As we only need an approximation of the Jacobian and even the Jacobian can be frozen for several integration steps, how exact is the Jacobian is not extremely important (that's also why we can use the coarse forward differentiation formula).
A suggestion to speed up the code is to remove those values of the Jacobian that are close to zero (negligible). This approach, could speed up the code because sparse solvers could be used in this case, instead of the full matrix solvers (and less memory would be used too). This idea has been implemented successfully in other fields (in machine learning when optimizing the AUC).
As we only need an approximation of the Jacobian and even the Jacobian can be frozen for several integration steps, how exact is the Jacobian is not extremely important (that's also why we can use the coarse forward differentiation formula).
A suggestion to speed up the code is to remove those values of the Jacobian that are close to zero (negligible). This approach, could speed up the code because sparse solvers could be used in this case, instead of the full matrix solvers (and less memory would be used too). This idea has been implemented successfully in other fields (in machine learning when optimizing the AUC).