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Module « scipy.optimize »

Fonction brenth - module scipy.optimize

Signature de la fonction brenth

def brenth(f, a, b, args=(), xtol=2e-12, rtol=np.float64(8.881784197001252e-16), maxiter=100, full_output=False, disp=True) 

Description

help(scipy.optimize.brenth)

Find a root of a function in a bracketing interval using Brent's
method with hyperbolic extrapolation.

A variation on the classic Brent routine to find a root of the function f
between the arguments a and b that uses hyperbolic extrapolation instead of
inverse quadratic extrapolation. Bus & Dekker (1975) guarantee convergence
for this method, claiming that the upper bound of function evaluations here
is 4 or 5 times that of bisection.
f(a) and f(b) cannot have the same signs. Generally, on a par with the
brent routine, but not as heavily tested. It is a safe version of the
secant method that uses hyperbolic extrapolation.
The version here is by Chuck Harris, and implements Algorithm M of
[BusAndDekker1975]_, where further details (convergence properties,
additional remarks and such) can be found

Parameters
----------
f : function
    Python function returning a number. f must be continuous, and f(a) and
    f(b) must have opposite signs.
a : scalar
    One end of the bracketing interval [a,b].
b : scalar
    The other end of the bracketing interval [a,b].
xtol : number, optional
    The computed root ``x0`` will satisfy ``np.allclose(x, x0,
    atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The
    parameter must be positive. As with `brentq`, for nice
    functions the method will often satisfy the above condition
    with ``xtol/2`` and ``rtol/2``.
rtol : number, optional
    The computed root ``x0`` will satisfy ``np.allclose(x, x0,
    atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The
    parameter cannot be smaller than its default value of
    ``4*np.finfo(float).eps``. As with `brentq`, for nice functions
    the method will often satisfy the above condition with
    ``xtol/2`` and ``rtol/2``.
maxiter : int, optional
    If convergence is not achieved in `maxiter` iterations, an error is
    raised. Must be >= 0.
args : tuple, optional
    Containing extra arguments for the function `f`.
    `f` is called by ``apply(f, (x)+args)``.
full_output : bool, optional
    If `full_output` is False, the root is returned. If `full_output` is
    True, the return value is ``(x, r)``, where `x` is the root, and `r` is
    a `RootResults` object.
disp : bool, optional
    If True, raise RuntimeError if the algorithm didn't converge.
    Otherwise, the convergence status is recorded in any `RootResults`
    return object.

Returns
-------
root : float
    Root of `f` between `a` and `b`.
r : `RootResults` (present if ``full_output = True``)
    Object containing information about the convergence. In particular,
    ``r.converged`` is True if the routine converged.

See Also
--------
fmin, fmin_powell, fmin_cg, fmin_bfgs, fmin_ncg : multivariate local optimizers
leastsq : nonlinear least squares minimizer
fmin_l_bfgs_b, fmin_tnc, fmin_cobyla : constrained multivariate optimizers
basinhopping, differential_evolution, brute : global optimizers
fminbound, brent, golden, bracket : local scalar minimizers
fsolve : N-D root-finding
brentq, ridder, bisect, newton : 1-D root-finding
fixed_point : scalar fixed-point finder

References
----------
.. [BusAndDekker1975]
   Bus, J. C. P., Dekker, T. J.,
   "Two Efficient Algorithms with Guaranteed Convergence for Finding a Zero
   of a Function", ACM Transactions on Mathematical Software, Vol. 1, Issue
   4, Dec. 1975, pp. 330-345. Section 3: "Algorithm M".
   :doi:`10.1145/355656.355659`

Examples
--------
>>> def f(x):
...     return (x**2 - 1)

>>> from scipy import optimize

>>> root = optimize.brenth(f, -2, 0)
>>> root
-1.0

>>> root = optimize.brenth(f, 0, 2)
>>> root
1.0

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