code: def root_finder(
      value_and_grad_func,
      initial_values,
      max_iterations=20,
      tolerance=2e-7,
      relative_tolerance=None,
      dtype=None,
      name="root_finder",
  ):
      """Finds roots of a scalar function using Newton's method.
  
      This method uses Newton's algorithm to find values `x` such that `f(x)=0` for
      some real-valued differentiable function `f`. Given an initial value `x_0` the
      values are iteratively updated as:
  
        `x_{n+1} = x_n - f(x_n) / f'(x_n),`
  
      for further details on Newton's method, see [1]. The implementation accepts
      array-like arguments and assumes that each cell corresponds to an independent
      scalar model.
  
      #### Examples
      ```python
      # Set up the problem of finding the square roots of three numbers.
      constants = np.array([4.0, 9.0, 16.0])
      initial_values = np.ones(len(constants))
      def objective_and_gradient(values):
        objective = values**2 - constants
        gradient = 2.0 * values
        return objective, gradient
  
      # Obtain and evaluate a tensor containing the roots.
      roots = tff.math.root_search.newton_root(objective_and_gradient,
        initial_values)
      print(root_values)  # Expected output: [ 2.  3.  4.]
      print(converged)  # Expected output: [ True  True  True]
      print(failed)  # Expected output: [False False False]
      ```
  
      #### References
      [1] Luenberger, D.G., 1984. 'Linear and Nonlinear Programming'. Reading, MA:
      Addison-Wesley.
  
      Args:
        value_and_grad_func: A python callable that takes a `Tensor` of the same
          shape and dtype as the `initial_values` and which returns a two-`tuple` of
          `Tensors`, namely the objective function and the gradient evaluated at the
          passed parameters.
        initial_values: A real `Tensor` of any shape. The initial values of the
          parameters to use (`x_0` in the notation above).
        max_iterations: positive `int`. The maximum number of
          iterations of Newton's method.
          Default value: 20.
        tolerance: positive scalar `Tensor`. The absolute tolerance for the root
          search. Search is judged to have converged  if
          `|f(x_n) - f(x_n-1)|` < |x_n| * `relative_tolerance` + `tolerance`
          (using the notation above), or if `x_n` becomes `nan`. When an element is
          judged to have converged it will no longer be updated. If all elements
          converge before `max_iterations` is reached then the root finder will
          return early. If None, it would be set according to the `dtype`,
          which is 4 * np.finfo(dtype.as_numpy_dtype(0)).eps.
          Default value: 2e-7.
        relative_tolerance: positive `double`, default 0. See the document for
          `tolerance`.
          Default value: None.
        dtype: optional `tf.DType`. If supplied the `initial_values` will be coerced
          to this data type.
          Default value: None.
        name: `str`, to be prefixed to the name of
          TensorFlow ops created by this function.
          Default value: 'root_finder'.
  
      Returns:
        A three tuple of `Tensor`s, each the same shape as `initial_values`. It
        contains the found roots (same dtype as `initial_values`), a boolean
        `Tensor` indicating whether the corresponding root results in an objective
        function value less than the tolerance, and a boolean `Tensor` which is true
        where the corresponding 'root' is not finite.
      """
      if tolerance is None:
          tolerance = utils.default_relative_root_tolerance(dtype)
      if relative_tolerance is None:
          relative_tolerance = utils.default_relative_root_tolerance(
              dtype
          )
  
      with tf.compat.v1.name_scope(
          name,
          default_name="newton_root_finder",
          values=[initial_values, tolerance],
      ):
          initial_values = tf.convert_to_tensor(
              initial_values, dtype=dtype, name="initial_values"
          )
  
          def _condition(counter, parameters, converged, failed):
              del parameters
              early_stop = tf.reduce_all(converged | failed)
              return ~((counter >= max_iterations) | early_stop)
  
          def _updater(counter, parameters, converged, failed):
              """Updates each parameter via Newton's method."""
              values, gradients = value_and_grad_func(parameters)
              deltas = tf.math.divide(values, gradients)
  
              converged = (
                  tf.abs(deltas)
                  < relative_tolerance * tf.abs(values) + tolerance
              )
  
              # Used to zero out updates to cells that have converged.
              update_mask = tf.cast(~converged, dtype=parameters.dtype)
              increment = -update_mask * deltas
              updated_parameters = parameters + increment
              failed = ~tf.math.is_finite(updated_parameters)
  
              return counter + 1, updated_parameters, converged, failed
  
          starting_position = (
              tf.constant(0, dtype=tf.int32),
              initial_values,
              tf.zeros_like(initial_values, dtype=tf.bool),
              tf.math.is_nan(initial_values),
          )
  
          return tf.while_loop(
              _condition,
              _updater,
              starting_position,
              maximum_iterations=max_iterations,
          )[1:]
  
label_desc: Use image node_id 1 for calling a global function with example usage: root_finder(value_and_grad_func, initial_values, max_iterations, tolerance, relative_tolerance, dtype, name) and returns: tf, unknown, unknown,  updated_parameters,  converged,  failed