Inclusion of higher-order and uncertainty heuristics for accelerated Jacobian-free inexact Newton convergence with the PIES code

We have investigated a novel Jacobian-free inexact Newton method for use with the Princeton Iterative Equilibrium Solver (PIES code). This enhanced method makes use of heuristics for uncertainties and higher-order terms in the non-linear objective function in order to optimize between (conventional) "compounding" and "reorthnormalization" adaptive preconditioning; equivalently, this method can be thought of as an enhanced Broyden's method, one which aims to produce a more accurate approximation of the Jacobian of the objective function based on the available numerical information. A speed-up over conventional methods is expected.