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I have the following system of non-linear equations which I want to find its roots:

  • x - exp(a x + b y) = 0
  • y - exp(c x + d y) = 0 The code I'm using to find its roots is:
equations = lambda x, kernel: np.array([x[0] - np.exp(kernel[0] * x[0] + kernel[2] * x[1]), x[1] - np.exp(kernel[1] * x[1] + kernel[3] * x[0])])

kernels = np.array([kernel0, kernel1, kernel2, kernel3])
x_init = np.array([x_init0, x_init1])
x_sol = fsolve(two_equations, x_init, args=(kernels))

From the equations I know that this system in some situations has two answers for each variable: (x_sol1, x_sol2) and (y_sol1, y_sol2).

Is there a clean way to pass multiple initial guesses to this fsolve function to get both roots for each variable? (instead of using a for loop) I know how to do it for a system of one equation only but I couldn't use that method for this case.

Nosrat Mohammadi
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    So I had a similar issue and found a solution that worked for my case. In terms of circumventing the `for` loop- if you find a way, please post it as that would optimize my code as well. Check out the solution [here](https://stackoverflow.com/questions/67249288/how-to-include-adjustable-parameter-in-fsolve). It is in the second part of the answer – t.o. Aug 03 '22 at 15:53

1 Answers1

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You can reduce the system to a single univariate equation by eliminating y.

y = (ln(x) - a x) / b

so that

(ln(x) - a x) / b - exp(c x + d (ln(x) - a x) / b) = 0