text: from __future__ import division, print_function, absolute_import
  
  import numpy as np
  from scipy.misc import factorial
  
  from scipy.lib.six import xrange
  
  
  __all__ = ["KroghInterpolator", "krogh_interpolate", "BarycentricInterpolator",
             "barycentric_interpolate", "PiecewisePolynomial",
             "piecewise_polynomial_interpolate", "approximate_taylor_polynomial",
             "PchipInterpolator", "pchip_interpolate", "pchip"]
  
  
  def _isscalar(x):
      """Check whether x is if a scalar type, or 0-dim"""
      return np.isscalar(x) or hasattr(x, 'shape') and x.shape == ()
  
  
  class _Interpolator1D(object):
      """
      Common features in univariate interpolation
  
      Deal with input data type and interpolation axis rolling.  The
      actual interpolator can assume the y-data is of shape (n, r) where
      `n` is the number of x-points, and `r` the number of variables,
      and use self.dtype as the y-data type.
  
      Attributes
      ----------
      _y_axis
          Axis along which the interpolation goes in the original array
      _y_extra_shape
          Additional trailing shape of the input arrays, excluding
          the interpolation axis.
      dtype
          Dtype of the y-data arrays. Can be set via set_dtype, which
          forces it to be float or complex.
  
      Methods
      -------
      __call__
      _prepare_x
      _finish_y
      _reshape_yi
      _set_yi
      _set_dtype
      _evaluate
  
      """
  
      __slots__ = ('_y_axis', '_y_extra_shape', 'dtype')
  
      def __init__(self, xi=None, yi=None, axis=None):
          self._y_axis = axis
          self._y_extra_shape = None
          self.dtype = None
          if yi is not None:
              self._set_yi(yi, xi=xi, axis=axis)
  
      def __call__(self, x):
          """
          Evaluate the interpolant
  
          Parameters
          ----------
          x : array-like
              Points to evaluate the interpolant at.
  
          Returns
          -------
          y : array-like
              Interpolated values. Shape is determined by replacing
              the interpolation axis in the original array with the shape of x.
  
          """
          x, x_shape = self._prepare_x(x)
          y = self._evaluate(x)
          return self._finish_y(y, x_shape)
  
      def _evaluate(self, x):
          """
          Actually evaluate the value of the interpolator.
          """
          raise NotImplementedError()
  
      def _prepare_x(self, x):
          """Reshape input x array to 1-D"""
          x = np.asarray(x)
          x_shape = x.shape
          return x.ravel(), x_shape
  
      def _finish_y(self, y, x_shape):
          """Reshape interpolated y back to n-d array similar to initial y"""
          y = y.reshape(x_shape + self._y_extra_shape)
          if self._y_axis != 0 and x_shape != ():
              nx = len(x_shape)
              ny = len(self._y_extra_shape)
              s = (list(range(nx, nx + self._y_axis))
                   + list(range(nx)) + list(range(nx+self._y_axis, nx+ny)))
              y = y.transpose(s)
          return y
  
      def _reshape_yi(self, yi, check=False):
          yi = np.rollaxis(np.asarray(yi), self._y_axis)
          if check and yi.shape[1:] != self._y_extra_shape:
              ok_shape = "%r + (N,) + %r" % (self._y_extra_shape[-self._y_axis:],
                                             self._y_extra_shape[:-self._y_axis])
              raise ValueError("Data must be of shape %s" % ok_shape)
          return yi.reshape((yi.shape[0], -1))
  
      def _set_yi(self, yi, xi=None, axis=None):
          if axis is None:
              axis = self._y_axis
          if axis is None:
              raise ValueError("no interpolation axis specified")
  
          yi = np.asarray(yi)
  
          shape = yi.shape
          if shape == ():
              shape = (1,)
          if xi is not None and shape[axis] != len(xi):
              raise ValueError("x and y arrays must be equal in length along "
                               "interpolation axis.")
  
          self._y_axis = (axis % yi.ndim)
          self._y_extra_shape = yi.shape[:self._y_axis]+yi.shape[self._y_axis+1:]
          self.dtype = None
          self._set_dtype(yi.dtype)
  
      def _set_dtype(self, dtype, union=False):
          if np.issubdtype(dtype, np.complexfloating) \
                 or np.issubdtype(self.dtype, np.complexfloating):
              self.dtype = np.complex_
          else:
              if not union or self.dtype != np.complex_:
                  self.dtype = np.float_
  
  
  class _Interpolator1DWithDerivatives(_Interpolator1D):
      def derivatives(self, x, der=None):
          """
          Evaluate many derivatives of the polynomial at the point x
  
          Produce an array of all derivative values at the point x.
  
