I partly meant to suggest that PowerSeries.jl
https://github.com/jwmerrill/PowerSeries.jl might also meet your needs for
2nd order forward automatic differentiation, and it also already works to
higher orders.
PowerSeries.jl works by computing truncated power series of functions. You
can read derivatives off the series coefficients because
f(x + e) = f(x) + f'(x) e + f''(x)/2 e^2 + f'''(x)/3! e^3 + ...
You can choose what order to compute to for any given application.
To borrow your README example:
julia> using PowerSeries
julia> t0 = series(1.5, 1.0, 0.0)
Series2{Float64}(1.5,1.0,0.0)
julia> f(x) = e^x / (sqrt(sin(x)^3 + cos(x)^3))
f (generic function with 1 method)
# f(1.5 + e) = 4.50 + 4.05e + 4.73e^2 + ...
julia> f(t0)
Series2{Float64}(4.497780053946162,4.05342789389862,4.731536840798301)
# First derivative
julia> polyder(f(t0))
Series1{Float64}(4.05342789389862,9.463073681596603)
# Second derivative
julia> polyder(polyder(f(t0)))
9.463073681596603
# If you start from the beginning with a higher order series,
# then you'll be able to take higher order derivatives at the end
julia> polyder(polyder(polyder(f(series(1.5, 1.0, 0.0, 0.0)))))
32.16790451368894
As far as I can tell (and I could well be wrong!), if you're interested in
differentiating programs to higher orders, truncated power series are a
more fit-to-purpose extension of Dual numbers than HyperDual numbers are.
This whole space is pretty lively right now. I think everyone is realizing
how easy it is to write new Number types in Julia, and there are *a lot* of
useful notions of number.
If PowerSeries.jl does end up fitting your purpose, there's plenty of room
for contribution/improvement. There's a good discussion going on in an
issue right now on figuring out how to combine the best aspects of
PowerSeries.jl and a new package called TaylorSeriesl.jl
https://github.com/jwmerrill/PowerSeries.jl/issues/7
Or if I've missed something and HyperDual numbers have some important
advantage, that would be good to know!
On Sunday, April 6, 2014 4:21:33 PM UTC-7, Rob J Goedman wrote:
>
> Hi John,
>
> Jeff has updated his source files with the MIT license and I've pasted
> those into the LICENSE file of the Julia package.
>
> Jason Merrill has also given good feedback that I'm still looking into. My
> interpretation of his feedback (a single package covering different hyper
> number types and orders) is substantial more work and will definitely take
> longer. So maybe we should publish the current version?
>
> Regards,
> Rob J. Goedman
> goe...@icloud.com <javascript:>
>
>
>
> On Apr 6, 2014, at 12:14 PM, Jeffrey Fike
> <jf...@alumni.stanford.edu<javascript:>>
> wrote:
>
> Rob,
>
> Thanks for your interest. I have been meaning to look into an actual open
> source license. I went with the MIT license. I have updated the code on
> the website to reflect this. Please let me know if you need any additional
> information.
>
> Jeff Fike
>
>
> On Mar 29, 2014, at 5:55 PM, John Myles White
> <johnmyl...@gmail.com<javascript:>>
> wrote:
>
> Thanks for looking into it, Rob. In the absence of a license, the code is
> technically not free to use. But I imagine the authors would like to share
> their code, so it should be easy to convince them to use something formal
> like the MIT or BSD licenses.
>
> — John
>
> On Mar 29, 2014, at 5:52 PM, Robert J Goedman
> <goe...@icloud.com<javascript:>>
> wrote:
>
> John,
>
> No license is mentioned on the c++ code nor on the matlab versions as far
> as I can see.
>
> I'll send the authors an email.
>
> Regards,
> Rob J. Goedman
> goe...@icloud.com <javascript:>
>
>
> On Mar 29, 2014, at 5:47 PM, John Myles White
> <johnmyl...@gmail.com<javascript:>>
> wrote:
>
> This looks really cool. Any idea what the license was on the original file?
>
> — John
>
> On Mar 29, 2014, at 5:43 PM, Robert J Goedman
> <goe...@icloud.com<javascript:>>
> wrote:
>
> Hi,
>
> As a first 'jump into the fray' exercise I've attempted to translate
> Jeffrey Fike's hyper-dual numbers code from c++ to Julia, more or less
> following the DualNumbers package.
>
> The c++ code can be found at http://adl.stanford.edu/hyperdual/hyperdual.h.
> The paper itself at
> http://adl.stanford.edu/hyperdual/Fike_AIAA-2011-886.pdf .
>
> The Julia package can be found at:
> https://github.com/goedman/HyperDualNumbers.jl.git .
>
> Of course, I'm pretty new at this so I'm sure there will be errors and
> poor practices. So any feedback is appreciated.
>
> Also, I'm wondering if the type should be called Hyper or a better name
> would be HyperDual.
>
> This work was triggered by the interesting threads around openPP,
> TaylorSeries.jl, Calculus2, PowerSeries.jl (and at some time I hope
> MCMC.jl).
>
> Rob J. Goedman
> goe...@icloud.com <javascript:>
>
>
>
>
