Type: New Feature
Affects Version/s: None
Fix Version/s: 4.X
The existing finite difference framework in commons math is a limiting because it accepts only fixed bandwidth parameters. Furthermore, the finite difference coefficients/descriptions are not exposed to the user in any reasonable fashion (e.g. a user doing a numerical ODE solve probably wants to just grab suitable coefficients from somewhere).
Conceptually, I think the work of finite difference can be broadly divided into three tasks:
1. Generation of finite difference coefficients. Again, one should be able to do this and get the results outside of the context of taking an actual derivative. Ideally, we could generate coefficients for any flavor (forward, central, backward) and order.
2. Selection of the bandwidth. This is, to be honest, the trickiest part of computing a numerical derivative. There is some "art" to picking a proper bandwidth that will generate an accurate numerical derivative - there are two competing sources of error (roundoff, due to the finite representation of floating points; and truncation, due to the inherent nature of finite differences). Ideally, we want to pick a bandwidth that will minimize the total error.
3. Actually computing the finite difference derivative estimate. This is really easy once you have 1. and 2.
4. Extend 1-3 to include support for multivariate finite differences.