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MathGroup Archive 1995

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Numerical Integration

  • To: mathgroup at christensen.cybernetics.net
  • Subject: [mg1982] Numerical Integration
  • From: Marcelo de Almeida Bueno - pos <bueno at ifi.unicamp.br>
  • Date: Mon, 4 Sep 1995 22:22:09 -0400

Hi,

             I would like to receive some hints (and tricks) about numerical integration with 
Mathematica. I don't know if the internal routines used by mathematica to resolve numerical 
integrations are poor (mainly, in dealing with singularities), or if I am not being able to 
use all Mathematica capabilities on this subject.
             Before I state my problem, let me show one (of many) strange result I have 
obtained::

In[1]:=
        Integrate[Cos[x1-x2]^2 Sqrt[Sin[x1-x2]^2],{x1,0,Pi},{x2,0,Pi}]

Out[1]:= 
        0 

In[2]:= 
        NIntegrate[Cos[x1-x2]^2 Sqrt[Sin[x1-x2]^2],{x1,0,Pi},{x2,0,Pi}]

Out[2]:=
        2.09439

             Take a look at Plot3D[Cos[x1-x2]^2 Sqrt[Sin[x1-x2]^2],{x1,0,Pi},{x2,0,Pi}], 
and you will see that the result in Out[1] is absurd!
             Another problem is the time to evaluate my integrals. I am using a DX4-100 Mhz with
40 M RAM (Mathematica 2.2.3 for Windows). 
             Consider the following function:


Out[3]:=
                           2    2      2
         (Sin[x1] (Sqrt[2 R  + w  - 2 R  Cos[x1 - x2]] - 
                      2     x1 - x2 2       2
              2 Sqrt[R  Sin[-------] ])) / R 
                               2
 
and the following action,

In[4]:= 
       R=1;
       w=.001;
       NIntegrate[%3,{x1,Pi/4,3 Pi/4},{x2,Pi/4,3 Pi/4},WorkingPrecision->17]

NIntegrate::slwcon: 
   Numerical integration converging too slowly; suspect one of the
     following: singularity, oscillatory integrand, or insufficient
     WorkingPrecision.
NIntegrate::slwcon: 
   Numerical integration converging too slowly; suspect one of the
     following: singularity, oscillatory integrand, or insufficient
     WorkingPrecision.
NIntegrate::slwcon: 
   Numerical integration converging too slowly; suspect one of the
     following: singularity, oscillatory integrand, or insufficient
     WorkingPrecision.
General::stop: 
   Further output of NIntegrate::slwcon
     will be suppressed during this calculation.
NIntegrate::ncvb: 
   NIntegrate failed to converge to prescribed accuracy after 13
     recursive bisections in x1 near {x1, x2} = {1.5707, 1.5707}.

Out[4]:=
         0.00001076

        After 34,684 s!!!
        Ok! The answer now is correct, but with RIWIAD (CERN) on the same machine I 
obtained the same answer in few minutes!!!
        Another example. Consider the function bellow:


Out[5]:=

-(R (5 Sin[x1] - 3 Sin[x1 - 2 x2] - 3 Sin[2 x1 - x2] - 5 Sin[x2]))
------------------------------------------------------------------ + 
                             2     x1 - x2 2
                     8 Sqrt[R  Sin[-------] ]
                                      2
    2    2      2
  (R  + w  - 3 R  Cos[x1 - x2]) (Sin[x1] - Sin[x2])
  ------------------------------------------------- - 
                   2    2      2
       2 R Sqrt[2 R  + w  - 2 R  Cos[x1 - x2]]
    2    2      2
  (R  + w  - 3 R  Cos[x1 - x2]) (-Sin[x1] + Sin[x2])
  -------------------------------------------------- + 
                   2    2      2
       2 R Sqrt[2 R  + w  - 2 R  Cos[x1 - x2]]
  R (-5 Sin[x1] + 3 Sin[x1 - 2 x2] + 3 Sin[2 x1 - x2] + 5 Sin[x2])
  ----------------------------------------------------------------
                              2     x1 - x2 2
                      8 Sqrt[R  Sin[-------] ]
                                       2

and,

In[6]:=
       R=1;
       w=.001;
       NIntegrate[%5,{x1,Pi/4,3 Pi/4},{x2,3 Pi/4,3 Pi/2}]
                                       1
Power::infy: Infinite expression -------------- encountered.
                                           -90
                                 Sqrt[0. 10   ]
Infinity::indet: 
                                   -57
                            -(0. 10    1 ComplexInfinity)
   Indeterminate expression ----------------------------- encountered.
                                          8
                                       1
Power::infy: Infinite expression -------------- encountered.
                                           -90
                                 Sqrt[0. 10   ]
Infinity::indet: 
                                 -57
                            0. 10    1 ComplexInfinity
   Indeterminate expression -------------------------- encountered.
                                        8
                                       1
Power::infy: Infinite expression -------------- encountered.
                                           -90
                                 Sqrt[0. 10   ]
General::stop: 
   Further output of Power::infy
     will be suppressed during this calculation.
Infinity::indet: 
                                   -57
                            -(0. 10    1 ComplexInfinity)
   Indeterminate expression ----------------------------- encountered.
                                          8
General::stop: 
   Further output of Infinity::indet
     will be suppressed during this calculation.
NIntegrate::rnderr: 
   Inexact arithmetic has caused {x1, x2} to take the value 
    {2.35619, 2.35619} where the integrand is singular.


Out[6]:=
                                 Pi  3 Pi        3 Pi  3 Pi
         NIntegrate[Out[5], {x1, --, ----}, {x2, ----, ----}]
                                  4    4           4     2


        And RIWIAD resolved it!


        Thanks in advance,

 
     _/_/  _/_/_/_/_/_/_/_/_/   | Department of Cosmic Rays and Chronology
    _/ _/_/ _/      _/      _/  | Institute of Physics "Gleb Wataghin"
   _/  _/  _/_/_/_/_/_/_/_/     | State University of Campinas, UNICAMP
  _/      _/      _/      _/    | 13083-970 Campinas, Sao Paulo, Brazil
 _/      _/      _/      _/     | e-mail:    bueno at ifi.unicamp.br
_/      _/      _/_/_/_/        | Home Page: http://www.ifi.unicamp.br/~bueno
                                | Tel.: (+55) (192) 398112 or 300646
Marcelo de Almeida Bueno        | FAX:  (+55) (192) 393127 or 300646




                                 
    



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