Coupled nonlinear equations

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Suppose we seek the solution to this set of equations:

\begin{align} y &=& x^2 \\ y &=& 8 - x^2 \end{align}

To solve this we need to setup a function that is equal to zero at the solution. We have two equations, so our function must return two values. There are two variables, so the argument to our function will be an array of values.

from scipy.optimize import fsolve

def objective(X):
    x, y = X            # unpack the array in the argument
    z1 = y - x**2       # first equation
    z2 = y - 8 + x**2   # second equation
    return [z1, z2]     # list of zeros

x0, y0 = 1, 1           # initial guesses
guess = [x0, y0]
sol = fsolve(objective, guess)
print sol

# of course there may be more than one solution
x0, y0 = -1, -1           # initial guesses
guess = [x0, y0]
sol = fsolve(objective, guess)
print sol
[ 2.  4.]
[-2.  4.]

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