class_docstr: The interpolating polynomial for a set of points
  
  Constructs a polynomial that passes through a given set of points.
  Allows evaluation of the polynomial, efficient changing of the y
  values to be interpolated, and updating by adding more x values.
  For reasons of numerical stability, this function does not compute
  the coefficients of the polynomial.
  
  The values yi need to be provided before the function is
  evaluated, but none of the preprocessing depends on them, so rapid
  updates are possible.
  
  Parameters
  ----------
  xi : array_like
      1-D array of x coordinates of the points the polynomial
      should pass through
  yi : array_like, optional
      The y coordinates of the points the polynomial should pass through.
      If None, the y values will be supplied later via the `set_y` method.
  axis : int, optional
      Axis in the yi array corresponding to the x-coordinate values.
  
  Notes
  -----
  This class uses a "barycentric interpolation" method that treats
  the problem as a special case of rational function interpolation.
  This algorithm is quite stable, numerically, but even in a world of
  exact computation, unless the x coordinates are chosen very
  carefully - Chebyshev zeros (e.g., cos(i*pi/n)) are a good choice -
  polynomial interpolation itself is a very ill-conditioned process
  due to the Runge phenomenon.
  
  Based on Berrut and Trefethen 2004, "Barycentric Lagrange Interpolation".
docstr: Add more x values to the set to be interpolated
  
  The barycentric interpolation algorithm allows easy updating by
  adding more points for the polynomial to pass through.
  
  Parameters
  ----------
  xi : array_like
      The x coordinates of the points that the polynomial should pass
      through.
  yi : array_like, optional
      The y coordinates of the points the polynomial should pass through.
      Should have shape ``(xi.size, R)``; if R > 1 then the polynomial is
      vector-valued.
      If `yi` is not given, the y values will be supplied later. `yi` should
      be given if and only if the interpolator has y values specified.
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code: def add_xi(self, xi, yi=None):
      """
      Add more x values to the set to be interpolated
  
      The barycentric interpolation algorithm allows easy updating by
      adding more points for the polynomial to pass through.
  
      Parameters
      ----------
      xi : array_like
          The x coordinates of the points that the polynomial should pass
          through.
      yi : array_like, optional
          The y coordinates of the points the polynomial should pass through.
          Should have shape ``(xi.size, R)``; if R > 1 then the polynomial is
          vector-valued.
          If `yi` is not given, the y values will be supplied later. `yi` should
          be given if and only if the interpolator has y values specified.
  
      """
      if yi is not None:
          if self.yi is None:
              raise ValueError("No previous yi value to update!")
          yi = self._reshape_yi(yi, check=True)
          self.yi = np.vstack((self.yi, yi))
      else:
          if self.yi is not None:
              raise ValueError("No update to yi provided!")
      old_n = self.n
      self.xi = np.concatenate((self.xi, xi))
      self.n = len(self.xi)
      self.wi **= -1
      old_wi = self.wi
      self.wi = np.zeros(self.n)
      self.wi[:old_n] = old_wi
      for j in range(old_n, self.n):
          self.wi[:j] *= self.xi[j] - self.xi[:j]
          self.wi[j] = np.multiply.reduce(self.xi[:j] - self.xi[j])
      self.wi **= -1
  
imports: import numpy
  from scipy.special import factorial
  from scipy._lib._util import _asarray_validated, float_factorial
label_desc: Use image node_id 3 for calling the BarycentricInterpolator obj's underlying member method code with example usage: obj.add_xi(xi, yi) without return types