Phil, Adaptive Quadrature is defined here ... Gilles mentioned this: http://en.wikipedia.org/wiki/Adaptive_quadrature We have discussed why AQ is better in the email chain. The differences have been discussed in this email chain and are quite minor.
Based on the tests done so far the question should be is there is ANY case where LGQ beats AQ. Here is a summary of what AQ has going for it relative to LGQ: 1. More efficient: It provides better error bounds per function evaluation. 2. Accurate: It can integrate much higher frequency functions (like the gaussian PDF). AQ will fail at some point too ... but its failure will be outside the region in which LGQ fails. 3. Not Buggy: I view the erroneous report of convergence to 1e-6 accuracy in the gaussian PDF with sigma=1000 to 0.03 when I was expecting 1.00 as a serious bug. This makes LGQ buggy and not merely inefficient. Gilles explains it as a numerical instability, if so please explain how AQ converges. 4. What are the measures of efficiency of an integrator by which LGQ does better than AQ? I'm deeply interested in use cases where AQ's accuracy per evaluation is not as good as LGQs or ... efficiency/bugs etc. Gilles solution test case for the problem of gaussian PDF by integrating from -5000 to 5000 only works in the case that the bounds of variability of the function are known. AQ / LGQ with Infinity.wrap(fn) in [-1,1] is much more generic. In summary outside of familiarity code design, is there any test case that shows why anyone should use LGQ over AQ? Cheers, Ajo. On Tue, Jul 2, 2013 at 7:56 AM, Phil Steitz <phil.ste...@gmail.com> wrote: > On 7/1/13 8:37 PM, Ajo Fod wrote: > > Hi Konstantin, > > > > Thanks for pointing out the inefficiency in AQ. I just improved the > > efficiency of AQ to 1.41x that of LGQ (up from 1.05x) - measured in > digits > > of accuracy per evaluation for integral of normal with sigma 1000 in > range > > [-5000, 5000] > > > > Please let me know if this doesn't answer your question about the > > discussion: > > > > In essence Gilles thinks there is no problem with LGQ because it > integrates > > the low frequency functions in his unit tests accurately. He thinks that > > the problem is with the function I provided because it has "numerical > > instabilities" while I think it is the high frequency nature of the > > function that LGQ can't handle because it divides intervals > > indiscriminately. So, it is not clear to me how Gilles explains why AQ > > converges to the right answer in the presence of these "numerical > > instabilities" ... after all, LGQ and AQ are being passed the same > function > > and identical limits. > > > > This problem should appear with any function that has sufficiently high > > frequency components. It would be better if LGQ threw an exception when > it > > encounters a high frequency function. Instead, it "converges" confidently > > to the wrong answer. I personally think that AQ will fail at some point > at > > a high enough frequency ... but that will be well beyond the point at > which > > LGQ fails. > > > > I've avoided the complex method of fetching weights used in the current > > schemes because the improvement in efficiency arises from the adaptive > > nature of the AQ method. You may notice that I'm using the weights that > you > > use in your code. I think Gilles requires that I use the weight > generation > > scheme he has worked with in the codebase in order to consider the code > > usable in Apache MATH. > > > > In summary, I feel the accuracy and versatility of AQ are being ignored > in > > favor of the familiarity of LGQ in the apache codebase. If there are > tests > > that AQ fails, I'll update my opinion. > > Thanks for digging in to this. What would be really great would be > some mathematics to support the claims of which method is better for > which class of functions. As Gilles stated, we favor standard > algorithms. The reason for this is that when we do that people can > rely on standard numerics references to understand what our > algorithms do and choose among them accordingly. There