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MathGroup Archive 2002

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Re: Integration problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg35528] Re: Integration problem
  • From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
  • Date: Thu, 18 Jul 2002 03:06:10 -0400 (EDT)
  • Organization: Universitaet Leipzig
  • References: <ah24r0$r3u$1@smc.vnet.net>
  • Reply-to: kuska at informatik.uni-leipzig.de
  • Sender: owner-wri-mathgroup at wolfram.com

Hi,

can you simply stop to mix numerical values like 81. with
analytical calculations ? Your problem is, that the integration
must find the roots of 1 - t^2 - 81 t^4

If you wish to use analytical calculations avoid inexact
numerical data like 0. or 81.

Regards
  Jens


Arnold Gregory Civ AFRL/SNAT wrote:
> 
> On Mathematica 4.1 w/ Win2000, I'm having the following problem with an integral:
> In[1]:= expr1 = (t)^2* UnitStep[1 - t^2 - 81. t^4]
> Out[1]= t^2 UnitStep[1 - t^2 - 81. t^4]
> 
> In[2]:= expr2 = (0.+ t)^2 UnitStep[1 - t^2 - 81. * t^4]
> Out[2]= (0.+ t)^2 UnitStep[1 - t^2 - 81. * t^4]
> 
> In[3]:= Integrate[Evaluate[expr1],{t,-1,1}]
> Out[3]= 0.0227181
> 
> In[4]:= Integrate[Evaluate[expr2],{t,-1,1}]
> Out[4]=0.666667
> 
> In[5]:= Integrate[Evaluate[Rationalize[expr2]],{t,-1,1}]
> Out[5]= (10*Sqrt[26*(-1 + 5*Sqrt[13])] -
>   2*Sqrt[-2 + 10*Sqrt[13]])/8748
> 
> In[6]:= N[%]
> Out[6]= 0.0227181
> 
> In[7]:= NIntegrate[Evaluate[Rationalize[expr2]],{t,-1,1}]
> NIntegrate::slwcon: Numerical integration converging too slowly; suspect one \
> of the following: singularity, value of the integration being 0, oscillatory \
> integrand, or insufficient WorkingPrecision. If your integrand is oscillatory \
> try using the option Method->Oscillatory in NIntegrate.
> NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after \
> 7 recursive bisections in t near t = -0.320313.
> Out[7]= 0.0225802
> 
> It doesn't matter if I put the expression directly into the integrals.  I can remove the difference by rationalizing either the expression t+0. or UnitStep[...] & both cases give the correct result.  I can also change to (t+eps) and then evaluate eps->0. and get the correct result.  In short, it looks like this particular form yields an unstable result & I'm looking for advice on approaches to mitigate this happening within a sequence of long & complex calculations.  It isn't clear to me that Rationalize is the best approach or that it will always guarantee correcting the problem.
> 
> Thanks!
> Greg


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