Mathematica 9 is now available
Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2004
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2004

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: Integrate vs. NIntegrate

  • To: mathgroup at smc.vnet.net
  • Subject: [mg51436] Re: Integrate vs. NIntegrate
  • From: carlos at colorado.edu (Carlos Felippa)
  • Date: Sun, 17 Oct 2004 03:05:37 -0400 (EDT)
  • References: <ckqnbf$nh0$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

carlos at colorado.edu (Carlos Felippa) wrote in message news:<ckqnbf$nh0$1 at smc.vnet.net>...
> Which one is the correct answer?
> 
> f=2+Cos[2 x]+ I*Sin[2 x];
> 
> Print[ N[Simplify[ Integrate[Log[f],{x,-Pi,Pi}]/(2*Pi)]] //InputForm];
> 0.34657359027997264 + 1.5707963267948966*I
> 
> Print[            NIntegrate[Log[f],{x,-Pi,Pi}]/(2*Pi)   //InputForm];
> 0.6931471805599428 + 0.*I
> 
> I believe the second one, have doubts about the first.

The 2nd answer  of Log[2] is verified by a brute force
Gauss-Legendre numerical integrator used in HWs:

NIntGauss3[fx_,x_,lower_,upper_,n_]:= Module[{fint,d,x1,x2,f,
   	g1=N[(1+Sqrt[3/5])/2],g2=N[1/2],g3=N[(1-Sqrt[3/5])/2]},
   	d=(upper-lower)/n; fint=0; f=fx;
   	Do[ x1=lower+(i-1)*d; x2=lower+i*d;  
	    fint +=(N[5*f/.x->N[x1*g1+x2*g3]]+
             N[8*f/.x->N[(x1+x2)/2]]+
             N[5*f/.x->N[x1*g3+x2*g1]])*N[d/18],
	   {i,1,n}];
	Return[fint]];

f=2+Cos[2 x]+ I*Sin[2 x];
Print[ NIntGauss3[Log[f],x,-Pi,Pi,200]/(2*Pi) //InputForm];
0.6931471805599455 - 3.969653887102225*^-17*I

Print[ NIntegrate[Log[f],{x,-Pi,Pi}]/(2*Pi)   //InputForm];
0.6931471805599428 + 0.*I

Print[ N[Log[2]] //InputForm];
0.6931471805599453

so NIntegrate[] gives 14 correct digits.


  • Prev by Date: Re: Concatenate matrices
  • Next by Date: Cursor is deactivated after Ctrl+Alt-combination
  • Previous by thread: Re: Integrate vs. NIntegrate
  • Next by thread: Re: Integrate vs. NIntegrate