Re: Re: Nested numerical integration

• To: mathgroup at smc.vnet.net
• Subject: [mg99358] Re: [mg99287] Re: Nested numerical integration
• From: Leonid Shifrin <lshifr at gmail.com>
• Date: Sun, 3 May 2009 05:25:07 -0400 (EDT)
• References: <200904280845.EAA28624@smc.vnet.net> <gt98ok\$oea\$1@smc.vnet.net>

```Hi Anton,

Thanks, I was not aware of that (despite having  worked with NIntegrate
quite a bit - there are always things to learn). Taking my words back, and
sorry for the confusion.

Regards,
Leonid

On Sat, May 2, 2009 at 2:56 AM, <antononcube at gmail.com> wrote:

> On Apr 29, 6:05 am, Leonid Shifrin <lsh... at gmail.com> wrote:
> > Hi,
> >
> > direct 2D integration:
> >
> > In[1] = NIntegrate[w^2/(s - w), {s, 1, 5}, {w, -5, -1}]
> >
> > Out[1] = 25.8364
> >
> > Step-by-step integration:
> >
> > In[2] =
> >
> > Clear[int];
> > int[w_?NumericQ] := NIntegrate[w^2/(s - w), {s, 1, 5}];
> > NIntegrate[int[w], {w, -5, -1}]
> >
> > Out[2] = 25.8364
> >
> > The second method is more flexible since you may use it for
> > non-rectangular domains.
>
> This statement is not correct. Using the multi-dimentional NIntegrate
> specfication gives more flexibility.
>
> 1. Multi-dimensional NIntegrate can be used non-rectangular domains.
> In the example below the second variable has functional boundary:
>
> In[2]:= NIntegrate[w^2/(s - w), {s, 1, 5}, {w, Sqrt[s], -1}]
>
> Out[2]= -5.07779
>
> You can use sampling points plot to see what the domain looks like:
>
> Needs["Integration`NIntegrateUtilities`"]
> NIntegrateSamplingPoints[NIntegrate[w^2/(s - w), {s, 1, 5}, {w, Sqrt
> [s], -1}]]
>
>
> 2. Multi-dimensional NIntegrate can be used for integration over parts
> of the integration region (or their exclusion).
> In the example below a disk with center {3,-3} and radius 1 is
> excluded from the integration:
>
> In[12]:= NIntegrate[w^2/(s - w)*Boole[ (s - 3)^2 + (w + 3)^2 > 1], {s,
> 1, 5}, {w, -5, -1}]
>
> Out[12]= 21.0579
>
> Again you can see the integration domain with
>
> NIntegrateSamplingPoints[
>  NIntegrate[
>  w^2/(s - w)*Boole[ (s - 3)^2 + (w + 3)^2 >= 1], {s, 1,
>   5}, {w, -5, -1}]]
>
>
> 3. The examples above used a mutlti-dimensional integration rule. If
> want to use one dimensional rule or, moreover, if you want to use
> different one-dimensional rule in each dimension, you can specify this
> with the Method option. This specification is closer to an integration
> with nested NIntegrate's.
>
> In[19]:= NIntegrate[w^2/(s - w), {s, 1, 5}, {w, -5, -1},
>  Method -> {"GaussKronrodRule", "LobattoKronrodRule"}]
>
> Out[19]= 25.8364
>
> Here are the sampling points:
>
> In[20]:= NIntegrateSamplingPoints[
>  NIntegrate[w^2/(s - w), {s, 1, 5}, {w, -5, -1},
>  Method -> {"GaussKronrodRule", "LobattoKronrodRule"}]]
>
>
> Anton Antonov
>
>

```

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