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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.