Numerical Jacobians

As noted in the W-method and the Rosenbrock Triple, Rosenbrock formulations require the explicit definition of the Jacobians for the ODEs. Analytical formulations for the Jacobians can be easily performed for simple ODE’s like the one presented in the examples earlier. However, the analytical definition of the Jacobians imposes a demanding restriction on the user to “create” such matrices and then program them. (Granted, one can easily avoid all the trouble of recreating the integration methodologies and resort to ODE subroutines from known sources such as Matlab, but what fun would that be…).

An alternative approach would be to numerically evaluate the Jacobians, as recommended by Shampine & Reichelt (The Matlab ODE Suite). Subroutine “fdjac” recommended by Press et al (Numerical Recipes) is one such algorithm that can be used for the numerical evaluation. “fdjac” uses a forward-difference approximation for the Jacobian calculations.