## About propagator derivative

### About propagator derivative

As far as I know, papers from Spinach show three ways to calculate 1st order propagator derivative: finite difference, commutator series and auxiliary method. I notice that the documentation says auxiliary method is better in accuracy and performance. But are there any published papers that talk about why "better"? Because I want to cite that in my paper as the motivation for auxiliary method

### Re: About propagator derivative

Well, better than finite difference because auxiliary matrix method is exact - it is not an approximation. And better than commutator series because the commutator series is a major pain to converge - there are many technical issues with double-precision arithmetic that require precise spectral radius estimation, etc. I would say that auxiliary matrix method is better logistically than the commutator series, just a lot less of a pain to use!

### Re: About propagator derivative

Thanks for answering!

If I understand correctly about the commutator series part, you are saying that it takes a lot of effort to solve or mitigate the rounding error in the evaluation of the commutator series. So It's nice to use the auxiliary method and apply the scaling and squaring method which has a systematic way to deal with the rounding errors?

If I understand correctly about the commutator series part, you are saying that it takes a lot of effort to solve or mitigate the rounding error in the evaluation of the commutator series. So It's nice to use the auxiliary method and apply the scaling and squaring method which has a systematic way to deal with the rounding errors?

### Re: About propagator derivative

Also, imagine you need the second derivative, like we do here (http://aip.scitation.org/doi/10.1063/1.4949534). Auxiliary matrix method remains simple, and a commutator series for the second derivative becomes an absolute horror.