Re: NIntegrate - Gaussian quadrature more exact than thought

*To*: mathgroup at smc.vnet.net*Subject*: [mg73893] Re: NIntegrate - Gaussian quadrature more exact than thought*From*: "ben" <benjamin.friedrich at gmail.com>*Date*: Sat, 3 Mar 2007 00:47:21 -0500 (EST)*References*: <es6dmv$s61$1@smc.vnet.net>

On 1 Mrz., 12:33, "janos" <janostothmeis... at gmail.com> wrote: > NIntegrate[x^4, {x, 0, 1}, Method->GaussKronrod, GaussPoints->2] > > gives 0.2, the exact result, although the Gauss quadrature should be > inexact on a polynomial of degree 2n+2 where n is the number of > GaussPoints. More exactly, we expected the same results as here: > > << NumericalMath`GaussianQuadrature` > gw = GaussianQuadratureWeights[2, 0, 1] > f[{x_, y_}] := x^4 y > Total[f /@ gw] > 0.194444 > > This is inexact, OK. > > Why is NIntegrate so good? > Something I may have missed. > > Thank you for your help. > > Janos Hi I wouldn't be surprised if Mathematica does some preprocessing of the integrand and realizes that the integral can be faster computed in closed form; however the implementation notes on NIntegrate do not state such a thing Bye Ben