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 ÎÎ³Ï?Î±Ï?Îµ: > 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 ÎÎ³Ï?Î±Ï?Îµ: > > 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.