e
antiderivative
roots

See Also
--------
CubicHermiteSpline : Piecewise-cubic interpolator.
Akima1DInterpolator : Akima 1D interpolator.
CubicSpline : Cubic spline data interpolator.
PPoly : Piecewise polynomial in terms of coefficients and breakpoints.

Notes
-----
The interpolator preserves monotonicity in the interpolation data and does
not overshoot if the data is not smooth.

The first derivatives are guaranteed to be continuous, but the second
derivatives may jump at :math:`x_k`.

Determines the derivatives at the points :math:`x_k`, :math:`f'_k`,
by using PCHIP algorithm [1]_.

Let :math:`h_k = x_{k+1} - x_k`, and  :math:`d_k = (y_{k+1} - y_k) / h_k`
are the slopes at internal points :math:`x_k`.
If the signs of :math:`d_k` and :math:`d_{k-1}` are different or either of
them equals zero, then :math:`f'_k = 0`. Otherwise, it is given by the
weighted harmonic mean

.. math::

    \frac{w_1 + w_2}{f'_k} = \frac{w_1}{d_{k-1}} + \frac{w_2}{d_k}

where :math:`w_1 = 2 h_k + h_{k-1}` and :math:`w_2 = h_k + 2 h_{k-1}`.

The end slopes are set using a one-sided scheme [2]_.


References
----------
.. [1] F. N. Fritsch and J. Butland,
       A method for constructing local
       monotone piecewise cubic interpolants,
       SIAM J. Sci. Comput., 5(2), 300-304 (1984).
       :doi:`10.1137/0905021`.
.. [2] see, e.g., C. Moler, Numerical Computing with Matlab, 2004.
       :doi:`10.1137/1.9780898717952`

c