          Parameters
          ----------
          x : array-like
              Point or points at which to evaluate the derivatives
  
          der : None or integer
              How many derivatives to extract; None for all potentially
              nonzero derivatives (that is a number equal to the number
              of points). This number includes the function value as 0th
              derivative.
  
          Returns
          -------
          d : ndarray
              Array with derivatives; d[j] contains the j-th derivative.
              Shape of d[j] is determined by replacing the interpolation
              axis in the original array with the shape of x.
  
          Examples
          --------
          >>> KroghInterpolator([0,0,0],[1,2,3]).derivatives(0)
          array([1.0,2.0,3.0])
          >>> KroghInterpolator([0,0,0],[1,2,3]).derivatives([0,0])
          array([[1.0,1.0],
                 [2.0,2.0],
                 [3.0,3.0]])
  
          """
          x, x_shape = self._prepare_x(x)
          y = self._evaluate_derivatives(x, der)
  
          y = y.reshape((y.shape[0],) + x_shape + self._y_extra_shape)
          if self._y_axis != 0 and x_shape != ():
              nx = len(x_shape)
              ny = len(self._y_extra_shape)
              s = ([0] + list(range(nx+1, nx + self._y_axis+1))
                   + list(range(1,nx+1)) +
                   list(range(nx+1+self._y_axis, nx+ny+1)))
              y = y.transpose(s)
          return y
  
      def derivative(self, x, der=1):
          """
          Evaluate one derivative of the polynomial at the point x
  
          Parameters
          ----------
          x : array-like
              Point or points at which to evaluate the derivatives
  
          der : integer, optional
              Which derivative to extract. This number includes the
              function value as 0th derivative.
  
          Returns
          -------
          d : ndarray
              Derivative interpolated at the x-points.  Shape of d is
              determined by replacing the interpolation axis in the
              original array with the shape of x.
  
          Notes
          -----
          This is computed by evaluating all derivatives up to the desired
          one (using self.derivatives()) and then discarding the rest.
  
          """
          x, x_shape = self._prepare_x(x)
          y = self._evaluate_derivatives(x, der+1)
          return self._finish_y(y[der], x_shape)
  
  
  class KroghInterpolator(_Interpolator1DWithDerivatives):
      """
      Interpolating polynomial for a set of points.
  
      The polynomial passes through all the pairs (xi,yi). One may
      additionally specify a number of derivatives at each point xi;
      this is done by repeating the value xi and specifying the
      derivatives as successive yi values.
  
      Allows evaluation of the polynomial and all its derivatives.
      For reasons of numerical stability, this function does not compute
      the coefficients of the polynomial, although they can be obtained
      by evaluating all the derivatives.
  
      Parameters
      ----------
      xi : array-like, length N
          Known x-coordinates. Must be sorted in increasing order.
      yi : array-like
          Known y-coordinates. When an xi occurs two or more times in
          a row, the corresponding yi's represent derivative values.
      axis : int, optional
          Axis in the yi array corresponding to the x-coordinate values.
  
      Notes
      -----
      Be aware that the algorithms implemented here are not necessarily
      the most numerically stable known. Moreover, 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. In general, even with well-chosen
      x values, degrees higher than about thirty cause problems with
      numerical instability in this code.
  
      Based on [1]_.
  
      References
      ----------
      .. [1] Krogh, "Efficient Algorithms for Polynomial Interpolation
          and Numerical Differentiation", 1970.
  
