> Hi Luc
I'm a bit confused about the purpose of [nabla]. You say you outline
three methods of computing the derivative of the function you
outlined. However in reality I think you have only described two.
Unless I am missing the point of your example, your idea is basically
performing symbolic differentiation, albeit at the bytecode level.
Presumably you mean to have some sort of interpreter or grammar that
can evaluate the bytecode sequence being executed and contains
mappings between functions you know about (e.g. functions in Math.*)
and their various derivatives f^(n)? If this is the case, I'm just
curious as to where this may be useful...if I wanted to do something
like this now, I would examine the function to be differentiated, and
either evaluate the derivate manually or via a package like
Mathematica (full symbolic differentiation) or even R (limited
symbolic differentiation via a simple tablebased approach). I would
then code the resulting derivative into my application explicitly. If
I do not have source code, then I cant do this...but if I dont have
source code, how can I isolate the derivative calculation?
Again, please excuse my ignorance of the crux of your approach if I
am way off in my assumptions...if there is something more here than
that, then this could be interesting...otherwise if it is just
symbolic diff (even deduced from bytecodes), then it doesnt seem much
different than the symbolic diff covered in SICP 20 years ago.
Cheers!
Rory
On Sun, Apr 13, 2008 at 12:22 PM, Luc Maisonobe <Luc.Maisonobe@free.fr>
>
> wrote:
>
>> Hello,
>>
>> I have played with an idea for a new project for a few months.
>> Asking for
>> a few advices both at the ApacheCon Europe and by direct contact, all
>> responses I received were quite positive and suggested me to set up a
>> component in sandbox. This message is the first public announcement
>> and is
>> intended to collect the opinion of all the commons community about
>> this
>> project. In short: can I play in sandbox with this or should I find
>> another
>> place for it ? Another possibility would be to put it inside
>> [math], but
>> that would be really strange.
>>
>> The project already has a name: Nabla, which is an operator used in
>> mathematics and physics for differentiation. It is a simple triangle
>> pointing downwards (see http://mathworld.wolfram.com/Nabla.html).
>> Lets
>> call the component I want to develop [nabla] from now, to match our
>> local
>> habits here. There is some code for it, but only developed by
>> myself on my
>> spare time with my personal computer and never distributed to
>> anyone. So I
>> can consider I developed it under Apache umbrella and put it on
>> sandbox with
>> the Apache headers and license. I am already a commons committer
>> and have
>> filed an Individual Contributor License Agreement to Apache.
>>
>> [nabla] will be a mathematics/physics library aimed at building the
>> symbolic differentiation of any function provided as a bytecode
>> compiled
>> function.
>>
>> Here is a typical use case for such a library. For some simulation
>> purposes, suppose I use a class with a method computing the
>> consumption of
>> performing an action as a function of its start time:
>>
>> public class DifficultComputation {
>> public double f(double t) {
>> // some lengthy equations here
>> }
>> }
>>
>> Now in addition to computing the consumption by itself, I want to
>> be able
>> to compute the sensitivity of this consumption to start time
>> changes. This
>> would allow me to say: if action is started at t = 10 seconds, then
>> consumption will be 1.2 kilograms, and this consumption will
>> increase by 10
>> grams for each second I delay the start. The value 10 grams per
>> second of
>> delay is computing by differentiating the original equation. There
>> are
>> several ways to do that.
>>
>> The first way relies on by mathematical transformations on the
>> equations
>> implemented in the function f. It it implies mathematical analysis
>> and new
>> development which is very errorprone (computing the differential
>> of a
>> function is much more complex than computing the function itself).
>> It is
>> only feasible if you know the equations or have the source code of
>> the
>> function. This approach may be used with symbolic computation
>> packages like
>> Mathematica, Axiom where you develop your equations using these
>> programs,
>> and have them generate the implementation for you. However, the
>> produce code
>> is only for some languages (typically fortran and C), it is awful
>> and cannot
>> be maintained (it is not intended to be), and needs to be
>> integrated with
>> the rest of the application which is already a difficult task.
>>
>> The second way is using numerical finitedifferences schemes. These
>> algorithms basically compute several values by changing the start
>> time by a
>> small known amount and looking at the various results. This implies
>> setting
>> up the step, which may be difficult if you don't already know the
>> behavior
>> of the function (should I use one microsecond or one century here,
>> in fact
>> it depends on the problem). This is also either quite computation
>> intensive
>> if you use high order schemes with 4, 6 or 8 points or inaccurate
>> if you
>> don't use them. It is also impossible to use too close to functions
>> boundaries which are often locations were we really want to explore.
>>
>> [nabla] provides a third way to get this result. It analyses the
>> bytecode
>> of the function at run time, performs the exact symbolic mathematical
>> transforms, and generates a new class implementing the differentiated
>> function. There is still a computation cost, but it is the same you
>> would
>> get from a manually differentiated code, plus a one time bytecode
>> differentiation overhead (but we can also cache results).
>>
>> This approach has the following benefits:
>>  derivation is exact
>>  there are no problemdependent step size to handle
>>  derivation can be computed even at domains boundaries
>>  there is no special handling of source
>> (no symbolic package with its own language, no source code
>> generation, no integration with the rest of application)
>>  one writes and maintains only the basic equation and get the
>> derivative for free
>>  it is effective even when source code is not available (but there
>> are licensing issues in this case of course, since what I do
>> automatically is really ... derived work)
>>
>> The only drawback I see is that functions calling native code
>> cannot be
>> handled. In this case, we have a fallback available with finite
>> differences
>> schemes.
>>
>> The existing implementation is not yet ready for production. A lot
>> of work
>> has been done, but there are many missing features. [nabla] can
>> handle
>> simple functions from end to end (i.e. up to creating an instance
>> of the
>> differentiated class that is fully functional). Making this code
>> available
>> in the sandbox would allow to let people look at it, comment on it,
>> participate if they are interested and make it go live.
>>
>> What do you think about it ?
>> Luc
>>
>>
>> 
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