is usually > not just one uniformly "best" method in numerical analysis. By > providing well-documented implementations of standard methods, we > make it easier for users to pick the method that works best for > their problem instance. When we modify standard algorithms or > introduce our own, it is incumbent on us to provide the numerical > analysis to support them. I see the beginnings of that above; but > it would be better if what we are seeing in the small number of > examples examined could be explained via some (ideally simple, > externally documented) error analysis. > > Phil > > > > Cheers, > > Ajo > > > > > > On Mon, Jul 1, 2013 at 6:49 PM, Konstantin Berlin <kber...@gmail.com> > wrote: > > > >> I am not understanding the discussion here. Adaptive integration is > >> designed for functions that have different behavior in different > >> regions. Some regions are smoother, some have higher frequeniesy. How > >> you integrate a divided region, Simpson rule or whatever is a separate > >> question. > >> > >> Adaptive integration saves of function evaluations by avoiding large > >> series approximation in smooth regions. Nothing to do with how you > >> compute the subdivided regions. > >> > >> On Jul 1, 2013, at 8:45 PM, Fod <ajo....@gmail.com> wrote: > >> > >>> Hi Gilles, > >>> > >>> Your accuracy concern made me wonder. So, I dropped the > >>> AdaptiveQuadrature.EPS to 1e-2 from 1e-9 in the code and ran the test > in > >>> the patch. > >>> > >>> I computed the log of the error per evaluation ...i.e a measure of the > >>> efficiency of the algorithm. > >>> And wait for it ... AQ beats LGQ by about 5% for the particular > >> formulation > >>> of the problem. > >>> > >>> Your request to use the Math classes falls under "coding style" IMHO. > If > >> it > >>> doesn't satisfy your standards, feel free to modify. I'm happy with it. > >>> Although as far as accuracy and convergence goes, I'd use AQ always. > >>> > >>> > >>> Got to compare apples to apples Gilles ! > >>> > >>> Cheers, > >>> -Ajo > >>> > >>> > >>> > >>> > >>> On Mon, Jul 1, 2013 at 4:16 PM, Gilles <gil...@harfang.homelinux.org> > >> wrote: > >>>> Hi. > >>>> > >>>> > >>>> On Mon, 1 Jul 2013 10:50:19 -0700, Ajo Fod wrote: > >>>> > >>>>> If you wanted to use the Math 3 codebase in AdaptiveQuadrature, you'd > >>>>> compute the calculations of Q1 and Q2 with something else. I'm not > >>>>> entirely > >>>>> familiar with the apache Math codebase [...] > >>>> You could file a "wish" request as a Commons Math's user. Then, if > >>>> and when some regular contributor finds some time, he will try to > >>>> implement the functionality. > >>>> > >>>> However, when you provide a patch for inclusion in the codebase, it > >>>> is necessary to be more informed about similar functionality that > >>>> would already exist in Commons Math, so that the contribution can be > >>>> merged gracefully (i.e. with "minor" changes which committers will > >>>> happily perform for you). > >>>> You are welcome to ask questions in order to be able to contribute. > >>>> As I tried to explain in more than one way, modifying your code is > >>>> far from being trivial. If the committer has to figure out how to > >>>> change/adapt/comment a significant part of the contribution, it > >>>> ends up being easier to implement the feature from scratch! > >>>> > >>>> > >>>> > >>>>> Each of the tests in the patch is integrating a UnivariateFunction in > >>>>> [-1,1]. Infinity.wrap(fn) just provides that UnivariateFunction. [In > >> the > >>>>> patches for MATH-995 the InfiniteIntegral was replaced by > >> Infinity.wrap() > >>>>> ]. So, if you are saying that the intent of > >>>>> IterativeLegendreGaussIntegrat**or (refered to as LGQ) was not to > >>>>> integrate > >>>>> this kind of UnivariateFunction in [-1,1], ... what kind of > univariate > >>>>> function would that be? > >>>> Again, it is not just any UnivariateFunction, it is a function that > >>>> maps the [-inf, +inf] interval into [-1, 1]. > >>>> It seems that the Gauss-Legendre quadrature is not appropriate for > >>>> this. This