NMinimize with numerical integrating

Question

I'm trying to find the coefficients of a function by minimizing an equation who I know is zero with Mathematica. My code is:

Clear[f];
Clear[g];
Clear[GetGood];
Clear[int];
Clear[xlist];
Xmax = 10;
n = 10;
dx = Xmax/n;
xlist = Table[i*dx, {i, n}];
A = 3.5;
slope = (A + 2)/3;
f[x_, a_, b_, c_, d_, e_] :=a/(1 + b*x + c*x^2 + d*x^3 + e*x^4)^(slope/4 + 2);
g[x_, a_, b_, c_, d_, e_] :=Derivative[1, 0, 0, 0, 0, 0][f][x, a, b, c, d, e];
int[a_?NumericQ, b_?NumericQ, c_?NumericQ, d_?NumericQ, e_?NumericQ] :=
     Module[{ans, i},ans = 0;Do[ans =ans + Quiet[NIntegrate[
    y^-slope*(f[Sqrt[xlist[[i]]^2 + y^2 + 2*xlist[[i]]*y*m], a, b,
         c, d, e] - f[xlist[[i]], a, b, c, d, e]), {m, -1, 1}, {y,
      10^-8, \[Infinity]}, MaxRecursion -> 30]], {i, 1, 
 Length[xlist]}];
ans
];
GetGood[a_?NumericQ, b_?NumericQ, c_?NumericQ, d_?NumericQ,e_?NumericQ] := 
    Module[{ans},ans = Abs[Sum[3*f[x, a, b, c, d, e] + x*g[x, a, b, c, d,e],
       {x,xlist}]+2*Pi*int[a, b, c, d, e]];
        ans
];
NMinimize[{GetGood[a, b, c, d, e], a > 0, b > 0, c > 0, d > 0, 
 e > 0}, {a, b, c, d, e}]

The error I get after the last line is:

Part::pspec: Part specification i$3002170 is neither an integer nor a list of integers. >>
NIntegrate::inumr: "The integrand (-(1.84529/(1+<<3>>+0.595769 Part[<<2>>]^4)^2.45833)+1.84529/(1+<<18>> Sqrt[Plus[<<3>>]]+<<1>>+<<1>>+0.595769 Plus[<<3>>]^2)^2.45833)/y^1.83333 has evaluated to non-numerical values for all sampling points in the region with boundaries {{-1,1},{\[Infinity],1.*10^-8}}"

Any ideas why I am getting an error?

Thanks

Answer 1

Change your NMinimize to be

NMinimize[{GetGood[a,b,c,d,e],a>0&&b>0&&c>0&&d>0&&e>0}, {a,b,c,d,e}]

to get your constraints to work correctly. Their help page should really show an example of using more than a single constraint. This old page does show an example.

http://reference.wolfram.com/legacy/v5_2/functions/AdvancedDocumentationNMinimize

If you change your int[] to

int[a_?NumericQ, b_?NumericQ, c_?NumericQ, d_?NumericQ, e_?NumericQ] :=
Module[{ans, i}, ans = 0; Do[
Print["First i=", i];
ans = ans + Quiet[NIntegrate[
    Print["Second i=", i]; 
    y^-slope*(f[Sqrt[xlist[[i]]^2 + y^2 + 2*xlist[[i]]*y*m], a,b,c,d,e] -
f[xlist[[i]], a,b,c,d,e]), {m,-1,1}, {y,10^-8, \[Infinity]}, MaxRecursion -> 30]],
{i, 1, Length[xlist]}];
ans];

you will see

First i=1
Second i=1
....
First i=10
Second i=i$28850

where the first debug print never says i=i$nnnn but the second debug print does often show that i has been unassigned a value only inside your NIntegrate, not outside it, and only after i has reached a value of 10, the length of your xlist, and at that point you can't subscript by a symbol and you get the error messages you have seen.

Nothing inside your NIntegrate is changing the value of i.

I think you may have stumbled onto a bug where Mathematica is writing over the value of i.

See if you can simplify the code without driving the bug into hiding. If you can make it simpler and still show the problem you might have more likelihood of success in getting Wolfram to admit you have found a bug.