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