Re: Error in integrals?
- To: mathgroup at smc.vnet.net
- Subject: [mg9499] Re: [mg9482] Error in integrals?
- From: "W. Meeussen" <w.meeussen.vdmcc at vandemoortele.be>
- Date: Sat, 8 Nov 1997 23:04:31 -0500
- Sender: owner-wri-mathgroup at wolfram.com
hi Clifford,
when in doubt, Plot it:
in the plot of ior[m] you see a nasty discontinuity at about m=0.26795
Your human input is required here (you're not out of a job soon
:Mathematica needs you).
again, the rule applies : PLOT IT
your function has three terms, begging you to show on screen :
Plot[{
1/3*2*Sqrt[3]*Sqrt[(m^2 - 1)^2/(m^2 + 1)^4] ,
1/3*Sqrt[(-Sqrt[3]*m^2 - 6*m + Sqrt[3])^2/(m^2 + 1)^4] ,
1/3*Sqrt[(-Sqrt[3]*m^2 + 6*m + Sqrt[3])^2/(m^2 + 1)^4]
},{m,0,1}]
and you see (by dropping term per term)that the second one has a nasty
bounce on zero at some m.
What m? let Mathematica find out :
Solve[ Sqrt[(-Sqrt[3]*m^2 - 6*m + Sqrt[3])^2/(m^2 + 1)^4]==0,m]
{{m -> -2 - Sqrt[3]}, {m -> 2 - Sqrt[3]}}
aha, it is at m -> 2 - Sqrt[3] that things go 'bump'. So, lets do the
old Integrate again, this time in two parts:
Integrate[ior[a],{a,0,2-Sqrt[3]}]
3 - 2 Sqrt[3]
--------------
-6 + 3 Sqrt[3]
Integrate[ior[a],{a,2-Sqrt[3],1}]
1 -3 + Sqrt[3]
-1 + ------- + ----------------
Sqrt[3] 3 (-2 + Sqrt[3])
Simplify[%+%%]
3 - 2 Sqrt[3]
-------------
-2 + Sqrt[3]
%//N
1.73205
So, NIntegrate did it right. You can do it symbolically if you
supplement the Integrate routines with some common sense.
I enjoyed it, altough some would be offended by the fact that we can not
yet "blindly" (without plotting it) trust Mathematica 's integration
packages. Blind trust ? no thanks!
wouter.
At 02:40 6-11-97 -0500, Clifford J. Nelson wrote:
>Mathematica 3.0 on a PowerMac 7200/90 88MB Mac OS 8 gives different
>results for Integrate vs. NIntegrate.
>
>Here is one example.
>
>ior[m_] := 1/3*(2*Sqrt[3]*Sqrt[(-1 + m^2)^2/(1 + m^2)^4] +
> Sqrt[(Sqrt[3] - 6*m - Sqrt[3]*m^2)^2/(1 + m^2)^4] + Sqrt[(Sqrt[3] +
>6*m - Sqrt[3]*m^2)^2/(1 + m^2)^4])
>
>In[31]:=
>N[Integrate[ior[m],{m,0,1}]]
>
>Out[31]=
>-0.42265
>
>In[32]:=
>NIntegrate[ior[m],{m,0,1}]
>
>Out[32]=
>1.73206
>
>What am I doing wrong ? Which answer is correct ?
>
>Cliff Nelson
>
>
>
>
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be