Implement ODE solvers

[ViralBShah]

A quick and easy way to get ode23 and ode45 like in MATLAB is:

http://www.netlib.org/ode/rksuite/

The state of the art is SUNDIALS, but it is no longer maintained. The lead developer went to develop video games! It is widely used in the national labs, and can give us ode15s.

https://computation.llnl.gov/casc/sundials/main.html

The long term strategy may simply be to implement these solvers in julia itself. The design of the Matlab ODE suite is described in:

http://www.mathworks.com/help/pdf_doc/otherdocs/ode_suite.pdf

[ViralBShah]

Move ode23 and ode45 from examples/ to j/ in commit 8475705. Can close this one once we have a basic ode15s implementation.

[rgbrown]

Just a comment -- since your initial post there has been a new release of SUNDIALS (in March this year -- the first in 3 years). Worth giving it another look?

[vtjnash]

@ViralBShah can we close this, as something for an eventual package contribution (or move it to a special place for issues pertaining to desirable packages)?

[ViralBShah]

I think we can close this. This will certainly be a package.

[stevengj]

There doesn't seem to be a package for this yet, and the ODE code mentioned above seems to have disappeared from Julia.

ODE solvers are bread and butter for numerics; this really needs to be done. It is especially important because this is an area where Julia can really shine: small systems of ODEs are difficult to vectorize, so in other dynamic languages you are basically forced to drop down to C if you need it to be fast.

[aviks]

The ODE code in extras is now in a package: https://github.com/vtjnash/ODE.jl

[pao]

As there is definitely a package, going to reclose this.

[stevengj]

Sorry, got fooled by the 0.1 package listing, which is missing ODE.jl.

Are there any benchmarks (accuracy vs. number of function evaluations) of how this compares with existing ODE packages (e.g. Matlab, GSL, SUNDIALS...)? i.e. Do we have a competitive implementation or something half-baked?

[pao]

I don't think so. Viral ported some of them from Octave; I added a few others from lecture notes from a dynamics simulation course I took. I think all we know is that they don't work particularly well in general (well, except for RK4). A good testsuite with appropriate problems would be a good place to start. Rewriting the whole thing is not out of the question; Julia's changed a lot in the past year or so.