Mathematica 9 is now available
Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2006
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2006

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: Question about DiracDelta

  • To: mathgroup at smc.vnet.net
  • Subject: [mg68662] Re: Question about DiracDelta
  • From: ab_def at prontomail.com
  • Date: Mon, 14 Aug 2006 06:44:46 -0400 (EDT)
  • References: <ebmth1$6g3$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

vasil michev wrote:
> Hi group, I need to plot this:
>
> Plot[NIntegrate[DiracDelta[3 - Cos[x] - Cos[y] - Cos[z] - e], {x, -Pi,
> Pi}, {y, -Pi, Pi}, {z, -Pi, Pi}, MaxPoints -> 1000000]/8/Pi^3, {e, 0,
> 6}, PlotPoints -> 30]
>
> the problem is that the function always returns 0, no matter what the
> value of 'e' is. I saw that other people have reported incorect
> evaluation/simplification related to DiracDelta, but I was wondering
> can anyone tell me if there's some workaround?
>
> Thanks in advance

NIntegrate doesn't attempt to process DiracDelta symbolically, it will
just evaluate it at the sampling points. The workaround is to use
PiecewiseIntegrate (
http://library.wolfram.com/infocenter/MathSource/5117/ ) and some
tricks. Integrate once to get rid of the delta function:

In[2]:= PiecewiseIntegrate[
  1/(8*Pi^3)*DiracDelta[3 - Cos[x] - Cos[y] - Cos[z] - e],
  {z, -Pi, Pi}]

Out[2]= If[e + Cos[x] + Cos[y] == 4, 1/(4*Pi^3*Sqrt[Abs[1 - (3 - e -
Cos[x] - Cos[y])^2]]), 0] + If[2 <= e + Cos[x] + Cos[y] <= 4,
1/(4*Pi^3*Sqrt[1 - (3 - e - Cos[x] - Cos[y])^2]), 0]

The result can be integrated numerically but it will be quite slow. A
better way is to perform some other symbolic transformations first.
Replace the integrand with an undefined function and integrate over y
and z:

In[3]:= PiecewiseIntegrate[
  PiecewiseIntegrate[% /.
      1/Sqrt[1 - (3 - e - Cos[x] - Cos[y])^2] :> f[e, x, y],
    {y, -Pi, Pi}],
  {x, -Pi, Pi}]

This way we don't attempt to integrate 1/Sqrt[1 - (3 - e - Cos[x] -
Cos[y])^2] symbolically but for each value of e we get a smooth
integrand.

In[4]:= % /.
      f[e_, x_, y_] :> 1/Sqrt[1 - (3 - e - Cos[x] - Cos[y])^2] /.
    Integrate[e_ + e2_, iter_] :>
      Integrate[e, iter] + Integrate[e2, iter] /.
  Integrate[Integrate[e_, iter_], iter2_] :>
    NIntegrate[e, iter2, iter,
      MaxRecursion -> 15, SingularityDepth -> 10^6, PrecisionGoal -> 2]

In[5]:= Plot[%, {e, 0, 6}, PlotDivision -> 3] // Timing

Out[5]= {3.078*Second, -Graphics-}

Large value of SingularityDepth prevents Mathematica from trying a
change of variables.

Maxim Rytin 
m.r at inbox.ru


  • Prev by Date: Hadamard -Sylvester Matrix Self-Similarity by substitution and reparatitioning
  • Next by Date: RE: RE: [heur] Scientific Notation on Y-axis of ListPlots
  • Previous by thread: Re: Question about DiracDelta
  • Next by thread: ContourPlot and finite approximation to level sets