At first it seems you are still compute redundant points. See my example
that I posted, where I propagate 3 functional values not two.
In regards to improvement. I am not an expert of different integration
strategies but:
The concept of adaptive quadrature is separate from how you integrate the
subinterval. In your implementation (and mine) we use Simpson's rule. This
is why putting comments on where your actual formulas come from is
important for public code.
double Q1 = delta / 6 * (va + 4 * vm + vb); //this is simpson's
rule of integration
It does not make adaptive integration better or worse than LegendreGauss
quadrature, rather they solve different problems.
2) The error tolerance should be an input to the quadrature, since
tolerance is a problem dependent value. This values should be subdivided
for each interval, such that the total sum would equal to the overall
desired tolerance. Look at the code I provided. In any case, like Gilles
said, the setup should be in the same spirit as other quadrature methods in
commons.
3) Looking at Commons numerical integrations it is a little hard to read. I
would have though that the basic quadratures, like trapezoidal or simpson's
rule, are adaptive already. I think its rare to use these basic formulas
without an adaptive quadrature. I would also think that Guass Kronrod would
be a better way of going about the integration than what is there now for
LGQ.
http://en.wikipedia.org/wiki/Gauss%E2%80%93Kronrod_quadrature_formula
On Mon, Jul 1, 2013 at 11:37 PM, Ajo Fod <ajo.fod@gmail.com> 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.
>
> Cheers,
> Ajo
>
>
> On Mon, Jul 1, 2013 at 6:49 PM, Konstantin Berlin <kberlin@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.fod@gmail.com> wrote:
> >
> > > Hi Gilles,
> > >
> > > Your accuracy concern made me wonder. So, I dropped the
> > > AdaptiveQuadrature.EPS to 1e2 from 1e9 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 <gilles@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 MATH995 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 GaussLegendre quadrature is not appropriate for
> > >> this. This is probably because the sample integration points do not
> > >> cover the _whole_ interval: for the 10point 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
> > >> GaussLegendre).
> > >>
> > >>
> > >>
> > >>> 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 MATH995 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 MATH995).
> > >>
> > >> Unless someone can point to something I'm missing in MATH995, 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 1e6
> > >> 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 <gilles@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 "highlevel"
> > >>>> 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 MATH994); 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 MATH995 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 MATH995 page.
> > >>>>
> > >>>>
> > >>>>
> > >>>> Cheers,
> > >>>>> Ajo.
> > >>>>>
> > >>>>>
> > >>>>> On Fri, Jun 28, 2013 at 11:07 AM, Ajo Fod <ajo.fod@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.fod@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 <
> > >>>>>>> gilles@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, email: devunsubscribe@commons.**apache.org<
> > devunsubscribe@commons.apache.org>
> > >> For additional commands, email: devhelp@commons.apache.org
> > >>
> > >>
> >
> > 
> > To unsubscribe, email: devunsubscribe@commons.apache.org
> > For additional commands, email: devhelp@commons.apache.org
> >
> >
>
