RE: Incorrect integral
- To: mathgroup at smc.vnet.net
- Subject: [mg43645] RE: [mg43623] Incorrect integral
- From: "Rasmusson, Lars" <larsr at exch.hpl.hp.com>
- Date: Sat, 27 Sep 2003 04:58:08 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Thanks again! This also explains the behavior I observed in my previous mail. Extra terms are added and/or dropped due to the disconinuous behaviour of the function. I'll work my way around it with Limit[...]. Best regards, Lars -----Original Message----- From: Daniel Lichtblau [mailto:danl at wolfram.com] To: mathgroup at smc.vnet.net Subject: [mg43645] Re: [mg43623] Incorrect integral "Rasmusson, Lars" wrote: > > Thank you for help. It seems to work. > > However, Mathematica does not seem to be working correctly when the > integration boundaries are given symbolicly. Is there a way to handle > that as well? > > To see that it is incorrect, compare the result of > > In[1]:= > InputForm[f[p_] = Integrate[UnitStep[k-a*p]*(k-a*p), > {k,K0,K1}, {a,A0,A1}, Assumptions->0<p<1]] ; > > In[2]:= > f[0.1]/.{K0|A0->0,K1|A1->1} > > Out[2]= > 0.453333 > > with the correct value 0.451667. > > Again, I'm very grateful for your good response to my question. > > Best regards, > Lars Rasmusson > > -----Original Message----- > From: Daniel Lichtblau [mailto:danl at wolfram.com] To: mathgroup at smc.vnet.net > Sent: Friday, September 26, 2003 7:47 AM > To: Lars Rasmusson > Cc: mathgroup at smc.vnet.net > Subject: [mg43645] Re: [mg43623] Incorrect integral > > Lars Rasmusson wrote: > > > > Hi, it seems like Integrate is not handling the Max[ ] function > > properly, or am I mistaken? Compare the two outputs In[1]:= > > > > fa[p_] := NIntegrate[Max[0, k - a p], {k, 0, 1}, {a, 0, 1}] fb[p_] > > := Integrate[Max[0, k - a p], {k, 0, 1}, {a, 0, 1}] fa[0.1] fb[0.1] > > > > Out[3]= > > 0.451667 > > Out[4]= > > 1.66667 > > > > The outputs differ. NIntegrate returns the correct value. > > > > In[5]:= $Version > > Out[5]= 5.0 for Microsoft Windows (June 11, 2003) > > > > Is there any way to get a correct behavior from Mathematica? I > > would be happy to receive the symbolic solution (to a more > > complicated integral of this kind). > > > > Thanks, > > > > Lars > > One method that Integrate can handle is to reformulate using UnitStep > instead of Max. > > In[9]:= InputForm[f[p_] = Integrate[UnitStep[k-a*p]*(k-a*p), > {k,0,1}, {a,0,1}, Assumptions->0<p<1]] > Out[9]//InputForm= (3 - 3*p + p^2)/6 > > In[11]:= f[.1] > Out[11]= 0.451667 > > Daniel Lichtblau > Wolfram Research This one is not so obvious. It has to do with the fact that you get different limiting behavior depending on how you arrive at the specified values. In particular we will at first refrain from performing the substitution for A0. In[30]:= InputForm[f[p] /. {A1|K1->1, K0->0, p->1/10}] Out[30]//InputForm= (5*((27*(-1 + A0/10)*(-1 + A0)*UnitStep[1 - A0/10]* (UnitStep[1 - A0] + UnitStep[-1 + A0]))/100 + UnitStep[1 - A0/10]*(((1 - 3*A0 + 3*A0^2)*UnitStep[-A0/10])/1000 - ((-1 + A0)^3*UnitStep[1 - A0]*UnitStep[A0/10])/1000)))/3 Now we do the substitution. In[4]:= ee /. A0->0 34 Out[4]= -- 75 But if you look at the prior expression you see a UnitStep[A0] term which indicates that we have discontinuous behavior at A0==0. We'll take a limit as we approach from the positive side (default behavior for Limit). In[5]:= Limit[ee, A0->0] 271 Out[5]= --- 600 So you have a path-dependent discontinuity at the intended values. Blind substitution cannot reliably work in this situation. Offhand I do not know how to work around this. The discontinuous behavior pretty much implies that you have to do things carefully at that point. Daniel Lichtblau Wolfram Research