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Re: how packages work

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  • Subject: [mg116686] Re: how packages work
  • From: Daniel Lichtblau <danl at>
  • Date: Thu, 24 Feb 2011 06:21:03 -0500 (EST)

Ted Sariyski wrote:
> Hi,
> I am confused how packages work. I have a simple function which I can plot
> and evaluate without any messages or warnings, e.g. 
> # f[3000, lam1, lam2]/.{lam1-> 4.*10^-7, lam2->5.*10^-7}
> Out: -411.32
> When I try FindRoot I get an obscure message followed by a correct result:
> # FindRoot[f[x, lam1, lam2] == 0, {x, 3000}] (*lam1=4.*10^-7;
> lam2=5.*10^-7*)
> Out: NIntegrate::inumr: The integrand 3.74177*10^-16/((-1+E^(<<21>>/(x
> z$36620))) z$36620^5) has evaluated to non-numerical values for all sampling
> points in the region with boundaries {{4.*10^-7,5.*10^-7}}. >>
> {x -> 4969.42}
> Function f is defined through two modules in myUtils` package:
> f[x_, lam1_, lam2_] := inbInt[x, lam1, lam2] - 420.;
> (*myUtils`*)
> inbInt[t_,l1_,l2_] :=
>     Module[ {z,bbint},
>         bbint = NIntegrate[pFun[t, z],{z,l1,l2}];
>         Return[bbint];
>     ];
> pFun[t_, x_] :=
>     Module[ {bbint},
>         bbint = (3.74*10^-16/ x^5)/(Exp[0.014/(x*t)]-1);
>         Return[bbint];
>     ]; 
> How is that Plot doesn't complain for non-numerical values but FindRoot
> does?
> Thanks in advance,
> --Ted

The message is not uninformative. It indicates that there is an 
integrand to an NIntegrate that has a non-numeric integrand (that is to 
say, contains parameters other than the variable of integration). 
Moreover it indicates that the variables it encounters are x and 
something like z$36620. So we expect that a "z" variable, localized in 
some way (hence the appended "$36620"), is in use. Clearly that is just 
the NIntegrate "z" variable of integration. We can conclude it is "x" 
that needs a value.

As has been mentioned in past on this forum, a good way to prevent this 
problem is to force the function passed to FindRoot (inbInt in this 
case) to be what is called a "black box", that is, undefined unless it 
has numeric arguments. That way a function like FindRoot will not be 
able to do any symbolic preprocessing that it might otherwise attempt 
(in this case, with warning-riddled and fruitless results).

The code below, a slight simplification of yours, does this.

f[x_, lam1_, lam2_] := inbInt[x, lam1, lam2] - 420.

inbInt[t_?NumericQ, l1_?NumericQ, l2_?NumericQ] :=
  NIntegrate[pFun[t, z], {z, l1, l2}]

pFun[t_, x_] := (3.74*10^(-16)/x^5)/(Exp[0.014/(x*t)] - 1)

I can now get a quiet evaluation of that FindRoot.

In[18]:= With[{lam1 = 4.*10^(-7), lam2 = 5.*10^(-7)},
  FindRoot[f[x, lam1, lam2] == 0, {x, 3000}]]

Out[18]= {x -> 2000.56}

I should point out that this is in no way related to packages, their 
design issues notwithstanding.

Daniel Lichtblau
Wolfram Research

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