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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)

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

Notes

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

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 :None:None:`func`.
fprime : callable ``f(x, *args)``, optional: A function to compute the Jacobian of :None:None:`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 :None:None:`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 :None:None:`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 :None:None:`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 :None:None:`mesg` for more information.

mesg : str

If no solution is found, :None:None:`mesg` details the cause of failure.

Find the roots of a function.

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[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])

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.optimize._minpack_py.fsolve scipy.optimize._zeros_py.bisect scipy.optimize._zeros_py.toms748

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