Get the roots of a Hermite interpolating polynomial

Question

I am using Python 3.7 to write a program that requires me to calculate the root of the Hermite interpolating polynomial, given two points $\epsilon_0$, $\epsilon_1$, the function ($d(\epsilon_0)$,$d(\epsilon_1)$) and the derivative values ($d'(\epsilon_1)$, $d'(\epsilon_1)$) at those points. I am using Scipy v1.3.0 and using the CubicHermiteSpline function from the scipy.interpolate library. The relevant extracts from the code are:

import numpy as np
from scipy.interpolate import BPoly,CubicHermiteSpline

#somewhere below inside a while loop with a counter variable k is this part

r=CubicHermiteSpline(eps[k-1:k+1],abs(l[k-1:k+1]), d1[k-1:k+1]).roots()
epsk=(np.abs(r - eps[k])).argmin()

Whereabs(l) contains the values for the polynomial and d1 contains the derivative values. The problem is the .roots() returns an empty array for the interval ($\epsilon_0$,$\epsilon_1$).

ValueError: attempt to get argmin of an empty sequence

This is because the interpolated polynomial from this interval looks like this:

How do I get all the three roots of the interpolated polynomial, which may not necessarily be inside the interval?

Edit: The numerical values: $$d(\epsilon_0)=1.00000188\\ d(\epsilon_1)=1.09393556\\ d'(\epsilon_0)=-4.30116854\\ d'(\epsilon_1)=-4.30428889 $$ Find the roots of the hermite intpolation polynomial. Interpolated polynomial graph:

Answer 1

The interpolated polynomial does not have roots. Considering that the behavior outside the interpolation region holds is termed extrapolation.

You can explicitly use the polynomial, given by (as I explained in this post)

$$f(x) \approx N_1(x) u_1 + N_2(x) u_2 + |J|(N_3(x) u'_1 + N_4(x) u'_2)\quad \forall x\in [a, b]$$

with $|J| = (b - a)/2$ the Jacobian determinant of the transformation, and, for $a=-1, b=1$,

\begin{align} N_1 (x) &= \frac{1}{4} (x - 1)^2 (2 + x)\\ N_2 (x) &= \frac{1}{4} (x + 1)^2 (2 - x)\\ N_3 (x) &= \frac{1}{4} (x - 1)^2 (x + 1)\\ N_4 (x) &= \frac{1}{4} (x + 1)^2 (x - 1)\, . \end{align}

Then, you can use the general formula for cubic equations or use a method like Newton-Raphson.