Module « scipy.optimize »

Fonction newton - module scipy.optimize

Signature de la fonction newton

def newton(func, x0, fprime=None, args=(), tol=1.48e-08, maxiter=50, fprime2=None, x1=None, rtol=0.0, full_output=False, disp=True) 

Description

newton.__doc__

    Find a zero of a real or complex function using the Newton-Raphson
    (or secant or Halley's) method.

    Find a zero of the function `func` given a nearby starting point `x0`.
    The Newton-Raphson method is used if the derivative `fprime` of `func`
    is provided, otherwise the secant method is used. If the second order
    derivative `fprime2` of `func` is also provided, then Halley's method is
    used.

    If `x0` is a sequence with more than one item, then `newton` returns an
    array, and `func` must be vectorized and return a sequence or array of the
    same shape as its first argument. If `fprime` or `fprime2` is given, then
    its return must also have the same shape.

    Parameters
    ----------
    func : callable
        The function whose zero is wanted. It must be a function of a
        single variable of the form ``f(x,a,b,c...)``, where ``a,b,c...``
        are extra arguments that can be passed in the `args` parameter.
    x0 : float, sequence, or ndarray
        An initial estimate of the zero that should be somewhere near the
        actual zero. If not scalar, then `func` must be vectorized and return
        a sequence or array of the same shape as its first argument.
    fprime : callable, optional
        The derivative of the function when available and convenient. If it
        is None (default), then the secant method is used.
    args : tuple, optional
        Extra arguments to be used in the function call.
    tol : float, optional
        The allowable error of the zero value. If `func` is complex-valued,
        a larger `tol` is recommended as both the real and imaginary parts
        of `x` contribute to ``|x - x0|``.
    maxiter : int, optional
        Maximum number of iterations.
    fprime2 : callable, optional
        The second order derivative of the function when available and
        convenient. If it is None (default), then the normal Newton-Raphson
        or the secant method is used. If it is not None, then Halley's method
        is used.
    x1 : float, optional
        Another estimate of the zero that should be somewhere near the
        actual zero. Used if `fprime` is not provided.
    rtol : float, optional
        Tolerance (relative) for termination.
    full_output : bool, optional
        If `full_output` is False (default), the root is returned.
        If True and `x0` is scalar, the return value is ``(x, r)``, where ``x``
        is the root and ``r`` is a `RootResults` object.
        If True and `x0` is non-scalar, the return value is ``(x, converged,
        zero_der)`` (see Returns section for details).
    disp : bool, optional
        If True, raise a RuntimeError if the algorithm didn't converge, with
        the error message containing the number of iterations and current
        function value. Otherwise, the convergence status is recorded in a
        `RootResults` return object.
        Ignored if `x0` is not scalar.
        *Note: this has little to do with displaying, however,
        the `disp` keyword cannot be renamed for backwards compatibility.*

    Returns
    -------
    root : float, sequence, or ndarray
        Estimated location where function is zero.
    r : `RootResults`, optional
        Present if ``full_output=True`` and `x0` is scalar.
        Object containing information about the convergence. In particular,
        ``r.converged`` is True if the routine converged.
    converged : ndarray of bool, optional
        Present if ``full_output=True`` and `x0` is non-scalar.
        For vector functions, indicates which elements converged successfully.
    zero_der : ndarray of bool, optional
        Present if ``full_output=True`` and `x0` is non-scalar.
        For vector functions, indicates which elements had a zero derivative.

    See Also
    --------
    brentq, brenth, ridder, bisect
    fsolve : find zeros in N dimensions.

    Notes
    -----
    The convergence rate of the Newton-Raphson method is quadratic,
    the Halley method is cubic, and the secant method is
    sub-quadratic. This means that if the function is well-behaved
    the actual error in the estimated zero after the nth iteration
    is approximately the square (cube for Halley) of the error
    after the (n-1)th step. However, the stopping criterion used
    here is the step size and there is no guarantee that a zero
    has been found. Consequently, the result should be verified.
    Safer algorithms are brentq, brenth, ridder, and bisect,
    but they all require that the root first be bracketed in an
    interval where the function changes sign. The brentq algorithm
    is recommended for general use in one dimensional problems
    when such an interval has been found.

    When `newton` is used with arrays, it is best suited for the following
    types of problems:

    * The initial guesses, `x0`, are all relatively the same distance from
      the roots.
    * Some or all of the extra arguments, `args`, are also arrays so that a
      class of similar problems can be solved together.
    * The size of the initial guesses, `x0`, is larger than O(100) elements.
      Otherwise, a naive loop may perform as well or better than a vector.

    Examples
    --------
    >>> from scipy import optimize
    >>> import matplotlib.pyplot as plt

    >>> def f(x):
    ...     return (x**3 - 1)  # only one real root at x = 1

    ``fprime`` is not provided, use the secant method:

    >>> root = optimize.newton(f, 1.5)
    >>> root
    1.0000000000000016
    >>> root = optimize.newton(f, 1.5, fprime2=lambda x: 6 * x)
    >>> root
    1.0000000000000016

    Only ``fprime`` is provided, use the Newton-Raphson method:

    >>> root = optimize.newton(f, 1.5, fprime=lambda x: 3 * x**2)
    >>> root
    1.0

    Both ``fprime2`` and ``fprime`` are provided, use Halley's method:

    >>> root = optimize.newton(f, 1.5, fprime=lambda x: 3 * x**2,
    ...                        fprime2=lambda x: 6 * x)
    >>> root
    1.0

    When we want to find zeros for a set of related starting values and/or
    function parameters, we can provide both of those as an array of inputs:

    >>> f = lambda x, a: x**3 - a
    >>> fder = lambda x, a: 3 * x**2
    >>> rng = np.random.default_rng()
    >>> x = rng.standard_normal(100)
    >>> a = np.arange(-50, 50)
    >>> vec_res = optimize.newton(f, x, fprime=fder, args=(a, ), maxiter=200)

    The above is the equivalent of solving for each value in ``(x, a)``
    separately in a for-loop, just faster:

    >>> loop_res = [optimize.newton(f, x0, fprime=fder, args=(a0,))
    ...             for x0, a0 in zip(x, a)]
    >>> np.allclose(vec_res, loop_res)
    True

    Plot the results found for all values of ``a``:

    >>> analytical_result = np.sign(a) * np.abs(a)**(1/3)
    >>> fig = plt.figure()
    >>> ax = fig.add_subplot(111)
    >>> ax.plot(a, analytical_result, 'o')
    >>> ax.plot(a, vec_res, '.')
    >>> ax.set_xlabel('$a$')
    >>> ax.set_ylabel('$x$ where $f(x, a)=0$')
    >>> plt.show()