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MathGroup Archive 2002

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Integration problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg35496] Integration problem
  • From: Arnold Gregory Civ AFRL/SNAT <Gregory.Arnold at wpafb.af.mil>
  • Date: Tue, 16 Jul 2002 04:50:05 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

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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