[Numpy-discussion] problems with numdifftools
Nicolai Heitz
nicolaiheitz@gmx...
Wed Oct 27 15:50:47 CDT 2010
m 27.10.2010 02:02, schrieb Sebastian Walter:
> On Wed, Oct 27, 2010 at 12:59 AM, Pauli Virtanen<pav@iki.fi> wrote:
>> Tue, 26 Oct 2010 14:24:39 -0700, Nicolai Heitz wrote:
>>>> http://mail.scipy.org/mailman/listinfo/scipy-user
>>> I contacted them already but they didn't responded so far and I was
>>> forwarded to that list which was supposed to be more appropriated.
>> I think you are thinking here about some other list -- scipy-user
>> is the correct place for this discussion (and I don't remember seeing
>> your mail there).
I was pretty sure that I put it there. Unfortunately it had a different
name there: Errors in calculation of the Hessian using numdifftools. But
I can't find the post by myself at the moment so maybe something went
wrong.
>> [clip]
>>> 1) Can I make it run/fix it, so that it is also going to work for the SI
>>> scaling?
>> Based on a brief look, it seems that uniform scaling will not help you,
>> as you have two very different length scales in the problem,
>>
>> 1/sqrt(m w^2)>> C
>>
>> If you go to CM+relative coordinates you might be able to scale them
>> separately, but that's fiddly and might not work for larger N.
>>
>> In your problem, things go wrong when the ratio between the
>> length scales approaches 1e-15 which happens to be the machine epsilon.
>> This implies that the algorithm runs into some problems caused by the
>> finite precision of floating-point numbers.
>>
>> What exactly goes wrong and how to fix it, no idea --- I didn't look into
>> how Numdifftools is implemented.
Probably you are right. I converted it to natural units and it worked
out in the 2 ion case. Increasing the number of ions leads to problems
again and I have no idea where those problems come from.
>>> 2) How can I be sure that increasing the number of ions or adding a
>>> somehow more complicated term to the potential energy is not causing the
>>> same problems even in natural units?
>>>
>>> 3) In which range is numdifftools working properly.
>> That depends on the algorithm and the problem. Personally, I wouldn't
>> trust numerical differentiation if the problem has significantly
>> different length scales, it is important to capture all of them
>> accurately, and it is not clear how to scale them to the same size.
>> Writing ND software that works as expected all the time is probably
>> not possible even in theory.
>>
>> Numerical differentiation is not the only game in the town. I'd look
>> into automatic differentiation (AD) -- there are libraries available
>> for Python also for that, and it is numerically stable.
>>
>> E.g.
>>
>> http://en.wikipedia.org/wiki/Automatic_differentiation#Software
>>
>> has a list of Python libraries. I don't know which of them would be
>> the best ones, though.
>>
> they all have their pro's and con's.
> Being (co-)author of some of these tools, my personal and very biased advice is:
> if you are on Linux, I would go for PYADOLC. it provides bindings to a
> feature-rich and well-tested C++ library.
> However, the installation is a little tricker than a "setup.py build"
> since you will need to compile ADOL-C and get Boost::Python to work.
> PYADOLC can also differentiate much more complicated code than your
> example in a relatively efficient manner.
Is there by chance any possibility to make PYADOLC run on a (lame)
windows xp engine. If not what else would u recommend (besides switching
to Linux, what I am going to do soon).
> For your example the code looks like:
>
> --------------------- code ---------------------------
>
> ....
>
> c=classicalHamiltonian()
> xopt = optimize.fmin(c.potential, c.initialposition(), xtol = 1e-10)
>
> import adolc; import numpy
>
> # trace the computation
> adolc.trace_on(0)
> x = adolc.adouble(c.initialposition())
> adolc.independent(x)
> y = c.potential(x)
> adolc.dependent(y)
> adolc.trace_off()
>
> hessian = adolc.hessian(0, xopt)
> eigenvalues = numpy.linalg.eigh(hessian)[0]
> normal_modes = c.normal_modes(eigenvalues)
> print 'hessian=\n',hessian
> print 'eigenvalues=\n',eigenvalues
> print 'normal_modes=\n',normal_modes
> --------------------- code ---------------------------
>
> and you get as output
> Optimization terminated successfully.
> Current function value: 0.000000
> Iterations: 81
> Function evaluations: 153
> hessian=
> [[ 5.23748399e-12 -2.61873843e-12]
> [ -2.61873843e-12 5.23748399e-12]]
> eigenvalues=
> [ 2.61874556e-12 7.85622242e-12]
> normal_modes=
> [ 6283185.30717959 10882786.30440101]
>
>
> Also, you should use an eigenvalue solver for symmetric matrices, e.g.
> numpy.linalg.eigh.
>
Your code example looks awesome and leads to the correct results. Thank
you very much. I try to make it work on my pc as well.
> regards,
> Sebastian
>
>> --
>> Pauli Virtanen
>>
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