sampled points by Method->Oscillatory

*To*: mathgroup at smc.vnet.net*Subject*: [mg69450] sampled points by Method->Oscillatory*From*: dimmechan at yahoo.com*Date*: Wed, 13 Sep 2006 04:01:15 -0400 (EDT)

Dear all, (I have converted everything to InputForm) Clear["Global`*"] $Version "5.2 for Microsoft Windows (June 20, 2005)" ***Let me consider the following code: sampledPoints={}; g[x_?NumberQ]:=(AppendTo[sampledPoints,x];1/x) ***The function g is just 1/x, except it keeps track of the points at which is evaluated, in a list called sampledPoints (see also http://groups.google.com/group/comp.soft-sys.math.mathematica/browse_thread/thread/11d40e17edbea9e9/b223cb3882539ac8?lnk=st&q=NIntegrate+SingularityDepth&rnum=3#b223cb3882539ac8). ***The function g[x]*Sin[x]=Sin[x]/x is a oscillatory function very slowly convergent to zero as it can be seen from the following plot Plot[g[x]*Sin[x],{x,0,30},PlotPoints->1000,PlotRange->{Automatic,{-0.3,1.1}}]; ***Analytically the integral of g[x]*Sin[x] over the range {x,0,Infinity} is equal to: Integrate[f[x],{x,0,8}]//Timing {0.04699999999999971*Second, Pi/2} N[%[[2]]] 1.5707963267948966 ***To get an approximation for this integral the option Method->Oscillatory is used. NIntegrate[g[x]*Sin[x],{x,0,8},Method->Oscillatory] 1.5707963267948966 ***Can someone explain me what is the concluding remarks from the following plots? Length[sampledPoints] 3784 ListPlot[sampledPoints]; ListPlot[Take[sampledPoints,100]]; ListPlot[Take[sampledPoints,{100,2000}]]; ListPlot[Take[sampledPoints,{1500,3000}]]; ***I have some ideas but I am a little in a mesh to put them in a row. ***I believe that the first ~30 points (0<x<3Pi) are the sampling points not used in acceleration algorithm. Am I right? ***There is also an apparent periodicity. What does this mean? ***What happen near the point numbered 1700? Thanks in advance for any assistance. Dimitris Anagnostou