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 >>> np.random.seed(4321) >>> x = np.random.randn(100) >>> a = np.arange(-50, 50) >>> vec_res = optimize.newton(f, x, fprime=fder, args=(a, ))
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()