are also updating QR factorization versions of Broyden I believe (instead of updating the inverse Jacobian approximation), but I'm not sure they can be made to work for sparse matrices. I think the dense version is described in Dennis and Schnabel.

[ChrisRackauckas]

Link?

I wonder if that's starting to get to the realm of NonlinearSolve.jl too.

[isaacsas]

https://epubs.siam.org/doi/book/10.1137/1.9781611971200

Section 8.3 references the updating QR I believe. (However, I doubt this is the best reference for immediately seeing a pseudo-code, there are probably better ones.)

I had searched around and someone had at least a subset of an updating QR in Julia I thought, but I'm having trouble finding it now.

[isaacsas]

Algorithm 3.4.1 gives pseudo-code.