      Examples
      --------
      To produce a polynomial that is zero at 0 and 1 and has
      derivative 2 at 0, call
  
      >>> KroghInterpolator([0,0,1],[0,2,0])
  
      This constructs the quadratic 2*X**2-2*X. The derivative condition
      is indicated by the repeated zero in the xi array; the corresponding
      yi values are 0, the function value, and 2, the derivative value.
  
      For another example, given xi, yi, and a derivative ypi for each
      point, appropriate arrays can be constructed as:
  
      >>> xi_k, yi_k = np.repeat(xi, 2), np.ravel(np.dstack((yi,ypi)))
      >>> KroghInterpolator(xi_k, yi_k)
  
      To produce a vector-valued polynomial, supply a higher-dimensional
      array for yi:
  
      >>> KroghInterpolator([0,1],[[2,3],[4,5]])
  
      This constructs a linear polynomial giving (2,3) at 0 and (4,5) at 1.
  
      """
  
      def __init__(self, xi, yi, axis=0):
          _Interpolator1DWithDerivatives.__init__(self, xi, yi, axis)
  
          self.xi = np.asarray(xi)
          self.yi = self._reshape_yi(yi)
          self.n, self.r = self.yi.shape
  
          c = np.zeros((self.n+1, self.r), dtype=self.dtype)
          c[0] = self.yi[0]
          Vk = np.zeros((self.n, self.r), dtype=self.dtype)
          for k in xrange(1,self.n):
              s = 0
              while s <= k and xi[k-s] == xi[k]:
                  s += 1
              s -= 1
              Vk[0] = self.yi[k]/float(factorial(s))
              for i in xrange(k-s):
                  if xi[i] == xi[k]:
                      raise ValueError("Elements if `xi` can't be equal.")
                  if s == 0:
                      Vk[i+1] = (c[i]-Vk[i])/(xi[i]-xi[k])
                  else:
                      Vk[i+1] = (Vk[i+1]-Vk[i])/(xi[i]-xi[k])
              c[k] = Vk[k-s]
          self.c = c
  
      def _evaluate(self, x):
          pi = 1
          p = np.zeros((len(x), self.r), dtype=self.dtype)
          p += self.c[0,np.newaxis,:]
          for k in range(1, self.n):
              w = x - self.xi[k-1]
              pi = w*pi
              p += pi[:,np.newaxis] * self.c[k]
          return p
  
      def _evaluate_derivatives(self, x, der=None):
          n = self.n
          r = self.r
  
          if der is None:
              der = self.n
          pi = np.zeros((n, len(x)))
          w = np.zeros((n, len(x)))
          pi[0] = 1
          p = np.zeros((len(x), self.r))
          p += self.c[0,np.newaxis,:]
  
          for k in xrange(1,n):
              w[k-1] = x - self.xi[k-1]
              pi[k] = w[k-1]*pi[k-1]
              p += pi[k,:,np.newaxis]*self.c[k]
  
          cn = np.zeros((max(der,n+1), len(x), r), dtype=self.dtype)
          cn[:n+1,:,:] += self.c[:n+1,np.newaxis,:]
          cn[0] = p
          for k in xrange(1,n):
              for i in xrange(1,n-k+1):
                  pi[i] = w[k+i-1]*pi[i-1]+pi[i]
                  cn[k] = cn[k]+pi[i,:,np.newaxis]*cn[k+i]
              cn[k] *= factorial(k)
  
          cn[n,:,:] = 0
          return cn[:der]
  
  
  def krogh_interpolate(xi,yi,x,der=0,axis=0):
      """
      Convenience function for polynomial interpolation.
  
      See `KroghInterpolator` for more details.
  
      Parameters
      ----------
      xi : array_like
          Known x-coordinates.
      yi : array_like
          Known y-coordinates, of shape ``(xi.size, R)``.  Interpreted as
          vectors of length R, or scalars if R=1.
      x : array_like
          Point or points at which to evaluate the derivatives.
      der : int or list
          How many derivatives to extract; None for all potentially
          nonzero derivatives (that is a number equal to the number
          of points), or a list of derivatives to extract. This number
          includes the function value as 0th derivative.
      axis : int, optional
          Axis in the yi array corresponding to the x-coordinate values.
  
