Speeding up series computations

[ericagol]

Right now the G_3 and H_1-H_8 functions are evaluated with series expansions when \gamma is small. These take a significant fraction of the time in the kepler+drift solver. Currently each of these functions is computed separately, and the coefficients are computed every time they are evaluated. We can greatly speed this up as follows:

• [ ] Write tests for evaluating these functions.
• [ ] Add these series to an appendix in the paper.
• [ ] Add the coefficients for each series to the Derivative structure.
• [ ] Pre-compute the coefficients once at the start of the simulation (and only if they have not been computed yet).
• [ ] Then evaluated all series simultaneously, so we save on the computation of \gamma^{2n}.

Some nuances:

• [ ] We don't know how many terms to include to reach a certain precision. Do we just compute a large number of terms and take the dot-product? Or do we have a while loop which terminates when machine precision is reached (this is what is currently being done?
• [ ] For the derivative-free integration we only need G_3, H_1, and H_2. With derivatives we need all series evaluated.
• [ ] Would it be useful to use StaticArrays for holding these coefficients?

[ericagol]

Well, after creating a function to pre-compute the coefficients (in the branch GHcoeff), store them in a structure, and then evaluate these three functions simultaneously, the end result is that it's faster to compute these directly! Not sure why... But closing this for now.