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)[source]#

Find the roots of a function.

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

funccallable f(x, *args)

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


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

argstuple, optional

Any extra arguments to func.

fprimecallable 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_outputbool, optional

If True, return optional outputs.

col_derivbool, optional

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

xtolfloat, optional

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

maxfevint, 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.

bandtuple, 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).

epsfcnfloat, 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.

factorfloat, optional

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

diagsequence, optional

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


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


A dictionary of optional outputs with the keys:


number of function calls


number of Jacobian calls


function evaluated at the output


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


upper triangular matrix produced by QR factorization of the same matrix


the vector (transpose(q) * fvec)


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


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

See also


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


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


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])