Numerical integration
- To: mathgroup at smc.vnet.net
- Subject: [mg79621] Numerical integration
- From: dimitris <dimmechan at yahoo.com>
- Date: Tue, 31 Jul 2007 06:13:18 -0400 (EDT)
In[3]:=
Integrate[Log[1 + z^2]*(BesselJ[1, z]^2/z), {z, 0, Infinity}]
N[%, 10]
Out[3]=
MeijerG[{{1/2}, {}}, {{0, 0}, {-1}}, 1]/(2*Sqrt[Pi])
Out[4]=
0.8732180258611361020606751916`10.
On another CAS I took,
convert("Integrate[Log[1 + z^2]*(BesselJ[1, z]^2/z), {z, 0,
Infinity}]",FromMma,evaluate);
evalf(%,20);
1/2
1/2 (2 Pi BesselI(0, 1) BesselK(0, 1)
1/2 / 1/2
+ 2 Pi BesselK(1, 1) BesselI(1, 1)) / Pi
/
0.87321802586113613925
Both CAS return the same symbolic result.
[An interesting challenge is to simplify the MeijerG
output of Mathematica to that of the other CAS]
I want to check this symbolic result with NIntegrate.
I have "played around" with the options but I could get
"more closely" than
In[16]:=
NIntegrate[Log[1+z^2]*(BesselJ[1,z]^2/z),{z,
0, },MaxRecursion\[Rule]18]//InputForm
>From In[16]:=
\!\(\*
RowBox[{\(NIntegrate::"slwcon
"\), \(\(:\)\(\ \)\), "\<\"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. \
\\!\\(\\*ButtonBox[\\\"More...\\\", ButtonStyle->\\\"RefGuideLinkText\\
\", \
ButtonFrame->None, ButtonData:>\\\"NIntegrate::slwcon\\\"]\\)\"\>"}]\)
>From In[16]:=
\!\(\*
RowBox[{\(NIntegrate::"ncvb"\), \(\(:\)\(\ \)\), "\<\"NIntegrate
failed
to converge to prescribed accuracy
after \\!\\(19\\) recursive bisections in \\!\\(z\\) near \\!\\(z\
\) = \
\\!\\(35857.55603944016`\\). \\!\\(\\*ButtonBox[\\\"More...\\\", \
ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \
ButtonData:>\\\"NIntegrate::ncvb\\\"]\\)\"\>"}]\)
Out[16]//InputForm=
0.8732193803103058
I would be very happy if someone pointed a tactic for perfroming
satisfactory numerical integration with mathematica for this integral.
I use mathematica 5.2 but you can use Mathematica 6 as well!
The integral arose in another forum. There it was pointed out that
the performance of Mathematica 6 is buggy as regards numerical
integration.
I look forward to seeing any replies.
Greetings from burning Greece!
Dimitris
PS
See here
http://groups.google.gr/group/sci.math.symbolic/browse_thread/thread/57af36ff6f540a0d/a076ffbc412f974a?hl=el#a076ffbc412f974a
for above mentioned thread.