# Increasing the accuracy of residue in FindRoot

I have solved a system of non-linear equations with findRoot. I tried to avoid inexact coefficients and also assign a high working precision. This is my snippet:

rho0=1.0; rho1=1.0;rho2=10^(-5);rhoint=10^(0);lambdaD=210;
fr={rho0*rho1/x-rho1*rho2/(y-x)-rho1*rhoint/(lambdaD-x)-rho0*rho1/(x^6)==0,
rho0*rho2/y+rho1*rho2/(y-x)-rho2*rhoint/(lambdaD-y)-rho0*rho2/(y^6)==0};
frprecise=SetPrecision[fr,48]
sol=FindRoot[frprecise,{{x,9/100},{y,8/10}},AccuracyGoal->20, PrecisionGoal->20,WorkingPrecision->48, MaxIterations->200000];
tab60 = Table[sol[[i]] // N[#, 60] &, {i, Length[sol]}]
Table[fr[[i, 1]] /. tab60, {i, Length[fr]}]


However, when I get the residue from the replacement of solutions in the non-linear equations, one of the residues is just zero. I want to know the exact residue. This 0 in the output is misleading. Can any body hint me on this?

Regards

• You do use inexact numbers when you set rho0=1.0; rho1=1.0;. Better use rho0 = 1; rho1 = 1; instead. Then you may also use FindRoot[fr,[...]] instead of FindRoot[frprecise,[...]]. – Henrik Schumacher Mar 30 at 20:17
• Your system of equations can be solved by Solve as well. If you use exact input (e.g., use rho0 = 1; rho1 = 1 as recommended by @Henrik), then you will get an exact answer that you can numericize to any desired precision. – Carl Woll Mar 30 at 20:32
• Thanks Henrik. That was a good hint. Mistakenly, I was still using inexact coefficients. – bobi Mar 30 at 22:53