      Returns
      -------
      d : ndarray
          If the interpolator's values are R-dimensional then the
          returned array will be the number of derivatives by N by R.
          If `x` is a scalar, the middle dimension will be dropped; if
          the `yi` are scalars then the last dimension will be dropped.
  
      See Also
      --------
      KroghInterpolator
  
      Notes
      -----
      Construction of the interpolating polynomial is a relatively expensive
      process. If you want to evaluate it repeatedly consider using the class
      KroghInterpolator (which is what this function uses).
  
      """
      P = KroghInterpolator(xi, yi, axis=axis)
      if der == 0:
          return P(x)
      elif _isscalar(der):
          return P.derivative(x,der=der)
      else:
          return P.derivatives(x,der=np.amax(der)+1)[der]
  
  
  def approximate_taylor_polynomial(f,x,degree,scale,order=None):
      """
      Estimate the Taylor polynomial of f at x by polynomial fitting.
  
      Parameters
      ----------
      f : callable
          The function whose Taylor polynomial is sought. Should accept
          a vector of `x` values.
      x : scalar
          The point at which the polynomial is to be evaluated.
      degree : int
          The degree of the Taylor polynomial
      scale : scalar
          The width of the interval to use to evaluate the Taylor polynomial.
          Function values spread over a range this wide are used to fit the
          polynomial. Must be chosen carefully.
      order : int or None, optional
          The order of the polynomial to be used in the fitting; `f` will be
          evaluated ``order+1`` times. If None, use `degree`.
  
      Returns
      -------
      p : poly1d instance
          The Taylor polynomial (translated to the origin, so that
          for example p(0)=f(x)).
  
      Notes
      -----
      The appropriate choice of "scale" is a trade-off; too large and the
      function differs from its Taylor polynomial too much to get a good
      answer, too small and round-off errors overwhelm the higher-order terms.
      The algorithm used becomes numerically unstable around order 30 even
      under ideal circumstances.
  
      Choosing order somewhat larger than degree may improve the higher-order
      terms.
  
      """
      if order is None:
          order = degree
  
      n = order+1
      # Choose n points that cluster near the endpoints of the interval in
      # a way that avoids the Runge phenomenon. Ensure, by including the
      # endpoint or not as appropriate, that one point always falls at x
      # exactly.
      xs = scale*np.cos(np.linspace(0,np.pi,n,endpoint=n % 1)) + x
  
      P = KroghInterpolator(xs, f(xs))
      d = P.derivatives(x,der=degree+1)
  
      return np.poly1d((d/factorial(np.arange(degree+1)))[::-1])
  
  
  class BarycentricInterpolator(_Interpolator1D):
      """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
          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".
  
      """
      def __init__(self, xi, yi=None, axis=0):
          _Interpolator1D.__init__(self, xi, yi, axis)
  
          self.xi = np.asarray(xi)
          self.set_yi(yi)
          self.n = len(self.xi)
  
          self.wi = np.zeros(self.n)
          self.wi[0] = 1
          for j in xrange(1,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
  
      def set_yi(self, yi, axis=None):
          """
          Update the y values to be interpolated
  
          The barycentric interpolation algorithm requires the calculation
          of weights, but these depend only on the xi. The yi can be changed
          at any time.
  
          Parameters
          ----------
          yi : array_like
              The y coordinates of the points the polynomial should pass through.
              If None, the y values will be supplied later.
          axis : int, optional
              Axis in the yi array corresponding to the x-coordinate values.
  
          """
          if yi is None:
              self.yi = None
              return
          self._set_yi(yi, xi=self.xi, axis=axis)
          self.yi = self._reshape_yi(yi)
          self.n, self.r = self.yi.shape
  
      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 xrange(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
  
      def __call__(self, x):
          """Evaluate the interpolating polynomial at the points x
  
          Parameters
          ----------
          x : array-like
              Points to evaluate the interpolant at.
  
          Returns
          -------
          y : array-like
              Interpolated values. Shape is determined by replacing
              the interpolation axis in the original array with the shape of x.
  
