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Re: Singularity-handling transformation employed by NIntegrate

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71250] Re: Singularity-handling transformation employed by NIntegrate
  • From: "Andrew Moylan" <andrew.j.moylan at gmail.com>
  • Date: Sun, 12 Nov 2006 06:47:31 -0500 (EST)
  • References: <eiut46$h63$1@smc.vnet.net><ej1q02$dvu$1@smc.vnet.net>

Thanks Anton. Your response answers my question.

Is it correct to assume that the {xmin, xmax} represent the subinterval
containing the suspected singularity, rather than the entire region of
numerical integration specified in NIntegrate?

Cheers,
Andrew


On Nov 10, 11:06 pm, "antononcube" <antononc... at gmail.com> wrote:
> 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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