is probably because the sample integration points do not > >>>> cover the _whole_ interval: for the 10-point rule, the first point > >>>> is at > >>>> -0.9739065285171717 > >>>> and the last point is at > >>>> 0.9994069665572084 > >>>> The interval in the original variable is thus [-18.908, 842.872]. This > >>>> is far from adequate for integrating a Gaussian function with > >> sigma=1000. > >>>> [And, as Phil pointed out from the outset, I suspect that the change > of > >>>> variable also introduces numerical errors since the result becomes > worse > >>>> when increasing the number of sample points. Increasing the requested > >>>> precision leads to a prohibitive increase of the number of > evaluations, > >>>> without improvement of the accuracy. In itself it is not sufficient to > >>>> indicate a bug of "IterativeGaussLegendre"; it could simply be a > >>>> limitation inherent to the algorithm.] > >>>> > >>>> > >>>> If it is indeed supposed to do the integration, > >>>>> then AQ clearly does a better job. > >>>> Adaptive methods are certainly useful, but we need examples where > >>>> its usage is appropriate. It is _not_ indicated for the improper > >>>> integral of a Gaussian (even though it indeed performs better than > >>>> Gauss-Legendre). > >>>> > >>>> > >>>> > >>>>> So, why does LGQ fail here? It is probably that the Adaptive division > >> of > >>>>> the integration domain (as opposed to the uniform division with LGQ) > >> gives > >>>>> AQ the critical edge. The test you have for LGQ so far are pretty > well > >>>>> behaved. > >>>> Cf. above. > >>>> > >>>> AFAIU, the problem reported by MATH-995 is not a bug in > >>>> "IterativeLegendreIntegrator": it correctly integrates a Gaussian > >>>> with a large sigma _if_ the integration interval is "large enough" > >>>> (cf. unit test referred to in my comment to MATH-995). > >>>> > >>>> Unless someone can point to something I'm missing in MATH-995, I'll > >>>> close that issue. > >>>> > >>>> > >>>> > >>>>> Summary: I'm demonstrating a clear bug/inefficiency with LGQ > >>>> I don't agree with that statement. > >>>> "IterativeGaussLegendre" produces the correct answer (at 1e-6 > >>>> accuracy) in less than 60 function evaluations. To achieve the same, > >>>> your code needs 995 evaluations. > >>>> > >>>> > >>>> and providing > >>>>> you with an alternative that is more accurate. > >>>> Cf. above (and my previous post), about how to contribute to > >>>> Commons Math. > >>>> Please open a new feature request. > >>>> > >>>> > >>>> Regards, > >>>> Gilles > >>>> > >>>> > >>>>> On Mon, Jul 1, 2013 at 8:22 AM, Gilles <gil...@harfang.homelinux.org > > > >>>>> wrote: > >>>>> > >>>>> Hi. > >>>>>> > >>>>>> > >>>>>> I just noticed your request to write the algorithm along the lines > of > >>>>>>> the > >>>>>>> wikipedia article. > >>>>>>> > >>>>>>> The only major difference between my code and the article on > >> Wikipedia > >>>>>>> is > >>>>>>> that I found it necessary to move the recursive stack in into a > data > >>>>>>> structure to avoid a StackOverflowException when the non polynomial > >>>>>>> curvature is concentrated in a corner of the domain of integration. > >>>>>>> Notice > >>>>>>> that the Stack objects stores a Stack of limits of integration. > >>>>>> There is a misunderstanding: I'm referring to the "high-level" > >>>>>> description of the algorithm that is the separation of concerns > >>>>>> between the quadrature method and the adaptive process. Your code > >>>>>> mixes the two. Moreover, it does not reuse any of the quadrature > >>>>>> schemes already implemented in CM, but implements a (new?) one > >>>>>> without any reference or comments. > >>>>>> [And this is even without delving into remarks concerning the > >>>>>> code structure itself.] > >>>>>> > >>>>>> Additionally, your patch also mixes two concepts: Adaptive > >>>>>> quadrature vs improper integral (which is also MATH-994); it is > >>>>>> hard to follow what problem this issue is supposed to point to, > >>>>>> and how the patch solves it. Indeed your unit tests shows a > >>>>>> problem with improper integrals which the class > >>>>>> "****IterativeGaussLegendreIntegrat****or" is _not_ meant to > >> handle.