          Notes
          -----
          Currently the code computes an outer product between x and the
          weights, that is, it constructs an intermediate array of size
          N by len(x), where N is the degree of the polynomial.
          """
          return _Interpolator1D.__call__(self, x)
  
      def _evaluate(self, x):
          if x.size == 0:
              p = np.zeros((0, self.r), dtype=self.dtype)
          else:
              c = x[...,np.newaxis]-self.xi
              z = c == 0
              c[z] = 1
              c = self.wi/c
              p = np.dot(c,self.yi)/np.sum(c,axis=-1)[...,np.newaxis]
              # Now fix where x==some xi
              r = np.nonzero(z)
              if len(r) == 1:  # evaluation at a scalar
                  if len(r[0]) > 0:  # equals one of the points
                      p = self.yi[r[0][0]]
              else:
                  p[r[:-1]] = self.yi[r[-1]]
          return p
  
  
  def barycentric_interpolate(xi, yi, x, axis=0):
      """
      Convenience function for polynomial interpolation.
  
      Constructs a polynomial that passes through a given set of points,
      then evaluates the polynomial. For reasons of numerical stability,
      this function does not compute the coefficients of the polynomial.
  
      This function 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.
  
      Parameters
      ----------
      xi : array_like
          1-d array of x coordinates of the points the polynomial should
          pass through
      yi : array_like
          The y coordinates of the points the polynomial should pass through.
      x : scalar or array_like
          Points to evaluate the interpolator at.
      axis : int, optional
          Axis in the yi array corresponding to the x-coordinate values.
  
      Returns
      -------
      y : scalar or array_like
          Interpolated values. Shape is determined by replacing
          the interpolation axis in the original array with the shape of x.
  
      See Also
      --------
      BarycentricInterpolator
  
      Notes
      -----
      Construction of the interpolation weights is a relatively slow process.
      If you want to call this many times with the same xi (but possibly
      varying yi or x) you should use the class `BarycentricInterpolator`.
      This is what this function uses internally.
  
      """
      return BarycentricInterpolator(xi, yi, axis=axis)(x)
  
  
  class PiecewisePolynomial(_Interpolator1DWithDerivatives):
      """Piecewise polynomial curve specified by points and derivatives
  
      This class represents a curve that is a piecewise polynomial. It
      passes through a list of points and has specified derivatives at
      each point. The degree of the polynomial may vary from segment to
      segment, as may the number of derivatives available. The degree
      should not exceed about thirty.
  
      Appending points to the end of the curve is efficient.
  
      Parameters
      ----------
      xi : array-like
          a sorted 1-d array of x-coordinates
      yi : array-like or list of array-likes
          yi[i][j] is the j-th derivative known at xi[i]   (for axis=0)
      orders : list of integers, or integer
          a list of polynomial orders, or a single universal order
      direction : {None, 1, -1}
          indicates whether the xi are increasing or decreasing
          +1 indicates increasing
          -1 indicates decreasing
          None indicates that it should be deduced from the first two xi
      axis : int, optional
          Axis in the yi array corresponding to the x-coordinate values.
  
      Notes
      -----
      If orders is None, or orders[i] is None, then the degree of the
      polynomial segment is exactly the degree required to match all i
      available derivatives at both endpoints. If orders[i] is not None,
      then some derivatives will be ignored. The code will try to use an
      equal number of derivatives from each end; if the total number of
      derivatives needed is odd, it will prefer the rightmost endpoint. If
      not enough derivatives are available, an exception is raised.
  
