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I have the following system

$$-a_1^2 + 2 a_0 a_2 - 2 b_0 b_2 =0\quad \text{and } \quad a_2^2 - 2 a_1 a_3 + 2 a_0 a_4 - b_2^2 - 2 b_0 b_4 =0$$

where the following $a_i, b_i$ are integral of some heavy function defined before in the notebook. I can give some example, but they basically depend on hypertrigonometric function.

$$ a_i= \int dk f_i(k,c,s) \quad b_i= \int dk g_i(k,c,s) $$

And I am interested in solving the system for $c,s$.

F[m_?NumericQ, p_?NumericQ] := Quiet[Chop[
    -NIntegrate[f1[k,c,s]/. { c -> m, s -> p}, {k, 0,
          Infinity} , WorkingPrecision -> 20, AccuracyGoal -> 50 ]*
      NIntegrate[
       f1[k,c,s]/. {c -> m, s -> p}, {k, 0, 
        Infinity}, WorkingPrecision -> 20, AccuracyGoal -> 50  ] + 
     2 NIntegrate[
      f0[k,c,s]/. { c -> m, s -> p}, {k, 0, 
        Infinity}, WorkingPrecision -> 20, AccuracyGoal -> 50 ] *
      NIntegrate[
      f2[k,c,s] /. {c -> m, s -> p}, {k, 0, 
        Infinity}, WorkingPrecision -> 20, AccuracyGoal -> 50  ] - 
     2 NIntegrate[
      g0[k,c,s] /. { c -> m, s -> p}, {k, 0, 
        Infinity}, WorkingPrecision -> 20, AccuracyGoal -> 50 ] *
      NIntegrate[
      g2[k,c,s] /. {c -> m, s -> p}, {k, 0, 
        Infinity}, WorkingPrecision -> 20, AccuracyGoal -> 50 ]
    ]];

G[m_?NumericQ, p_?NumericQ] :=    Quiet[Chop[
    NIntegrate[
      f2[k,c,s] /. {c -> m, s -> p}, {k, 0, 
        Infinity}, WorkingPrecision -> 20, AccuracyGoal -> 50]*
      NIntegrate[
       f2[k,c,s] /. { c -> m, s -> p}, {k, 0, 
        Infinity}, WorkingPrecision -> 20, AccuracyGoal -> 50] - 
     2 NIntegrate[
      f1[k,c,s]/. { c -> m, s -> p}, {k, 0, 
        Infinity}, WorkingPrecision -> 20, AccuracyGoal -> 50 ] *
      NIntegrate[
     f3[k,c,s] /. { c -> m, s -> p}, {k, 0, 
        Infinity}] + 
     2 NIntegrate[
       f0[k,c,s] /. {c -> m, s -> p}, {k, 0, 
        Infinity}, WorkingPrecision -> 20, AccuracyGoal -> 50] *
      NIntegrate[
      f4[k,c,s] /. { c -> m, s -> p}, {k, 0, 
        Infinity}] -
     NIntegrate[
      g2[k,c,s] /. { c -> m, s -> p}, {k, 0, 
        Infinity}, WorkingPrecision -> 20, AccuracyGoal -> 50]*
      NIntegrate[
      g2[k,c,s] /. { c -> m, s -> p}, {k, 0, 
        Infinity}, WorkingPrecision -> 20, AccuracyGoal -> 50] - 
     2*NIntegrate[
     g0[k,c,s]/. { c -> m, s -> p}, {k, 0, 
        Infinity}, WorkingPrecision -> 20, AccuracyGoal -> 50]*
      NIntegrate[
       g4[k,c,s]/. {c -> m, s -> p}, {k, 0, 
        Infinity}, WorkingPrecision -> 30, AccuracyGoal -> 50]
    ]];

FindRoot[{Chop[F[c, s]] == 0,    Chop[G[c, s]] ==0}, {{c, 0.8}, {s, 0.01}}

but FindRoot struggle to find a solution (but I know one should exist around 0.81 and 0.0128), I think it is because of the operations between numerical integrations. Unfortunately, it is not possible to solve analytically, is there a way to combine FindRoot with multiplication and soustraction of NIntegrate?

FindRoot: The line search decreased the step size to within tolerance specified \ by AccuracyGoal and PrecisionGoal but was unable to find a sufficient \ decrease in the merit function. You may need more than
MachinePrecision digits of working precision to meet these tolerances.

I already used FindRoot with a single NIntegrate==0 inside, and it worked well, but here product of NIntegrate must propagate a big error.

Improve this question
$\endgroup$
2
  • $\begingroup$ You should take a look at a basic tutorial about Mathematica's syntax. For example, you probably want f[1][k,m,p] or f1[k,m,p] instead of f_1(k,c,s)/. {c -> m, s -> p} After you've fixed that, do a single example to make sure that your numerical integrations are returning something sensible. Also, try plotting your F[c,s] to see if FindRoot has a chance of finding a root. $\endgroup$ Commented Aug 25 at 15:54
  • $\begingroup$ I have modified my answer sorry, you are right. I mixed up the Latex notation with mathematica on the forum $\endgroup$ Commented Aug 25 at 19:53

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