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I try to solve this system:

e1 = H1[0, y] - H1[0, Y]
e2 = H2[0, y] - H2[0, Y]

with

H1[x_, y_] = 8 (-0.1` x - 1.2` y) + (4 x - 0.8` y)^2 + 4 y^2;
H2[x_, y_] = (
  1 - 2 (1.5` - 0.5` x - 4.553076848028063`*^-15 y)^2 + 
   4 (-1.0344827586206884` - x + y)^2)/(1.5` - 0.5` x - 
    4.553076848028063`*^-15 y)^4;

I use Solve like this:

Solve[{e1, e2} == 0, {y, Y}]

but I found this message

Solve::ratnz: Solve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result. >>

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  • $\begingroup$ You need a semicolon between your lines. $\endgroup$
    – David G. Stork
    Aug 5, 2021 at 16:17
  • $\begingroup$ To form the exact system use Rationalize. To force rationalization of all numeric values use Rationalize[#, 0]& $\endgroup$
    – Bob Hanlon
    Aug 5, 2021 at 16:46
  • $\begingroup$ It says it solved the system and gave an answer....It gave me another warning, in effect that Y == y is solution in which y is a free variable. In short, it worked! :) -- That said, coefficients that vary in magnitude by a factor of 10^15 is asking for numerical trouble. That's an issue with how the problem is posed. Ratioinalize won't fix it, but it will show the uncertainty in a couple of the solutions, if you compare their numerical values. (I don't know how to fix it. How was H2 derived?) $\endgroup$
    – Michael E2
    Aug 5, 2021 at 19:15
  • $\begingroup$ This will let you compare solutions side-by-side: Sort /@ {{y, Y} /. Solve[{e1, e2} == 0, {y, Y}], {y, Y} /. Solve[Rationalize[{e1, e2}, 0] == 0, {y, Y}] // N} // Transpose // Grid $\endgroup$
    – Michael E2
    Aug 5, 2021 at 19:18

1 Answer 1

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There is an extremely small number $a=4.553076848028063\times10^{-15}$ involved, which I'd treat with a lot of caution. Rationalizing all numbers except $a$, we have

H1[x_, y_] = 8 (-1/10 x - 12/10 y) + (4 x - 8/10 y)^2 + 4 y^2;
H2[x_, y_] = (1 - 2 (3/2 - 1/2 x - a y)^2 + 4 (-30/29 - x + y)^2)/(3/2 - 1/2 x - a y)^4;

and then

sol = Solve[{H1[0, y] - H1[0, Y] == 0, H2[0, y] - H2[0, Y] == 0}, {y, Y}];

The first solution is $y=Y$,

sol[[1]]
(*    {Y -> y}    *)

The remaining four solutions can be series-expanded for small $a$ to see their behavior:

Assuming[a > 0, Series[{y, Y} /. sol[[2]], {a, 0, 1}]] // FullSimplify
(*    {(30/29 + Sqrt[5]/4) - (9 Sqrt[5] a)/29 + O[a]^2,
       (30/29 - Sqrt[5]/4) + (9 Sqrt[5] a)/29 + O[a]^2}    *)

from which you can see that inserting an extremely small value of $a$ will give stable results $y\approx\frac{30}{29}+\frac{\sqrt{5}}{4}$ and $Y\approx\frac{30}{29}-\frac{\sqrt{5}}{4}$.

The third solution sol[[3]] is the same with $y$ and $Y$ interchanged.

The fourth solution diverges as $a\to0$,

Assuming[a > 0, Series[{y, Y} /. sol[[4]], {a, 0, 1}]] // FullSimplify
(*    (3 I)/(2 a) + (30/29 - (30 I)/29) + (3 I a)/4 + O[a]^2,
      -((3 I)/(2 a)) + (30/29 + (30 I)/29) - (3 I a)/4 + O[a]^2}    *)

and the fifth as well (with $y$ and $Y$ interchanged).

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  • $\begingroup$ @MichaelE2 I don't think your three-argument form of Solve gives an actual solution of the equations: back-substituting the suggested {{Y -> 1/29 (60 - 29 y)}, {Y -> y}} into the equations gives no match. $\endgroup$
    – Roman
    Aug 6, 2021 at 5:13
  • $\begingroup$ That may be; but notice that the solutions for $a=0$ are not the same as the solutions for $\lim_{a\to0^+}$. This is one of those cases where perturbing the equations is not the same as perturbing the solutions. $\endgroup$
    – Roman
    Aug 6, 2021 at 6:46

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