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MathGroup Archive 2007

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Re: NIntegrate - Gaussian quadrature more exact than thought

  • To: mathgroup at smc.vnet.net
  • Subject: [mg73987] Re: NIntegrate - Gaussian quadrature more exact than thought
  • From: Peter Pein <petsie at dordos.net>
  • Date: Sun, 4 Mar 2007 02:04:07 -0500 (EST)
  • References: <es6dmv$s61$1@smc.vnet.net>

janos schrieb:
> 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 Janos,

obviously a subdivision of the interval takes place:


Length[xvals = Reap[result = NIntegrate[x^4, {x, 0, 1},
       Method -> GaussKronrod, GaussPoints -> 2, EvaluationMonitor :>
        Sow[x]]][[2,1]]]
result

--> 155
    0.2

With Listplot[xvals] you can see, where the evaluations have been done.

At least a warning can be forced by additionally setting MaxRecursion to 0:

Length[xvals = Reap[result = NIntegrate[x^4, {x, 0, 1},
       Method -> GaussKronrod, GaussPoints -> 2, MaxRecursion -> 0,
       EvaluationMonitor :> Sow[x]]][[2,1]]]
Abs[result - 1/5]

NIntegrate::ncvb :
 <NIntegrate failed to converge to prescribed accuracy after
  1 recursive bisections in x near x = 0.25`. More...

--> 15
     0.

Peter


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