Use of FastAnonymous.jl for ODE solvers.

[amitjamadagni]

This aims at speeding up the current implementation of ODE solvers using FastAnonymous.jl.

WIP : Perform benchmarks, Add documentation, Update dependencies in README.

[amitjamadagni]

Benchmarks :
Using Fast Anon : https://github.com/amitjamadagni/Notes-Notebooks/blob/master/Notebooks/ODE%20Solvers%20using%20Fast%20Anon.ipynb

Without Using Fast Anon : https://github.com/amitjamadagni/Notes-Notebooks/blob/master/Notebooks/ODE%20Solvers%20without%20Fast%20Anon.ipynb

Though there is a speed up with respect to ode45, not much efficiency is observed with ode78.

@acroy any thoughts on this would be helpful ! Thanks !

[acroy]

You can try to do scale!(coeffs(op)*y, -im), which should save one copy and some multiplications.

[amitjamadagni]

@acroy here are the results. There is a considerable speed up by the combination of @anon and scale!. Is this ready to go ?!

[acroy]

The speed up is nice, but mainly due to anon. Would be interesting to see a comparison with the other methods. I guess you haven't updated the docs etc?

[amitjamadagni]

@acroy I have re run the benchmarks with a branch check (using REQUIRE file as diff) and here are the benchmarks

1. Using fast anon : https://github.com/amitjamadagni/Notes-Notebooks/blob/master/Notebooks/Benchmarks%20using%20fast%20anon.ipynb
2. Without fast anon : https://github.com/amitjamadagni/Notes-Notebooks/blob/master/Notebooks/Benchmarks%20without%20fast%20anon.ipynb

For smaller systems use of anon is not really useful as the timing efficiency decreases, but as the system size increases there is an increase in timing efficiency.

[acroy]

Thanks! For smaller systems (N<=6) the degradation of performance is quite significant, which is a bit disappointing (factor 100 or so matters). I guess the construction of the "fast anonymous function" takes too much time. Could you maybe try without anon, but with scale!?

[amitjamadagni]

Here are the results without anon but scale!,

https://github.com/amitjamadagni/Notes-Notebooks/blob/master/Notebooks/Use%20of%20scale%20but%20not%20including%20Fast%20Anon.ipynb

seems like the original implementation is faster without anon and scale! for smaller systems, for larger systems anon with scale! is a better case (I must have mixed up the statements :(

[acroy]

It becomes much slower than the old version, which seems to be strange. Anyways, it is not clear what one should do now. We certainly don't want to spoil performance for small systems. Maybe we should leave this open until we understand the side-effects better?

[amitjamadagni]

yup ! I guess we need a better understanding of the individual additions i.e., scale!, anon and without scale! and also on the combination of these additions.