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

Out:=
0

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

Out:=
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 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:=
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:=
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:=
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:=

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

_/_/  _/_/_/_/_/_/_/_/_/   | 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
| Tel.: (+55) (192) 398112 or 300646
Marcelo de Almeida Bueno        | FAX:  (+55) (192) 393127 or 300646

```

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