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Paweł Biernat
@pwl
have you looked at high order Taylor methods?
Christopher Rackauckas
@ChrisRackauckas
I mean, I know how they're derived.
Paweł Biernat
@pwl
There was some package on github (now removed), that implemented arbitrary order Taylor methods, which I hooked up to ODE to test the backend API and the method, they worked!
Christopher Rackauckas
@ChrisRackauckas
Did you have to provide the derivatives?
Paweł Biernat
@pwl
ForwardDiff does it for you
Christopher Rackauckas
@ChrisRackauckas
But for really high derivatives, that doesn't give more overhead?
Paweł Biernat
@pwl
it does, but on the other hand you can make longer stepsizes
Christopher Rackauckas
@ChrisRackauckas
Depending on the function being "nice"
Paweł Biernat
@pwl
so it sort of balances out
Christopher Rackauckas
@ChrisRackauckas
With automatic differentiation... that's interesting. I'd like to test that out vs other methods.
Paweł Biernat
@pwl
I tested it with a hyperbolic PDE that I work on and it worked as quick as RK
Christopher Rackauckas
@ChrisRackauckas
I thought of using symbolic differentiation before with Taylor methods.
Paweł Biernat
@pwl
at least the same order of magnitude
the fun thing with this method is that you can prescribe an error much lower then the machine epsilon
like reltol 1e-30 with Float64
Christopher Rackauckas
@ChrisRackauckas
Probably scales better than extrapolation.
Paweł Biernat
@pwl
but this is a niche application
Christopher Rackauckas
@ChrisRackauckas
Well you'd need to go to other numbers to avoid weird truncation error problems at that point.
But a battle between high-order extrapolation, RK16, and 16th order Taylor series via automatic differentiation.
I'd want to try that out just for fun.
People in astrodynamics may find that useful.
Mauro
@mauro3
@PerezHz looks into taylor methods in his PhD with Benet & Sanders: https://github.com/JuliaLang/ODE.jl/issues/91#issue-139761284
Christopher Rackauckas
@ChrisRackauckas
But @pwl said the package is no longer available?
Paweł Biernat
@pwl
unfortunately the package disappeared
I still have a hard copy on my hd
Christopher Rackauckas
@ChrisRackauckas
Please share :smile:
It will have to be updated for sure.
I know ForwardDiff no longer exports the derivative functions, so someone will have to go in and change those kinds of things.
Paweł Biernat
@pwl
I don't know how @PerezHz would feel about it, maybe you should write him first?
Christopher Rackauckas
@ChrisRackauckas
Okay, I'll email him.
Thanks for the heads up on this
Paweł Biernat
@pwl
if you won't be able to reach him you can try to write @lbenet, he will know what the status of the package is
I think he is his advisor, but I'm not so sure about it now:-)
Jorge Pérez
@PerezHz
Hi everyone! @ChrisRackauckas thanks for contacting us, glad to know you're interested in what we've been working on! :smile: Yes, we did take it down, we're re-organizing it (it's pretty much mixed with some celestial mechanics stuff) and planning to re-upload the generic parts of the code
Christopher Rackauckas
@ChrisRackauckas
Okay cool!
I am currently wrapping a bunch of packages into DifferentialEquations.jl, and would like to have that as one of the options to choose from.
Just let me know when you got something you want to show.
Paweł Biernat
@pwl
Are there any plans to release the interval arithmetic ODE solver? I saw something like that on @lbenet presentation.
Jorge Pérez
@PerezHz
Sure Chris, we'll let you know!
Yes @pwl there are plans, we're still working on it; the thing is, the leap towards interval arithmetics is non-trivial enough, @lbenet can give you more details about that
Christopher Rackauckas
@ChrisRackauckas
Are you going to be using ArbReals? I know dpsanders is in on that project.
Jorge Pérez
@PerezHz
AFAIK, we'll be using @dpsanders ValidatedNumerics.jl
Christopher Rackauckas
@ChrisRackauckas
I see
This application would be perfect for ArbReals though.
Though Interval{Float64} would be faster for small calculations, Interval{BigFloat} would do better than ArbReals if you need 500+ bits of accuracy.
But in the middle, 64-~500 bits, ArbReals is the way to go.
I kow Sanders and Benet have been on the emails with Jeffrey Sarnoff on ArbFloats/ArbReals, so I wonder if that's why.
Felix Henneke
@fhenneke
I might be mistaken, but should the name fehlberg in runge-kutta-fehlberg not be spelled with an E instead of a U?
Christopher Rackauckas
@ChrisRackauckas
Yes.
Where's do I have the error?