Derivatives using backward differences are a method used in numerical methods to approximate the derivative of a function. The idea is to use a sequence of poin...
Derivatives using backward differences are a method used in numerical methods to approximate the derivative of a function. The idea is to use a sequence of points that are spaced closer and closer together to get a better approximation of the derivative.
The backward difference method works by approximating the derivative as the difference between the values at positions t_i and t_i+h, where h is a small positive number.
The error in this approximation decreases as the step size h gets smaller. This method is also known as the "forward difference method" because it is the opposite of the forward difference method used for approximating the first derivative.
To use the backward difference method, we first need to choose a set of points t_i, where i = 1, 2, 3, ..., N. We then evaluate the function f(x) at these points and store the values in a list.
Once we have the list of function values, we can calculate the derivative as the difference between the values at positions t_i and t_i+h.
The error in this approximation is bounded by the error in the forward difference method, which is given by the formula:
where n is the order of the derivative. The lower this value, the more accurate the approximation