[1] > >>>>>> > >>>>>> To be clear, hopefully, you are demonstrating a problem that > >>>>>> occurs when combining Commons Math code with code which you > >>>>>> created. > >>>>>> The first step is to create a unit test demonstrating whether > >>>>>> an issue exists with "****IterativeGaussLegendreIntegrat****or" code > >>>>>> > >>>>>> only (i.e. without relying on your "InfiniteIntegral" class).[1] > >>>>>> If no independent issue exist, then MATH-995 should be replaced > >>>>>> by an appropriate feature request. > >>>>>> Also, it would certainly be helpful to pinpoint the reason why > >>>>>> the combination of "****IterativeGaussLegendreIntegrat****or" and > >>>>>> > >>>>>> "InfiniteIntegral" is not legitimate (if that's the case). > >>>>>> > >>>>>> > >>>>>> Regards, > >>>>>> Gilles > >>>>>> > >>>>>> [1] Cf. also my latest comment on the MATH-995 page. > >>>>>> > >>>>>> > >>>>>> > >>>>>> Cheers, > >>>>>>> Ajo. > >>>>>>> > >>>>>>> > >>>>>>> On Fri, Jun 28, 2013 at 11:07 AM, Ajo Fod <ajo....@gmail.com> > wrote: > >>>>>>> > >>>>>>> BTW, it is possible that I'm not using LGQ correctly. If so, please > >>>>>>> show > >>>>>>> > >>>>>>>> how to pass the tests I've added. I'd much rather use something > >> that is > >>>>>>>> better tested than my personal code. > >>>>>>>> > >>>>>>>> -Ajo. > >>>>>>>> > >>>>>>>> > >>>>>>>> On Fri, Jun 28, 2013 at 11:04 AM, Ajo Fod <ajo....@gmail.com> > >> wrote: > >>>>>>>> I just posted a patch on this issue. Feel free to edit as > necessary > >> to > >>>>>>>>> match your standards. There is a clear issue with LGQ. > >>>>>>>>> > >>>>>>>>> Cheers, > >>>>>>>>> Ajo. > >>>>>>>>> > >>>>>>>>> > >>>>>>>>> On Fri, Jun 28, 2013 at 10:54 AM, Gilles < > >>>>>>>>> gil...@harfang.homelinux.org> > >>>>>>>>> **wrote: > >>>>>>>>> > >>>>>>>>> > >>>>>>>>> Ted, > >>>>>>>>> > >>>>>>>>>> > >>>>>>>>>> > >>>>>>>>>> Did you read my other (rather more lengthy) post? Is that > >>>>>>>>>> "jumping"? > >>>>>>>>>> > >>>>>>>>>>> > >>>>>>>>>>>> Yes. You jumped on him rather than helped him be productive. > >> The > >>>>>>>>>>> general > >>>>>>>>>>> message is "we have something in the works, don't bother us > with > >>>>>>>>>>> your > >>>>>>>>>>> ideas". > >>>>>>>>>>> > >>>>>>>>>>> > >>>>>>>>>>> Then please read all the messages pertaining to those issues > more > >>>>>>>>>> carefully: > >>>>>>>>>> I never wrote such a thing (neither now nor in the past). > >>>>>>>>>> I pointed to a potential problem in the usage of the CM code. > >>>>>>>>>> I pointed (several times and in details) to problems in > candidate > >>>>>>>>>> contributions, > >>>>>>>>>> with arguments that go well beyond "bad formatting". > >>>>>>>>>> I pointed out how we could improve the functionality _together_ > >> (i.e. > >>>>>>>>>> by > >>>>>>>>>> using > >>>>>>>>>> what we have, instead of throwing it out without even trying to > >>>>>>>>>> figure > >>>>>>>>>> out how > >>>>>>>>>> good or bad it is). > >>>>>>>>>> > >>>>>>>>>> IMHO, these were all valid suggestions to be productive in > >> helping CM > >>>>>>>>>> to > >>>>>>>>>> become > >>>>>>>>>> better, instead of merely larger. The former indeed requires > more > >>>>>>>>>> effort > >>>>>>>>>> than > >>>>>>>>>> the latter. > >>>>>>>>>> > >>>>>>>>>> > >>>>>>>>>> > >>>>>>>>>> Gilles > >>>> > >>>> > >> > ------------------------------**------------------------------**--------- > >>>> To unsubscribe, e-mail: dev-unsubscribe@commons.**apache.org< > >> dev-unsubscr...@commons.apache.org> > >>>> For additional commands, e-mail: dev-h...@commons.apache.org > >>>> > >>>> > >> --------------------------------------------------------------------- > >> To unsubscribe, e-mail: dev-unsubscr...@commons.apache.org > >> For additional commands, e-mail: dev-h...@commons.apache.org > >> > >> > > > --------------------------------------------------------------------- > To unsubscribe, e-mail: dev-unsubscr...@commons.apache.org > For additional commands, e-mail: dev-h...@commons.apache.org > >
