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Re: question about DiracDelta
*To*: mathgroup at smc.vnet.net
*Subject*: [mg69653] Re: question about DiracDelta
*From*: dimmechan at yahoo.com
*Date*: Tue, 19 Sep 2006 05:45:26 -0400 (EDT)
*References*: <eel230$cng$1@smc.vnet.net>
Maybe if somebody work with proper delta sequences.
Î?/Î? Andrzej Kozlowski ÎÎ³Ï?Î±Ï?Îµ:
> On 17 Sep 2006, at 19:58, dimmechan at yahoo.com wrote:
>
> > Dear everyone,
> >
> > Mathematica is able to integrate numerically functions with peaks.
> > E.g.
> >
> >
> > Clear["Global*"]
> >
> > Plot[Exp[-x^2], {x, -10, 10}, PlotRange -> All]
> >
> > Integrate[Exp[-x^2], {x, -1000, 1000}]
> > N[%]
> > NIntegrate[Exp[-x^2], {x, -1000, 1000}]
> > Sqrt[Pi]*Erf[1000]
> > 1.7724538509055159
> > 1.7724538509054621
> >
> > With the default settings for all options, NIntegrate misses the peak
> > at x=0 here.
> >
> > NIntegrate[Exp[-x^2], {x, -10000, 10000}]
> > NIntegrate::ploss: Numerical integration stopping due to loss of
> > precision. \
> > Achieved neither the requested PrecisionGoal nor AccuracyGoal; suspect
> > one of \
> > the following: highly oscillatory integrand or the true value of the
> > integral \
> > is 0. If your integrand is oscillatory on a (semi-)infinite interval
> > try \
> > using the option Method->Oscillatory in NIntegrate.
> > 0.
> >
> > However,
> >
> > NIntegrate[Exp[-x^2], {x, -10000, 10000}, MinRecursion -> 6,
> > MaxRecursion -> 12]
> > NIntegrate[Exp[-x^2], {x, -10000, -10, 0, 10, 10000}]
> > 1.7724538509054268
> > 1.7724538509055054
> >
> > My question is if, somehow, can NIntegrate treat functions with
> > "spikes" like DiracDelta?
> >
> > Integrate[DiracDelta[x - 1], {x, -1, 3}]
> > Block[{Message},NIntegrate[DiracDelta[x - 1], {x, -1,1, 3}]]
> > Block[{Message},NIntegrate[DiracDelta[x - 1], {x, -1, 1, 3},
> > WorkingPrecision -> 50, MinRecursion -> 10, MaxRecursion -> 20]]
> > 1
> > 0.
> > 0.
> >
> > I know that I am asking essentially Mathematica to do a numerical
> > integral of a function that will be zero everywhere except x=1. So,
> > unless NIntegrate causes the function to be sampled at x=1, it will
> > evaluate to zero.
> >
> > Anyway, here are the sampled points used by NIntegrate.
> >
> > Block[{Message}, ListPlot[Reap[a1=NIntegrate[DiracDelta[x - 1], {x,
> > -1,
> > 1, 3}, EvaluationMonitor :> Sow[x]]][[2,1]]]]
> > Block[{Message}, ListPlot[a2=Reap[NIntegrate[DiracDelta[x - 1], {x,
> > -1,
> > 1, 3}, WorkingPrecision->50,MinRecursion -> 10, MaxRecursion -> 20,
> > EvaluationMonitor :> Sow[x]]][[2,1]]]]
> >
> > Length[Select[a1, 0.95 < #1 < 1.05 & ]]
> > 2
> > Length[Select[a2, 0.95 < #1 < 1.05 & ]]
> > 1688
> >
> > Thanks in advance for any help.
> >
> > Dimitris Anagnostou
> >
>
> Numerical functions are not suitable for dealing with distributions
> for rather obvious reasons. So you have to use
> Integrate and not NIntegrate when you have an expression involving
> DiracDelta.
>
>
> Integrate[DiracDelta[x - 1], {x, -1, 3}]
>
> 1
>
> However, you will get much better results if you download from
> MathSource Maxim Rytin's package Piecewise and use his function
> PieciwiseIntegrate. It pretty much leaves Integrate in the dust in
> this kind of problems.
>
> Andrzej Kozlowski
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