I agree that the choice of quadrature rules is orthogonal to Adaptive strategy. But, can one quadrature rule be more efficient than another by points evaluated per digit of accuracy?
It seems like in GKQ, the G and K nodes are different points at each level of resolution. However in CCQ, the high frequency coefficients for one level involve the computation of the same values of f() as the one order lower frequency coefficients at a higher resolution (take a look at how a_k is computed for CCQ). Since the points that are computed at each level are completely orthogonal to each other, I think this method is more efficient. Cheers, -Ajo On Wed, Jul 3, 2013 at 7:52 AM, Gilles <gil...@harfang.homelinux.org> wrote: > On Wed, 3 Jul 2013 07:39:00 -0700, Ajo Fod wrote: > >> I wonder if Clenshaw-Curtis Quadrature (CCQ) is more adapted for Adaptive >> Quadrature than Gauss-Kronrod (GKQ). >> > > As Konstantin already pointed out, the choice of a quadrature rule > is orthogonal to the choice of an adaptive strategy. > > > It seems like with CCQ the points at >> one level of resolution can be reused for the next level. >> > > This is also the purpose of Gauss-Kronrod: > > http://en.wikipedia.org/wiki/**Gauss–Kronrod_quadrature_**formula<http://en.wikipedia.org/wiki/Gauss%E2%80%93Kronrod_quadrature_formula> > > > The error bounds >> for a given level of resolution is the comparable to GKQ, so I'm guessing >> that from an accuracy/evaluation perspective, CCQ is 2 times as efficient >> as GKQ. >> > > According to the Wikipedia article which you site here: > > > http://en.wikipedia.org/wiki/**Clenshaw–Curtis_quadrature<http://en.wikipedia.org/wiki/Clenshaw%E2%80%93Curtis_quadrature> >> > > your statement is wrong. [They state that the _weights computation_ is > faster than with Gauss-Kronrod.] > > > Gilles > > >> Cheers, >> Ajo >> >> >> On Tue, Jul 2, 2013 at 2:09 PM, Gilles <gil...@harfang.homelinux.org> >> wrote: >> >> On Tue, 2 Jul 2013 12:54:12 -0400, Konstantin Berlin wrote: >>> >>> IterateiveLegendreGaussIntegra****tor should be replaced by adaptive >>>> >>>> guass-kronrod. The simpson and trapezoidal methods should be replaced by >>>> their adaptive versions, or at least adaptive wrapper provided for them. >>>> >>>> >>> [I mentioned Gauss-Kronrod very early in the discussion about MATH-995.] >>> >>> See also >>> >>> https://issues.apache.org/****jira/browse/MATH-830<https://issues.apache.org/**jira/browse/MATH-830> >>> <https://**issues.apache.org/jira/browse/**MATH-830<https://issues.apache.org/jira/browse/MATH-830> >>> > >>> >>> >>> Identified sub-tasks can be filed there. >>> >>> In a library like Commons Math, we aim to provide "low-level" tools which >>> knowledgeable users can mix and match to solve a wide range of problems. >>> >>> Patches welcome. >>> >>> > > > ------------------------------**------------------------------**--------- > To unsubscribe, e-mail: > dev-unsubscribe@commons.**apache.org<dev-unsubscr...@commons.apache.org> > For additional commands, e-mail: dev-h...@commons.apache.org > >
