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: > 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? 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