Backward Differencing Formula
If you’ve calculated slopes before, the formula might look familiar: it’s a variation on the theme. There are four parts to the formula:
- xk: The x-value you’re estimating at.
- f(xk): The function value at xk.
- xk-1: The x-value, less the step size. For example, if you’re estimating at x = 2 with a step size of 1, then xk-1 is 2 -1 = 1.
- f(xk-1): The function value at xk-1.
Example question: Approximate the derivative of f(x) = x2 + 2x at x = 3 using backward differencing with a step size of 1.
Step 1: Identify xk. This is given in the question as x = 3.
Step 2: Calculate f(xk), the function value at the given point. For this example, that’s at x = 3. Inserting that value into the formula we’re given in the question: (f(x) = x2 + 2x ), we get:
f(3) = 32 + 2(3) = 15
Step 3: Identify xk-1. This your x-value from Step 1, minus 1:
3 – 1 = 2.
Step 4: Find f(xk-1), the function value one step behind. We’re given that the step size is 1 in the example problem, so we’re calculating the value at x = 2:
f(2) = 22 + 2(2) = 8
Step 5: Insert your values into the formula and solve:
Other Types of Differencing
Other types of differencing are forward differencing, and central differencing.
- Forward differencing: When h (the distance between the two points) is greater than zero (i.e. h > 0).
- Backward differencing: When h < 0.
- Central differencing: An average of the two methods (using three points).
Andasari, V. (2020). Numerical Differentiation. Retrieved September 21, 2020 from: http://people.bu.edu/andasari/courses/Fall2015/LectureNotes/Lecture7_24Sept2015.pdf
Kutz, J. (2013). Data-Driven Modeling & Scientific Computation. Methods for Complex Systems & Big Data. OUP Oxford.
Stephanie Glen. "Backward Differencing" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/backward-differencing/
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