      """
  
      def __init__(self, xi, yi, orders=None, direction=None, axis=0):
          _Interpolator1DWithDerivatives.__init__(self, axis=axis)
  
          if axis != 0:
              try:
                  yi = np.asarray(yi)
              except ValueError:
                  raise ValueError("If yi is a list, then axis must be 0")
  
              preslice = ((slice(None,None,None),) * (axis % yi.ndim))
              slice0 = preslice + (0,)
              slice1 = preslice + (slice(1, None, None),)
          else:
              slice0 = 0
              slice1 = slice(1, None, None)
  
          yi0 = np.asarray(yi[slice0])
          self._set_yi(yi0)
  
          self.xi = [xi[0]]
          self.yi = [self._reshape_yi(yi0)]
          self.n = 1
          self.r = np.prod(self._y_extra_shape, dtype=np.int64)
  
          self.direction = direction
          self.orders = []
          self.polynomials = []
          self.extend(xi[1:],yi[slice1],orders)
  
      def _make_polynomial(self,x1,y1,x2,y2,order,direction):
          """Construct the interpolating polynomial object
  
          Deduces the number of derivatives to match at each end
          from order and the number of derivatives available. If
          possible it uses the same number of derivatives from
          each end; if the number is odd it tries to take the
          extra one from y2. In any case if not enough derivatives
          are available at one end or another it draws enough to
          make up the total from the other end.
          """
          n = order+1
          n1 = min(n//2,len(y1))
          n2 = min(n-n1,len(y2))
          n1 = min(n-n2,len(y1))
          if n1+n2 != n:
              raise ValueError("Point %g has %d derivatives, point %g has %d derivatives, but order %d requested" % (x1, len(y1), x2, len(y2), order))
          if not (n1 <= len(y1) and n2 <= len(y2)):
              raise ValueError("`order` input incompatible with length y1 or y2.")
  
          xi = np.zeros(n)
          yi = np.zeros((n, self.r), dtype=self.dtype)
  
          xi[:n1] = x1
          yi[:n1] = y1[:n1].reshape((n1, self.r))
          xi[n1:] = x2
          yi[n1:] = y2[:n2].reshape((n2, self.r))
  
          return KroghInterpolator(xi,yi,axis=0)
  
      def append(self, xi, yi, order=None):
          """
          Append a single point with derivatives to the PiecewisePolynomial
  
          Parameters
          ----------
          xi : float
              Input
          yi : array_like
              `yi` is the list of derivatives known at `xi`
          order : integer or None
              a polynomial order, or instructions to use the highest
              possible order
  
          """
          yi = self._reshape_yi(yi, check=True)
          self._set_dtype(yi.dtype, union=True)
  
          if self.direction is None:
              self.direction = np.sign(xi-self.xi[-1])
          elif (xi-self.xi[-1])*self.direction < 0:
              raise ValueError("x coordinates must be in the %d direction: %s" % (self.direction, self.xi))
  
          self.xi.append(xi)
          self.yi.append(yi)
  
          if order is None:
              n1 = len(self.yi[-2])
              n2 = len(self.yi[-1])
              n = n1+n2
              order = n-1
  
          self.orders.append(order)
          self.polynomials.append(self._make_polynomial(
              self.xi[-2], self.yi[-2],
              self.xi[-1], self.yi[-1],
              order, self.direction))
          self.n += 1
  
      def extend(self, xi, yi, orders=None):
          """
          Extend the PiecewisePolynomial by a list of points
  
          Parameters
          ----------
          xi : array_like
              A sorted list of x-coordinates.
          yi : list of lists of length N1
              ``yi[i]`` (if ``axis == 0``) is the list of derivatives known
              at ``xi[i]``.
          orders : int or list of ints
              A list of polynomial orders, or a single universal order.
          direction : {None, 1, -1}
              Indicates whether the `xi` are increasing or decreasing.
  
                  +1 indicates increasing
  
                  -1 indicates decreasing
  
              None indicates that it should be deduced from the first two `xi`.
  
