Numerical Differentiation: Comparing Central & Other Formulas

In summary, the speaker is discussing a different central differentiation formula for a first derivative that they have found. They are unsure if this formula is the same as the ones they are used to seeing. They are also having trouble finding numerical differentiation formulas in scholarly articles and have found one, but it looks different from what they are used to. The speaker mentions a slide show that provides examples of multipoint approximations in simpler notation.
  • #1
t0mm02
49
0
Homework Statement
Central, Forward, and Backward difference equations.
Relevant Equations
Central, Forward, and Backward difference equations.
Hello there, I have found a different central differentiation formula for a first derivate from what I am used to seeing and I was wondering if they were the same one. I am struggling to find the Numerical Differentiation formulas (forward, backward and central) in scholarly articles and I have found one with the supposed formulas but they look so different from how I normally see them.
This is the formula I have seen:
,k,yyui,.png


This is the formula I normally see:
6th5h.png
 
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  • #2
t0mm02 said:
This is the formula I have seen:
View attachment 276503

I'm not sure what the ##d## symbol denotes, but this looks like a formula for a "multipoint" approximation of ##f'##. For example, the slide show https://www3.nd.edu/~gtryggva/CFD-Course/2011-Lecture-8.pdf gives examples of multipoint approximations in simpler notation.
 

1. What is numerical differentiation?

Numerical differentiation is a method used to approximate the derivative of a function at a given point by using a set of discrete data points. It involves using a mathematical formula or algorithm to estimate the slope of a curve at a specific point.

2. How is central difference formula different from other numerical differentiation formulas?

The central difference formula uses two points on either side of the point of interest to calculate the slope, while other formulas may use more or fewer points. This makes the central difference formula more accurate than other formulas since it takes into account the curvature of the function at the point of interest.

3. What are the advantages of using numerical differentiation?

Numerical differentiation allows for the estimation of derivatives when the function is not known analytically or is too complex to differentiate by hand. It also provides a way to approximate derivatives at points that are not defined in the original function.

4. What are some potential sources of error in numerical differentiation?

Some potential sources of error in numerical differentiation include rounding errors in the data points, truncation errors from using a finite number of points, and the choice of step size used in the formula. Additionally, if the function being differentiated is not smooth or has discontinuities, this can also introduce errors.

5. How can we determine the accuracy of a numerical differentiation method?

The accuracy of a numerical differentiation method can be determined by comparing the results to the exact derivative of the function at the given point. The smaller the difference between the estimated derivative and the exact derivative, the more accurate the method is. Additionally, the choice of step size can also affect the accuracy, with smaller step sizes typically resulting in more accurate estimates.

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