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MathGroup Archive 2006

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Re: CauchyPrincipalValue questions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg69689] Re: CauchyPrincipalValue questions
  • From: dimmechan at yahoo.com
  • Date: Thu, 21 Sep 2006 07:29:07 -0400 (EDT)
  • References: <eeof5i$23f$1@smc.vnet.net><eeqqtl$oro$1@smc.vnet.net>

After private communacation with Daniel Huber I have some things to
state.

I wondered myself for long time why we can give for Integrate a list
argument.
Because Integrate does not have the Listable argument.

Attributes[Integrate]
{Protected, ReadProtected}

lst = {1/x, 1/x^2};
Integrate[lst, {x, 1, 2}]
{Log[2], 1/2}

After I learn that for NIntegrate or Plot you should use Evaluate,
because otherwise (here) they do not evaluate their arguments to
numeric values.

{Attributes[NIntegrate], Attributes[Plot]}
{{HoldAll, Protected}, {HoldAll, Protected}}
NIntegrate[Evaluate[lst], {x, 1, 2}]
{0.693147180559947, 0.5000000000000211}
Plot[Evaluate[lst], {x, 1, 2}]

However I kept wondering how they become "listable".
I thought it was something too trivial to ask in the forum.
But now I read your message (...that this works nevertheless with
NIntegrate is astonishing...) I feel from one side "happy" (these
commands normally should not share the listability identity) because I
was right but from the other side I am confused...
Mathematica sometimes do not work how I expected.

But ok most of the cases is due to me!

Dimitris Anagnostou

Î?/Î? dimmechan at yahoo.com έγÏ?αÏ?ε:
> Dear Daniel,
>
> Thanks for your response.
>
> As regards my second question I also take notice of the appearance of
> these "x" for other functions. What is more strange is that evaluating
> by yourself the integral, these "x" do not appear.
> Ok it may be a quirck of CauchyPrincipalValue but this package use
> Nntegrate (or something I miss).
>
> eps = 10^(-5);
> aa = Reap[NIntegrate[1/x, {x, -1, 0 - eps}, EvaluationMonitor :>
> Sow[x]] + NIntegrate[1/x, {x, 0 + eps, 2},
> EvaluationMonitor :> Sow[x]]];
> bb = Block[{Message}, Reap[CauchyPrincipalValue[1/x, {x, -1, {0}, 2},
> EvaluationMonitor :> Sow[x]]]];
>
> aa[[1]]
> Length[aa[[2,1]]]
> Cases[aa[[2,1]], x]
> bb[[1]]
> Length[bb[[2,1]]]
> Cases[bb[[2,1]], x]
> 0.6931471806032992
> 506
> {}
> 0.6931471805602387
> 55
> {x, x, x, x, x, x, x, x, x, x, x}
>
> Note also the big difference in the number of sampled points.
>
> As regards my first question, maybe I was not too clear.
> I apologised for this.
>
> Let me state it again. First define a list of two functions.
>
> lst = {1/(x - Pi/2), Tan[x]};
>
> Then
>
> NIntegrate[Evaluate[lst], {x, 0, Pi/4}]
> {-0.6931471805599468, 0.3465735902799742}
>
> because NIntegrate has the attribute HoldAll (as you know without
> Evaluate they appeared the NIntegrate::inum: messages and the integral
> returns unevaluated).
> However both of the following commands fail (they appeared the
> NIntegrate::inum: messages).
>
> CauchyPrincipalValue[lst, {x, 0, {Pi/2}, 3*(Pi/4)}, AccuracyGoal -> 20]
> CauchyPrincipalValue[Evaluate[lst], {x, 0, {Pi/2}, 3*(Pi/4)},
> AccuracyGoal -> 20]
>
> The question is why.
> Do I miss something here?
> Thanks for your time
>
> Regards
> Dimitris
>
>
> Î?/Î? dimmechan at yahoo.com έγÏ?αÏ?ε:
> > Hello to all.
> >
> > I have two questions about the CauchyPrincipalValue package.
> >
> > Needs["NumericalMath`CauchyPrincipalValue`"]
> >
> > First
> >
> > lst = {1/Sqrt[x], Sin[x]/x};
> > N[Integrate[lst, {x, 0, 10}]]
> > {6.324555320336759, 1.658347594218874}
> >
> > However
> >
> > NIntegrate[lst, {x, 0, 10}]
> > \!\(NIntegrate::"inum" : "Integrand \!\({1\/\@x,
> >     Sin[x]\/x}\) is not numerical at \!\({x}\) = {5.`}"\)
> > NIntegrate[lst, {x, 0, 10}]
> >
> > because NIntegrate has the attribute HoldAll
> >
> > ({#1, Attributes[#1]} & ) /@ {Integrate, NIntegrate}
> > {{Integrate, {Protected, ReadProtected}}, {NIntegrate, {HoldAll,
> > Protected}}}
> >
> > As it well known someone must enclose the argument of NIntegrate in
> > Evaluate here.
> >
> > NIntegrate[Evaluate[lst], {x, 0, 10}]
> > {6.324555320387874, 1.6583475942188746}
> >
> > However why the following command fails, since, as the HelpBrowser
> > says, CauchyPrincipalValue uses internally the NIntegrate?
> >
> > lst2 = {1/(x - Pi/2), Tan[x]};
> > CauchyPrincipalValue[Evaluate[lst2], {x, 0, {Pi/2}, Pi}]
> > NIntegrate::inum : Integrand (the rest of the messages is not
> > displayed...)
> > CauchyPrincipalValue[{1/Sqrt[x], Sin[x]/x}, {x, 0, {Pi/2}, Pi}]
> >
> > My query become even bigger considering that
> >
> > Attributes[CauchyPrincipalValue]
> > {Protected}
> >
> > My second question follows immediately.
> >
> > Consider the (numerical evaluation) of CPV of 1/x over {x,-1,2}.
> >
> > aa = Reap[CauchyPrincipalValue[1/x, {x, -1, {0}, 2}, AccuracyGoal ->
> > 20,
> > WorkingPrecision -> 30, EvaluationMonitor :> Sow[x]]];
> >
> > This is the value of the integral
> >
> > aa[[1]]
> > 0.69314718055994530942
> >
> > This the total number of sampled points by NIntegrate
> >
> > Length[aa[[2,1]]]
> > 273
> >
> > And here are the last 25 sampled points by NIntegrate.
> >
> > Take[aa[[2,1]], -25]
> > {1.492698661756523787149586238081974793655503742219`29.99999998468416,
> >   1.382301338243476212850413761918025206344496257781`29.96663056969523,
> > 1.465413938559055852040904875211822497115793411141`29.999999992110578,
> > 1.409586061440944147959095124788177502884206588859`29.983131270623772,
> >   x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x}
> >
> > >From where appeared all these "x" in the last result?
> > It seems that NIntegrate (although the result is not affected
> > as it is demonstrated below) fails to sampled the integrand for
> > same values of x. Why this happens?
> >
> > N[Integrate[1/x, {x, -1, 2}, PrincipalValue -> True], 20]
> > 0.69314718055994530941723212145817656807`20.
> > 
> > Thanks in advance for any help.


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