          """
          if self._y_axis == 0:
              # allow yi to be a ragged list
              for i in xrange(len(xi)):
                  if orders is None or _isscalar(orders):
                      self.append(xi[i],yi[i],orders)
                  else:
                      self.append(xi[i],yi[i],orders[i])
          else:
              preslice = (slice(None,None,None),) * self._y_axis
              for i in xrange(len(xi)):
                  if orders is None or _isscalar(orders):
                      self.append(xi[i],yi[preslice + (i,)],orders)
                  else:
                      self.append(xi[i],yi[preslice + (i,)],orders[i])
  
      def _evaluate(self, x):
          if _isscalar(x):
              pos = np.clip(np.searchsorted(self.xi, x) - 1, 0, self.n-2)
              y = self.polynomials[pos](x)
          else:
              m = len(x)
              pos = np.clip(np.searchsorted(self.xi, x) - 1, 0, self.n-2)
              y = np.zeros((m, self.r), dtype=self.dtype)
              if y.size > 0:
                  for i in xrange(self.n-1):
                      c = pos == i
                      y[c] = self.polynomials[i](x[c])
          return y
  
      def _evaluate_derivatives(self, x, der=None):
          if der is None and self.polynomials:
              der = self.polynomials[0].n
          if _isscalar(x):
              pos = np.clip(np.searchsorted(self.xi, x) - 1, 0, self.n-2)
              y = self.polynomials[pos].derivatives(x,der=der)
          else:
              m = len(x)
              pos = np.clip(np.searchsorted(self.xi, x) - 1, 0, self.n-2)
              y = np.zeros((der,m,self.r), dtype=self.dtype)
              if y.size > 0:
                  for i in xrange(self.n-1):
                      c = pos == i
                      y[:,c] = self.polynomials[i].derivatives(x[c],der=der)
          return y
  
  
  def piecewise_polynomial_interpolate(xi,yi,x,orders=None,der=0,axis=0):
      """
      Convenience function for piecewise polynomial interpolation.
  
      Parameters
      ----------
      xi : array_like
          A sorted list of x-coordinates.
      yi : list of lists
          ``yi[i]`` is the list of derivatives known at ``xi[i]``.
      x : scalar or array_like
          Coordinates at which to evalualte the polynomial.
      orders : int or list of ints, optional
          A list of polynomial orders, or a single universal order.
      der : int or list
          How many derivatives to extract; None for all potentially
          nonzero derivatives (that is a number equal to the number
          of points), or a list of derivatives to extract. This number
          includes the function value as 0th derivative.
      axis : int, optional
          Axis in the `yi` array corresponding to the x-coordinate values.
  
      Returns
      -------
      y : ndarray
          Interpolated values or derivatives. If multiple derivatives
          were requested, these are given along the first axis.
  
      See Also
      --------
      PiecewisePolynomial
  
      Notes
      -----
      If `orders` is None, or ``orders[i]`` is None, then the degree of the
      polynomial segment is exactly the degree required to match all i
      available derivatives at both endpoints.  If ``orders[i]`` is not None,
      then some derivatives will be ignored.  The code will try to use an
      equal number of derivatives from each end; if the total number of
      derivatives needed is odd, it will prefer the rightmost endpoint. If
      not enough derivatives are available, an exception is raised.
  
      Construction of these piecewise polynomials can be an expensive process;
      if you repeatedly evaluate the same polynomial, consider using the class
      PiecewisePolynomial (which is what this function does).
  
      """
  
      P = PiecewisePolynomial(xi, yi, orders, axis=axis)
      if der == 0:
          return P(x)
      elif _isscalar(der):
          return P.derivative(x,der=der)
      else:
          return P.derivatives(x,der=np.amax(der)+1)[der]
  
  
  class PchipInterpolator(PiecewisePolynomial):
      """PCHIP 1-d monotonic cubic interpolation
  
      x and y are arrays of values used to approximate some function f,
      with ``y = f(x)``.  The interpolant uses monotonic cubic splines
      to find the value of new points.
  
      Parameters
      ----------
      x : ndarray
          A 1-D array of monotonically increasing real values.  `x` cannot
          include duplicate values (otherwise f is overspecified)
      y : ndarray
          A 1-D array of real values.  `y`'s length along the interpolation
          axis must be equal to the length of `x`.
      axis : int, optional
          Axis in the yi array corresponding to the x-coordinate values.
  
      Notes
      -----
      Assumes x is sorted in monotonic order (e.g. ``x[1] > x[0]``).
  
