Why can't NSolve find values that FindInstance can?

Question

I am trying to solve for \[Epsilon], given three input paramters n, x and eps. Only FindInstance provides a solution for me whereas NSolve and NSolveValues fail.

Why does this happen?

$PreRead = (# /. 
     s_String /; 
       StringMatchQ[s, NumberString] && 
        Precision@ToExpression@s == MachinePrecision :> s <> "`50." &);

kldFunc[x_, y_] := x Log[x/y] + (1 - x) Log[(1 - x)/(1 - y)];

n = 10.^13.; x = 6. 10.^9.; eps = 10.^-20.; 

Clear[\[Epsilon], \[Mu], p]; 
\[Mu] = x + n \[Epsilon];
p = \[Mu]/n; 
\[Epsilon] = \[Epsilon] /. 
    Flatten[FindInstance[
      Exp[-kldFunc[p - \[Epsilon], p] n] == eps && 
       1. > \[Epsilon] > 0., \[Epsilon], Reals] ]

Clear[\[Epsilon], \[Mu], p]; 
\[Mu] = x + n \[Epsilon];
p = \[Mu]/n; 
\[Epsilon] = 
   NSolveValues[
    Exp[-kldFunc[p - \[Epsilon], p] n] == eps && 
     1. > \[Epsilon] > 0., \[Epsilon], Reals] 

Clear[\[Epsilon], \[Mu], p]; 
\[Mu] = x + n \[Epsilon];
p = \[Mu]/n; 
\[Epsilon] = 
   NSolve[Exp[-kldFunc[p - \[Epsilon], p] n] == eps && 
     1. > \[Epsilon] > 0., \[Epsilon], Reals] 

EDIT

I tried plotting the two function to find the intersection

$PreRead = (# /. 
     s_String /; 
       StringMatchQ[s, NumberString] && 
        Precision@ToExpression@s == MachinePrecision :> 
      s <> "`100." &);
kldFunc[x_, y_] := x Log[x/y] + (1 - x) Log[(1 - x)/(1 - y)];
n = 10.^13.; x = 6.  10.^9.; eps = 10.^-20.; 



Clear[\[Epsilon], \[Mu], p]; 
\[Mu] = x + n \[Epsilon];
p = \[Mu]/n; 
Plot[{10^-20, Exp[-kldFunc[p - \[Epsilon], p] n]}, {\[Epsilon], 
  7 10^-8, 8  10^-8}, WorkingPrecision -> 50]

I tried to extract the intersection point using

plot1 = Plot[
   Exp[-kldFunc[p - \[Epsilon], p] n], {\[Epsilon], 7 10^-8, 
    8  10^-8}, WorkingPrecision -> 50];
plot2 = Plot[10^-20, {\[Epsilon], 7 10^-8, 8  10^-8}, 
   WorkingPrecision -> 50];


intersections = Graphics`Mesh`FindIntersections[{plot2, plot1}]

but it does'nt give an output.(I'm running on 13.3. See related)

Answer 1

Use FindRoot

Clear["Global`*"]

kldFunc[x_, y_] := x Log[x/y] + (1 - x) Log[(1 - x)/(1 - y)];

n = 10^13; x = 6*10^9; eps = 10^-20;
μ = x + n ϵ;
p = μ/n;

expr = N[Exp[-kldFunc[p - ϵ, p] n] - eps, 100] // Simplify;

With FindRoot you need to provide a starting value and only a single result is returned irrespective of the number of roots.

(sol1 = FindRoot[expr == 0, {ϵ, 10^-8}, WorkingPrecision -> 80]) // N

(* {ϵ -> 7.43192*10^-8} *)

expr /. sol1

(* 0.*10^-94 *)

(sol2 = FindRoot[expr == 0, {ϵ, 10^-7}, WorkingPrecision -> 80]) // N

(* {ϵ -> 7.43192*10^-8} *)

expr /. sol2

(* 0.*10^-94 *)

With two starting values, FindRoot avoids the use of derivatives

(sol3 = FindRoot[expr == 0, {ϵ, 10^-9, 10^-6}, 
    WorkingPrecision -> 80]) // N

(* {ϵ -> 7.43192*10^-8} *)

expr /. sol3

(* 0.*10^-94 *)

EDIT: For your revised values given in the comment

n = 10^15; x = 6*10^9; eps = 10^-20;
μ = x - n ϵ;
p = μ/n;

Increase the precision for expr

expr = N[Exp[-kldFunc[p - ϵ, p] n] - eps, 105] // Simplify;

The starting value must be close to the actual value

(sol4 = FindRoot[expr == 0, {ϵ, 7*10^-10}, WorkingPrecision -> 80]) //
  N

(* {ϵ -> 7.43321*10^-10} *)

expr /. sol4

(* 0.*10^-92 *)

Answer 2

NSolve work if we use infinite precision number n,x,eps.

Clear["Global`*"];
kldFunc[x_, y_] := x Log[x/y] + (1 - x) Log[(1 - x)/(1 - y)];
n = 10^13; x = 6*10^9; eps = 10^-20;
μ = x + n ϵ;
p = μ/n;
NSolve[{Exp[-kldFunc[p - ϵ, p] n] == eps, 
  0 < ϵ < 10^-1}, ϵ]

{{ϵ -> 7.43192*10^-8}}

• test another values.

NSolve[{Exp[-kldFunc[p - ϵ, p] n] == 10^-3, 
  0 < ϵ < 10^-1}, ϵ]

{{ϵ -> 2.8783*10^-8}}