Progressive Paths has four important advantages. First, the building blocks of the method, time paths of particular economies, are "natural" concepts, not artificially introduced for computational purposes. Second, the method exploits the user's knowledge of linear systems of ordinary differential equations. The analysis relies only on systems of ODE's - no PDE's or functional equations. Third, Progressive Paths can be used to study decentralized dynamic equilibria because the focus is on a system of differential equations - not on a Bellman equation or value function. Finally, the programming required is quite minimal; actual application of the method needs only for the user to write loops that iterate difference equations! Therefore, the prerequisites are relatively modest, especially from the computer science and numerical mathematics departments.
The paper has a self-contained discussion of implicit finite difference methods as applied to systems of ODE's as well as two MATLAB programs that implement some of the methods.