Discrete Newton: Newton's method for discrete functions

This function is a modified Newton method that works with discrete data, i.e., computes the derivative of the data numerically. The derivative at any point is taken as the mean of the slope of the function to the right and the left.
The secant method is also programmed in the function and the user can select either of them. The code is heavily commented to make it easy to understand and modifiy. A lot of variable validation, printing and plotting is employed as well.