How to use NIntegrate in a purely numerical mode?

Question

I have a function that is defined as the smallest root of a given polynomial. Something like this:

f[param1_,param2_,param3]:= Module[{roots},
  roots = getRoots[param1,param2,param3];
  Return[Min[roots]];
]

The function getRootswould build a polynomial and use Solve[] to solve it.

I need to integrate fnumerically, but I keep getting errors:

 NIntegrate::inumr: "The integrand Min[<<1>>] has evaluated to non-numerical values for all sampling points in the region with boundaries {{0,300000}}"

Using Manipulate, I can calculate values along the integration range, that is, the function indeed has numerical values.

I suspect that the symbolic preprocessor tries to analyze the argument of Min[], but it get stuck because the argument is a list of roots of a polynomial.

Since the function f[]can be evaluated numerically, it should be possible to integrate it in a purely numerical way, without the symbolic analysis. Is there any way, any choice of integration method to do that?

Answer 1

Here is an example: let us take one equation, simple enough for a trial. Let it be

 Clear[x, y];
eq = y^5 - y^3 + x == 0;
sl = Solve[eq, y]

The result is:

(*  {{y -> Root[x - #1^3 + #1^5 &, 1]}, {y -> 
   Root[x - #1^3 + #1^5 &, 2]}, {y -> 
   Root[x - #1^3 + #1^5 &, 3]}, {y -> 
   Root[x - #1^3 + #1^5 &, 4]}, {y -> Root[x - #1^3 + #1^5 &, 5]}}  *)

The first of them, for example, is real at 0<=x<=1. Let us check it:

Plot[sl[[1, 1, 2]], {x, 0, 1}] 

Here it is shown:

Now let us integrate:

NIntegrate[sl[[1, 1, 2]], {x, 0, 1}]

The answer is:

(* -1.14189 *)