No converging solution with sympy.nonlinsolve() in Python-CodePudding

I'm trying to solve a system with three nonlinear equations in Python 3.8. I'm using the function sympy.nonlinsolve(). However, I received the error message "convergence to root failed; try n < 15 or maxsteps > 50".

This is my code:

import sympy as sp

x_1 = 0.0
z_1 = 1.0
x_2 = 15.81
z_2 = 0.99
x_3 = 23.8
z_3 = 0.98

r, x_m, z_m = sp.symbols('r, x_m, z_m', real=True)
Eq_1 = sp.Eq((x_1 - x_m) ** 2   (z_1 - z_m) ** 2 - r ** 2, 0)
Eq_2 = sp.Eq((x_2 - x_m) ** 2   (z_2 - z_m) ** 2 - r ** 2, 0)
Eq_3 = sp.Eq((x_3 - x_m) ** 2   (z_3 - z_m) ** 2 - r ** 2, 0)
ans = sp.nonlinsolve([Eq_1, Eq_2, Eq_3], [r, x_m, z_m])

I would welcome every help. Thanks in advance.

CodePudding user response：

I get an answer from solve:

In [56]: sp.solve([Eq_1, Eq_2, Eq_3], [r, x_m, z_m])
Out[56]: 
[(-5.71609538434502e 18, -4.80343980343979e 15, -5.71609336609336e 18), (-19222.9235141152, -4.2537
0843989772, -19221.9230434783), (19222.9235141152, -4.25370843989772, -19221.9230434783), (5.716095
38434502e 18, -4.80343980343979e 15, -5.71609336609336e 18)]

I'm not sure why nonlinsolve works but from the large numbers in the answer I guess that this isn't well conditioned.

If you use exact rational numbers then you can get the same solution from both solve and nonlinsolve:

In [59]: import sympy as sp
    ...: 
    ...: x_1 = 0
    ...: z_1 = 1
    ...: x_2 = sp.Rational('15.81')
    ...: z_2 = sp.Rational('0.99')
    ...: x_3 = sp.Rational('23.8')
    ...: z_3 = sp.Rational('0.98')
    ...: 
    ...: r, x_m, z_m = sp.symbols('r, x_m, z_m', real=True)
    ...: Eq_1 = sp.Eq((x_1 - x_m) ** 2   (z_1 - z_m) ** 2 - r ** 2, 0)
    ...: Eq_2 = sp.Eq((x_2 - x_m) ** 2   (z_2 - z_m) ** 2 - r ** 2, 0)
    ...: Eq_3 = sp.Eq((x_3 - x_m) ** 2   (z_3 - z_m) ** 2 - r ** 2, 0)
    ...: ans = sp.solve([Eq_1, Eq_2, Eq_3], [r, x_m, z_m])

In [60]: ans
Out[60]: 
⎡⎛-√564927076558939081   -8316   -44210423 ⎞  ⎛√564927076558939081  -8316   -44210423 ⎞⎤
⎢⎜─────────────────────, ──────, ──────────⎟, ⎜───────────────────, ──────, ──────────⎟⎥
⎣⎝        39100           1955      2300   ⎠  ⎝       39100          1955      2300   ⎠⎦