Points sampled by NIntegrate

*To*: mathgroup at smc.vnet.net*Subject*: [mg67431] Points sampled by NIntegrate*From*: andrew.j.moylan at gmail.com*Date*: Fri, 23 Jun 2006 04:32:10 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Hi, [I apologise for the length of this post; I have tried to state my problem as clearly as possible, and the result is long.] I am trying to understand how NIntegrate decides at which points to sample a function when estimating its improper integral over e.g. {1, Infinity}. Please consider the following code: sampledPoints = {}; f[r_?NumericQ] := ( AppendTo[sampledPoints, r]; 1/(r^2 + Sin[r]) ) NIntegrate[f[r], {r, 1, Infinity}, PrecisionGoal -> 4] Take[Sort[sampledPoints], -10] Length[sampledPoints] The function f is just 1/(r^2 + Sin[r]), except it keeps track of the points at which it is evaluated, in a list called sampledPoints. After NIntegrate is called on f over {1, Infinity}, the last two lines returns the 10 largest values of r for which f[r] was evaluated, and the number of values of r for which f[r] is evaluated. In the case above, the following are returned: {69.3345, 85.2695, 125.67, 130.17, 170.539, 251.341, 341.078, 502.682, 1005.36, 2010.73} and 99. Now consider the following code, which is identical except that the PrecisionGoal option to NIntegrate has been increased from 4 to 5: sampledPoints = {}; f[r_?NumericQ] := ( AppendTo[sampledPoints, r]; 1/(r^2 + Sin[r]) ) NIntegrate[f[r], {r, 1, Infinity}, PrecisionGoal -> 5] Take[Sort[sampledPoints], -10] Length[sampledPoints] The following are now returned by the last two lines: {1005.363620876827`, 1518.2201755137608`, 2010.727241753654`, \ 20094.148354998237`, 34180.18350952692`, 6.859835603121333`*^7, \ 1.066219395284406`*^10, 1.931379671660648`*^19, 2.22745842812734`*^55, \ 8.429342764501086`*^109} and 176. To me, sampling 176 points instead of 99 seems perfectly reasonable; but I can't understand the presence of such large numbers as ~10^110 in the list of sampled points of f. Can anyone explain why they suddenly emerge as the PrecisionGoal increases? Now, since 1/(r^2 + Sin[r]) is not (I assume) any more expensive to compute at ~10^110 than at around ~1, these enormous values at which f is sampled do not prevent NIntegrate from converging rapidly. However, I wish to integrate elements of a class of functions that _are_ more expensive to evaluate at large values of their argument. The functions I am interested in integrating over {1, Infinity} go to zero as their argument goes to Infinity (as they must, if they are to converge) and do so monotonically. Their values at ~10^100 might as well be zero. By a hack, I could define them to be zero for values of their argument larger than some cutoff; or I could use only low options for PrecisionGoal in NIntegrate; but I would prefer to understand what it is that is making NIntegrate suddenly choose to sample the integrand at very large values of the argument. Thanks for any help. Cheers, Andrew.