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Re: Gauss-Kronrod algorithm

• To: mathgroup at smc.vnet.net
• Subject: [mg75468] Re: Gauss-Kronrod algorithm
• From: dimitris <dimmechan at yahoo.com>
• Date: Wed, 2 May 2007 03:48:34 -0400 (EDT)
• References: <f04nlc$8kr$1@smc.vnet.net><f16pjl$70u$1@smc.vnet.net>

Although this is from the documentation of the upcoming version of
Mathematica
I suggest you to have a look!

http://reference.wolfram.com/mathematica/ref/NIntegrate.html

Now let's see some things.

1) Some useful links

http://en.wikipedia.org/wiki/Gaussian_quadrature
http://numericalmethods.eng.usf.edu/topics/gauss_quadrature.html
(amazing site!)
http://www.sitmo.com/eqcat/13
http://scholar.google.com/scholar?hl=en&lr=&q=%22Calculation+of+Gauss=
-Kronrod+Quadrature+Rules%22
(the first article is of special notice)

2) Some Mathematica code

In[64]:=
oo[x_] = Exp[-5*x^4] + (1/2)*x;

In[71]:=
Integrate[oo[x], {x, -3, 5}]
{N[%], ({#1, NIntegrate[oo[x], {x, -3, 5}, Method -> #1]} & ) /@
{Trapezoidal, DoubleExponential, GaussKronrod}}

Out[71]=
4 + (8*Gamma[5/4] - Gamma[1/4, 405] - Gamma[1/4, 3125])/(4*5^(1/4))

Out[72]=
{5.212295737874928, {{Trapezoidal, 5.212295737874928},
{DoubleExponential, 5.212295737874928},
   {GaussKronrod, 5.212295737874985}}}

In[73]:=
(ListPlot[Reap[NIntegrate[oo[x], {x, -3, 3}, Method -> #1,
EvaluationMonitor :> Sow[x]]][[2,1]], PlotLabel -> #1] & ) /@
{Trapezoidal, DoubleExponential, GaussKronrod};

I hope I help you a bit!

Dimitris


=CF/=C7 acmilan =DD=E3=F1=E1=F8=E5:
> On Apr 24, 3:21 pm, dimitris <dimmec... at yahoo.com> wrote:
> > Hello.
> >
> > As a first hint I suggest you searching in the archives about posts
> > of V. Antonov of WRI.
> >
> > Dimitris
> >
> > =CF/=C7 acmilan =DD=E3=F1=E1=F8=E5:
> >
> >
> >
> > > On Apr 18, 5:16 pm, acmilan <fanvn_Acmi... at yahoo.com> wrote:
> > > > Dear All,
> > > > I am looking for the algorithm to calculate the general Gauss-Kronr=
od
> > > > points and weights  in Mathematica( which is similar with the built=
-in
> > > > function
> > > > GaussianQuadratureWeights[n, a, b, prec]  for Gauss points and
> > > > weights)
> > > > Thank you very much for your help.
> > > > Best regards,
> > > > Milan
> >
> > > Hi all,
> > > Anybody helps me on this Mathematica Gauss-Kronrod points and weights
> > > code?
> > > Thanks and regards,
> > > Khoi Khoa- Hide quoted text -
> >
> > - Show quoted text -
>
> Dear Dimitris,
> I did try to search all the archives but could not find what I want.
> Do you have any other suggestions?
> Thanks and regards,
> Milan