# 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)[source]

Find the roots of a function.

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

Parameters
funccallable f(x, *args)

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

x0ndarray

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.

Returns
xndarray

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

infodictdict

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)

ierint

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

mesgstr

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

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.

>>> from scipy.optimize import fsolve
>>> def func(x):
...     return [x * np.cos(x) - 4,
...             x * x - x - 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])


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