code: def newton(
      func,
      x0,
      fprime=None,
      args=(),
      tol=1.48e-8,
      maxiter=50,
      fprime2=None,
      x1=None,
      rtol=0.0,
      full_output=False,
      disp=True,
  ):
      """
      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()
  
      """
      if tol <= 0:
          raise ValueError("tol too small (%g <= 0)" % tol)
      maxiter = operator.index(maxiter)
      if maxiter < 1:
          raise ValueError("maxiter must be greater than 0")
      if np.size(x0) > 1:
          return _array_newton(
              func, x0, fprime, args, tol, maxiter, fprime2, full_output
          )
  
      # Convert to float (don't use float(x0); this works also for complex x0)
      p0 = 1.0 * x0
      funcalls = 0
      if fprime is not None:
          # Newton-Raphson method
          for itr in range(maxiter):
              # first evaluate fval
              fval = func(p0, *args)
              funcalls += 1
              # If fval is 0, a root has been found, then terminate
              if fval == 0:
                  return _results_select(
                      full_output, (p0, funcalls, itr, _ECONVERGED)
                  )
              fder = fprime(p0, *args)
              funcalls += 1
              if fder == 0:
                  msg = "Derivative was zero."
                  if disp:
                      msg += (
                          " Failed to converge after %d iterations, value is %s."
                          % (itr + 1, p0)
                      )
                      raise RuntimeError(msg)
                  warnings.warn(msg, RuntimeWarning)
                  return _results_select(
                      full_output, (p0, funcalls, itr + 1, _ECONVERR)
                  )
              newton_step = fval / fder
              if fprime2:
                  fder2 = fprime2(p0, *args)
                  funcalls += 1
                  # Halley's method:
                  #   newton_step /= (1.0 - 0.5 * newton_step * fder2 / fder)
                  # Only do it if denominator stays close enough to 1
                  # Rationale: If 1-adj < 0, then Halley sends x in the
                  # opposite direction to Newton. Doesn't happen if x is close
                  # enough to root.
                  adj = newton_step * fder2 / fder / 2
                  if np.abs(adj) < 1:
                      newton_step /= 1.0 - adj
              p = p0 - newton_step
              if np.isclose(p, p0, rtol=rtol, atol=tol):
                  return _results_select(
                      full_output, (p, funcalls, itr + 1, _ECONVERGED)
                  )
              p0 = p
      else:
          # Secant method
          if x1 is not None:
              if x1 == x0:
                  raise ValueError("x1 and x0 must be different")
              p1 = x1
          else:
              eps = 1e-4
              p1 = x0 * (1 + eps)
              p1 += eps if p1 >= 0 else -eps
          q0 = func(p0, *args)
          funcalls += 1
          q1 = func(p1, *args)
          funcalls += 1
          if abs(q1) < abs(q0):
              p0, p1, q0, q1 = p1, p0, q1, q0
          for itr in range(maxiter):
              if q1 == q0:
                  if p1 != p0:
                      msg = "Tolerance of %s reached." % (p1 - p0)
                      if disp:
                          msg += (
                              " Failed to converge after %d iterations, value is %s."
                              % (itr + 1, p1)
                          )
                          raise RuntimeError(msg)
                      warnings.warn(msg, RuntimeWarning)
                  p = (p1 + p0) / 2.0
                  return _results_select(
                      full_output, (p, funcalls, itr + 1, _ECONVERGED)
                  )
              else:
                  if abs(q1) > abs(q0):
                      p = (-q0 / q1 * p1 + p0) / (1 - q0 / q1)
                  else:
                      p = (-q1 / q0 * p0 + p1) / (1 - q1 / q0)
              if np.isclose(p, p1, rtol=rtol, atol=tol):
                  return _results_select(
                      full_output, (p, funcalls, itr + 1, _ECONVERGED)
                  )
              p0, q0 = p1, q1
              p1 = p
              q1 = func(p1, *args)
              funcalls += 1
  
      if disp:
          msg = (
              "Failed to converge after %d iterations, value is %s."
              % (itr + 1, p)
          )
          raise RuntimeError(msg)
  
      return _results_select(
          full_output, (p, funcalls, itr + 1, _ECONVERR)
      )
  
imports: import warnings
  from collections import namedtuple
  import operator
  from .None import _zeros
  import numpy
label_desc: Use image node_id 3 for calling a global function with example usage: newton(func, x0, fprime, args, tol, maxiter, fprime2, x1, rtol, full_output, disp) and returns: _results_select, _array_newton, _results_select, _results_select, _results_select, _results_select, _results_select