All right, it took me longer than I had anticipated, but I have this project assembled in a more complete form now. To recap, the idea is to simplify allow for accurate evaluation of integrals of discontinuous functions, for which the form of the discontinuity is known beforehand. There is a paper describing the concept available here: http://authors.elsevier.com/a/1QsPV508HRs4u (this link expires June 2). I have also broken out the integration software, which is available in my GitHub: https://github.com/woodscn/sympy_discontinuous_integration
As it stands right now, this package uses symbolic processing to convert multidimensional integrals into the correct form for SciPy's nquad integration routine. For discontinuous integrand functions, it also fits the multidimensional discontinuity, which eliminates the error that would normally be introduced into the integral. I see no reason why this idea could not be adapted to use the built-in Sympy integration routines; that's just not how I developed it. Please let me know what you think, and whether this would be a good fit for Sympy. Nathan Woods On Tuesday, February 24, 2015 at 11:26:18 AM UTC-7, Aaron Meurer wrote: > > This sounds like it could be useful for lambdify. I'd like to see > lambdification of integrals be improved. > > Aaron Meurer > > On Tue, Feb 24, 2015 at 11:48 AM, Nathan Woods <charle...@gmail.com > <javascript:>> wrote: > > I'll need to break it out of a more specialized package first, but this > all > > sounds very promising. It'll be a few weeks before I can really get > started > > (dissertation writing...), but I think it should only take a day or so > after > > that, at least to have something to share. Maybe longer to flesh out the > > feature set and add a cleaner interface. Mostly I just want it to be > > available in a place where it'll get use. > > > > N > > > > On Tuesday, February 24, 2015 at 10:28:13 AM UTC-7, Jason Moore wrote: > >> > >> Nathan, > >> > >> Can you show us the code? It may help us understand what you are doing. > >> > >> Also, we have code in sympy that optional depends on scipy as do we > other > >> packages: cython, numpy, theano, matplotlib, etc. The code generation > and > >> tightly coupled symboli/numeric code is in that blurry zone about what > we'd > >> add. > >> > >> But if you make a working separate package it can be a good start. For > >> example the PyDy project depends on sympy but deals with generating > specific > >> numerical codes. If you make the package and we want to include it, > then > >> great. If not, you have an easily installable package that others can > use > >> and we can link to it from our docs/website. > >> > >> > >> Jason > >> moorepants.info > >> +01 530-601-9791 > >> > >> On Tue, Feb 24, 2015 at 9:08 AM, Nathan Woods <charle...@gmail.com> > wrote: > >>> > >>> The decoupling is actually pretty easy. Unfortunately, the > Sympy-specific > >>> stuff isn't especially useful without the context of a numerical > integrator. > >>> > >>> Maybe an example will help. Say you want to numerically integrate a > step > >>> function H(x) over the interval [-1, 1]. Most numerical integrators > will try > >>> to fit a polynomial to this, but polynomials can't handle the jump at > x=0 > >>> gracefully, so you end up wasting a lot of processing time trying to > refine > >>> it. The fix is to break up the integral, and do it in two pieces, one > from > >>> [-1, 0), and one from [0, 1]. The integrator has no trouble with this, > so > >>> you get a very fast evaluation. Many integrators, and in particular > >>> scipy.integrate.quad, have this capability. > >>> > >>> What I've done is figure out a way to get the same benefits for > >>> multidimensional functions. So, you could integrate H(x**2 + y**2 - z) > over > >>> a volume, and get the same kind of efficiency. Without this, not only > is > >>> evaluation of the integral very slow, but the results can sometimes be > very > >>> inaccurate too. > >>> > >>> Anyway, from what I can tell, a similar effect could be achieved using > >>> mpmath, rather than SciPy. Maybe that project is a better venue. > >>> > >>> Thanks again! > >>> > >>> N > >>> > >>> > >>> On Tuesday, February 24, 2015 at 9:17:41 AM UTC-7, Joachim Durchholz > >>> wrote: > >>>> > >>>> Am 24.02.2015 um 17:03 schrieb Nathan Woods: > >>>> > Anyway, I would like to package this up in a way that would be > >>>> > publicly > >>>> > useful, but I'm not sure where it fits. Sympy seemed a likely > guess, > >>>> > but > >>>> > the SciPy dependence is problematic. > >>>> > >>>> Out of the box, a separate project with a dependency on both SymPy > and > >>>> SciPy would probably fit best. > >>>> > >>>> > Alternatively, the > >>>> > discontinuity-processing could be decoupled and used with any > iterated > >>>> > integrator that supports manually specified points of > discontinuity. > >>>> > >>>> Decoupling and putting the modules into their respective projects > would > >>>> work, too. > >>>> It's much more work though. > >>>> > >>>> > If > >>>> > Sympy isn't a good fit for this, I would appreciate suggestions > about > >>>> > other > >>>> > places that might be better. > >>>> > >>>> I did not fully understand your description of what your code does, > so I > >>>> can't be very specific. > >>>> In general, anything that analyzes mathematical objects for > properties > >>>> of interest that go beyond a one-shot task would be a worthy addition > to > >>>> SymPy. > >>> > >>> -- > >>> You received this message because you are subscribed to the Google > Groups > >>> "sympy" group. > >>> To unsubscribe from this group and stop receiving emails from it, send > an > >>> email to sympy+un...@googlegroups.com. > >>> To post to this group, send email to sy...@googlegroups.com. > >>> Visit this group at http://groups.google.com/group/sympy. > >>> To view this discussion on the web visit > >>> > https://groups.google.com/d/msgid/sympy/125f4bc9-7c1d-441f-bf41-5df851379a7d%40googlegroups.com. > > > >>> > >>> For more options, visit https://groups.google.com/d/optout. > >> > >> > > -- > > You received this message because you are subscribed to the Google > Groups > > "sympy" group. > > To unsubscribe from this group and stop receiving emails from it, send > an > > email to sympy+un...@googlegroups.com <javascript:>. > > To post to this group, send email to sy...@googlegroups.com > <javascript:>. > > Visit this group at http://groups.google.com/group/sympy. > > To view this discussion on the web visit > > > https://groups.google.com/d/msgid/sympy/bed69b8c-fd47-4cba-a3fa-e800e34acea8%40googlegroups.com. > > > > > > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "sympy" group. 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