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Re: Singularity-handling transformation employed by NIntegrate


NIntegrate uses the so called IMT Rule (the Iri, Moriguti, Takasawa
rule) that changes the integration variable with this transformation
over the interval [xmin, xmax]:

Clear[\[Phi]]
\[Phi][{xmin_, xmax_}, t_] := xmin + (xmax - xmin)*E^(1 - 1/t);
\[Phi][t_] := \[Phi][{0, 1}, t]


It is instructive to look at the plot of the transformation:

Plot[\[Phi][t], {t, 0, 1}, PlotRange -> All, AxesOrigin -> {-0.05,
-0.05}]

and at the plot of its derivative (Jacobian):

Plot[Evaluate[D[\[Phi][t], t]], {t, 0, 1}, PlotRange -> All]

The IMT rule "flattens" the singularity, but the points are clustered
closer to the singular point. That is why in many(most) cases extra
precision is required for the evaluation of the integrand.

Anton Antonov,
Wolfram Research, Inc.


Andrew Moylan wrote:
> Hi all,
>
> Further to a previous question: Can anyone tell me what transformation
> NIntegrate employs when a singularity is detected (i.e., when the
> number of recursive subdivisions specified by SingularityDepth is
> reached)?
> 
> Cheers,
> Andrew


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