CauchyPrincipalValue questions

*To*: mathgroup at smc.vnet.net*Subject*: [mg69644] CauchyPrincipalValue questions*From*: dimmechan at yahoo.com*Date*: Tue, 19 Sep 2006 05:44:52 -0400 (EDT)

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.