9 | Numerical Calculus: Differentiation
In this section, we focus on finding derivatives using numerical techniques. We will look at operating on functions and on discrete empirical data. Throughout this lecture, I will use the terms numerical differentiation, numerical differencing and finite differencing interchangeably.
At the end of this section you should be able to:
Employ forward, backward, and central differentiation to numerically differentiate functions or data of one variable
Utilize various orders of accuracy in numerical differentiation
Describe the advantages and disadvantages of the different methods and orders of accuracy
Describe how noise in the data will impact numerical differentiation
Differentiation of Data
Numerical Differentiation: we begin by looking at the calculus definition of derivative and how we translate that into a numerical definition. We then look at the concepts of forward, backward and central differencing.
Numerical Differentiation: General Finite Differencing Formulae. We look at the general equations to use the different types of differencing. Expanded coefficient tables can be found at https://en.wikipedia.org/wiki/Finite_difference_coefficient.
Numerical Differentiation of Data: how the derivative of experimentally collected data can be found numerically and under what conditions we can use finite differencing.
Differentiation of Data Example 1: Using and comparing forward, backward and central differentiation on a small set of empirical data.
Differentiation of Functions Example 2: using higher-order central differentiation on a function. Here, we combine the use of forward, backward and central differencing to find the derivative at every point.
Differentiation of Noisy Data Example 3: using numerical differentiation on numerical data with noise. Noisy data generally yields poor derivative results.