# scipy.optimize.fsolve

scipy.optimize.fsolve(func, x0, args=(), fprime=None, full_output=0, col_deriv=0, xtol=1.49012e-08, maxfev=0, band=None, epsfcn=None, factor=100, diag=None)

  Find the roots of a function.

  Return the roots of the (non-linear) equations defined by `func(x) = 0` given a starting estimate.

  Parameters:
    func callable `f(x, *args)`

      A function that takes at least one (possibly vector) argument, and returns a value of the same length.

    x0 ndarray

      The starting estimate for the roots of `func(x) = 0`.

    args tuple, optional

      Any extra arguments to func.

    fprime callable `f(x, *args)`, optional

      A function to compute the Jacobian of func with derivatives across the rows. By default, the Jacobian will be estimated.

    full_output bool, optional

      If True, return optional outputs.

    col_deriv bool, optional

      Specify whether the Jacobian function computes derivatives down the columns (faster, because there is no transpose operation).

    xtol float, optional

      The calculation will terminate if the relative error between two consecutive iterates is at most xtol.

    maxfev int, optional

      The maximum number of calls to the function. If zero, then `100*(N+1)` is the maximum where N is the number of elements in x0.

    band tuple, optional

      If set to a two-sequence containing the number of sub- and super-diagonals within the band of the Jacobi matrix, the Jacobi matrix is considered banded (only for `fprime=None`).

    epsfcn float, optional

      A suitable step length for the forward-difference approximation of the Jacobian (for `fprime=None`). If epsfcn is less than the machine precision, it is assumed that the relative errors in the functions are of the order of the machine precision.

    factor float, optional

      A parameter determining the initial step bound (`factor * || diag * x||`). Should be in the interval `(0.1, 100)`.

    diag sequence, optional

      N positive entries that serve as a scale factors for the variables.

  Returns:
    x ndarray

      The solution (or the result of the last iteration for an unsuccessful call).

    infodict dict

      A dictionary of optional outputs with the keys:

      `nfev`

        number of function calls

      `njev`

        number of Jacobian calls

      `fvec`

        function evaluated at the output

      `fjac`

        the orthogonal matrix, q, produced by the QR factorization of the final approximate Jacobian matrix, stored column wise

      `r`

        upper triangular matrix produced by QR factorization of the same matrix

      `qtf`

        the vector `(transpose(q) * fvec)`

    ier int

      An integer flag. Set to 1 if a solution was found, otherwise refer to mesg for more information.

    mesg str

      If no solution is found, mesg details the cause of failure.

  See also

  `root`

    Interface to root finding algorithms for multivariate functions. See the `method='hybr'` in particular.

  Notes

  `fsolve` is a wrapper around MINPACK’s hybrd and hybrj algorithms.

  Examples

  Find a solution to the system of equations: `x0*cos(x1) = 4, x1*x0 - x1 = 5`.

      >>> import numpy as np
      >>> from scipy.optimize import fsolve
      >>> def func(x):
      ...     return [x[0] * np.cos(x[1]) - 4,
      ...             x[1] * x[0] - x[1] - 5]
      >>> root = fsolve(func, [1, 1])
      >>> root
      array([6.50409711, 0.90841421])
      >>> np.isclose(func(root), [0.0, 0.0])  # func(root) should be almost 0.0.
      array([ True,  True])