Using points on either side of a point x0 results in the best approximations of the derivative. Unfortunately, often in engineering, especially in real-time applications, we must use only past data to estimate the derivative at a point. For example, it may be necessary to use the current rate-of-change of an incoming signal to adjust a system, but only past points are available. Formula which use a technique similar to that in 13.1 and use only points ≤ x0 to approximate the derivative at x0 are termed backward divided-difference formula.
There are corresponding formulae using points greater than or equal to x0, but the derivation of these are left as an exercise to the reader.
Useful background for this topic includes:
- Bradie, Section 6.2, Numerical Differentiation, Part II, p.438.
- Mathews, Section 6.2, Numerical Differentiation Formulas, p.343.
- Chapra, Section 23.1, High-accuracy Differentiation Formaulas, p.634.
To generate a backward divided-difference formula, keep the points to the left of x, for example, f(x - 3*h) to f(x).
Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.