NLEIGS uses a cache to store factorization objects, e.g., in the code in src/rk_helper/linsolvercache.jl. In NewLinSolvers.jl the FactorizeLinSolverCreator provides the same functionality.
Therefore, I propose we remove the functionality in nleigs and base it on FactorizeLinSolver.
This has to be done carefully so we don't loose performance.