I am using scipy.optimize's function fsolve
to solve for two unknowns in two equations. The equations that I am trying to solve in the end are (much) more complex but I already struggle understanding the following basic example.
import scipy.optimize as scopt
def fun(variables) :
(x,y) = variables
eqn_1 = x ** 2 + y - 4
eqn_2 = x + y ** 2 +3
return [eqn_1, eqn_2]
result = scopt.fsolve(fun, (0.1, 1))
print(result)
This gives the result [-2.08470396 -0.12127194]
, however when I plug those numbers back into the function (one time assuming the first is meant as x, one time assuming the first is y), I get results very different from zero.
print((-2.08470396)**2 - 0.12127194 - 4)
print((-2.08470396) + (- 0.12127194) ** 2 + 3)
Result 0.22 and 0.93.
print((-0.12127194)**2 -2.08470396 - 4)
print((-0.12127194) + (-2.08470396) ** 2 + 3)
Result -6.06 and 7.22.
What am I missing here?
Did you notice the warning that is generated when you run result = scopt.fsolve(fun, (0.1, 1))
? That warning tells you that something failed:
In [35]: result = scopt.fsolve(fun, (0.1, 1))
/Users/warren/a202111/lib/python3.9/site-packages/scipy/optimize/minpack.py:175: RuntimeWarning: The iteration is not making good progress, as measured by the
improvement from the last ten iterations.
warnings.warn(msg, RuntimeWarning)
The problem is that there is no solution to fun(variables) = (0, 0)
. The first equation gives y = 4-x**2
, and then the second equation can be written x + (4-x**2)**2 + 3 = 0
, which has no real solution (you can plot the left side or do some algebra to convince yourself of that).
If you use, say, eqn_2 = x + y ** 2 - 3
, fsolve
gives a valid numerical solution:
In [36]: def fun(variables) :
...: (x,y) = variables
...: eqn_1 = x ** 2 + y - 4
...: eqn_2 = x + y ** 2 - 3
...: return [eqn_1, eqn_2]
...:
In [37]: result = scopt.fsolve(fun, (0.1, 1))
In [38]: result
Out[38]: array([-1.38091841, 2.09306436])
In [39]: fun(result)
Out[39]: [0.0, 0.0]
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