       Re: NIntegrate and FindRoot error

• To: mathgroup at smc.vnet.net
• Subject: [mg75687] Re: NIntegrate and FindRoot error
• From: Szabolcs <szhorvat at gmail.com>
• Date: Tue, 8 May 2007 06:03:15 -0400 (EDT)
• Organization: University of Bergen
• References: <20070507050017.M68851@phys.cts.nthu.edu.tw> <463EC1B7.1080102@smc.vnet.net> <f1msjl\$rok\$1@smc.vnet.net>

```Judith Bunder wrote:
> Hi,
>
> I have to perform a double integral. To evaluate the integrand I have to use
> FindRoot and this seems to work as I can evaluate the integrand at
> specific points and even make a 3D plot over the whole range. However, when
> I try to use NIntegrate I get a FindRoot::"nlnum" error message:
>
> Here is a simplified version of my code:
>
> Ly = 3;
> w = 5;
> t2 = 1;
> t1[kx_] := 2 t2 Cos[kx/2];
>
> py[kx_, m_ /; m >= 1 && m <= Ly] := p/. FindRoot[(2 Ly + 1) p +ArcTan[ Tan
> [p] (t1[kx] - t2)/(t1[kx] + t2)]==m Pi,{p,1}]
>
> WG[kx1_, kx2_] := Sum[Sum[py[kx1, m1] py[kx2, m2],{m1,1,Ly-1}],{m2,1,Ly-1}]
>
> Plot3D[Re[Exp[ I w (a2 - a1)] WG[a1, a2]], {a2, -Pi, Pi}, {a1, -Pi, Pi}]
> Plot3D[Im[Exp[ I w (a2 - a1)] WG[a1, a2]], {a2, -Pi, Pi}, {a1, -Pi, Pi}]
>
> NIntegrate[Exp[ I w (a2 - a1)] WG[a1, a2], {a2, -Pi, Pi}, {a1, -Pi, Pi}]
>
> The NIntegrate command fails. It appears to be objecting to my Cos[2 kx]
> function in the FindRoot command. I don't understand why it would object as
> Plot3D over the same integrand worked.
>
> Thanks a lot,
> Judy Bunder

The problem is that while Plot3D holds its first argument, NIntegrate
evaluates it immediately. So WG[a1, a2] is evaluated before a1 and a2
get a numeric value.

Change the definition of WG like this:

WG[kx1_?NumericQ, kx2_?NumericQ] := ...

and make sure that when you evaluate WG[x,y] (symbolic arguments), it
returns unevaluated. (First you need to clear the old definition of WG,
otherwise the WG[kx1_, kx2_] and the WG[kx1_?NumericQ, kx2_?NumericQ]
definitions will coexist.)

Szabolcs

```

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