Re: Singularity-handling transformation employed by NIntegrate

• To: mathgroup at smc.vnet.net
• Subject: [mg71309] Re: Singularity-handling transformation employed by NIntegrate
• From: "antononcube" <antononcube at gmail.com>
• Date: Tue, 14 Nov 2006 05:06:31 -0500 (EST)
• References: <eiut46\$h63\$1@smc.vnet.net><ej71qt\$ifa\$1@smc.vnet.net>

```Andrew Moylan wrote:
>
> 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?

Yes that is correct. If a sub-region is obtained after SingularityDepth
bisections of the original integration region, the IMT rule is apllied
to that sub-region.

Anton Antonov
Wolfram Research, Inc.

>
> 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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