      """
      def __init__(self, x, y, axis=0):
          x = np.asarray(x)
          y = np.asarray(y)
  
          axis = axis % y.ndim
  
          xp = x.reshape((x.shape[0],) + (1,)*(y.ndim-1))
          yp = np.rollaxis(y, axis)
  
          data = np.empty((yp.shape[0], 2) + yp.shape[1:], y.dtype)
          data[:,0] = yp
          data[:,1] = PchipInterpolator._find_derivatives(xp, yp)
  
          s = list(range(2, y.ndim + 1))
          s.insert(axis, 1)
          s.insert(axis, 0)
          data = data.transpose(s)
  
          PiecewisePolynomial.__init__(self, x, data, orders=3, direction=None,
                                       axis=axis)
  
      @staticmethod
      def _edge_case(m0, d1, out):
          m0 = np.atleast_1d(m0)
          d1 = np.atleast_1d(d1)
          mask = (d1 != 0) & (m0 != 0)
          out[mask] = 1.0/(1.0/m0[mask]+1.0/d1[mask])
  
      @staticmethod
      def _find_derivatives(x, y):
          # Determine the derivatives at the points y_k, d_k, by using
          #  PCHIP algorithm is:
          # We choose the derivatives at the point x_k by
          # Let m_k be the slope of the kth segment (between k and k+1)
          # If m_k=0 or m_{k-1}=0 or sgn(m_k) != sgn(m_{k-1}) then d_k == 0
          # else use weighted harmonic mean:
          #   w_1 = 2h_k + h_{k-1}, w_2 = h_k + 2h_{k-1}
          #   1/d_k = 1/(w_1 + w_2)*(w_1 / m_k + w_2 / m_{k-1})
          #   where h_k is the spacing between x_k and x_{k+1}
  
          y_shape = y.shape
          if y.ndim == 1:
              # So that _edge_case doesn't end up assigning to scalars
              x = x[:,None]
              y = y[:,None]
  
          hk = x[1:] - x[:-1]
          mk = (y[1:] - y[:-1]) / hk
          smk = np.sign(mk)
          condition = ((smk[1:] != smk[:-1]) | (mk[1:] == 0) | (mk[:-1] == 0))
  
          w1 = 2*hk[1:] + hk[:-1]
          w2 = hk[1:] + 2*hk[:-1]
          # values where division by zero occurs will be excluded
          # by 'condition' afterwards
          with np.errstate(divide='ignore'):
              whmean = 1.0/(w1+w2)*(w1/mk[1:] + w2/mk[:-1])
          dk = np.zeros_like(y)
          dk[1:-1][condition] = 0.0
          dk[1:-1][~condition] = 1.0/whmean[~condition]
  
          # For end-points choose d_0 so that 1/d_0 = 1/m_0 + 1/d_1 unless
          #  one of d_1 or m_0 is 0, then choose d_0 = 0
          PchipInterpolator._edge_case(mk[0],dk[1], dk[0])
          PchipInterpolator._edge_case(mk[-1],dk[-2], dk[-1])
  
          return dk.reshape(y_shape)
  
  
  def pchip_interpolate(xi, yi, x, der=0, axis=0):
      """
      Convenience function for pchip interpolation.
  
      See `PchipInterpolator` for details.
  
      Parameters
      ----------
      xi : array_like
          A sorted list of x-coordinates, of length N.
      yi : list of lists
          yi[i] is the list of derivatives known at xi[i]. Of length N.
      x : scalar or array_like
          Of length M.
      der : integer or list
          How many derivatives to extract; None for all potentially
          nonzero derivatives (that is a number equal to the number
          of points), or a list of derivatives to extract. This number
          includes the function value as 0th derivative.
      axis : int, optional
          Axis in the yi array corresponding to the x-coordinate values.
  
      See Also
      --------
      PchipInterpolator
  
      Returns
      -------
      y : scalar or array_like
          The result, of length R or length M or M by R,
  
      """
      P = PchipInterpolator(xi, yi, axis=axis)
      if der == 0:
          return P(x)
      elif _isscalar(der):
          return P.derivative(x,der=der)
      else:
          return P.derivatives(x,der=np.amax(der)+1)[der]
  
  # Backwards compatibility
  pchip = PchipInterpolator
  