Error propagation of exponentials

In summary, there is a discrepancy between two methods for error propagation. While in Q = (a)(b)(c) the relative error in Q is simply the square root of the sum of the squares of each term, in Q = (a)(a)(a) the error is not simply the square root of the sum of the squares of a, as the terms are no longer independent. This can be seen in functions like S=(A+B)^2 where the correlation between A and B affects the spread in the distribution of S. This can make it difficult to account for correlations in variables with random uncertainties when calculating experimental uncertainties.
  • #1
TheCanadian
367
13
I am just wondering why there is a discrepancy between two different methods for error propagation. For example, if you have ## Q = (a)(b)(c) ## then the relative error in Q is simply the square root of the sum of the squares of each of the terms being multiplied together, correct? But what if ## Q = (a)(a)(a) ##. Why isn't the relative error in Q now simply once again the square root of the sum of the squares of a (which in this case would be 3 terms)? I understand the derivation for the relative error in ## Q = a^3 ## being ## 3 \frac {\Delta a}{a} ## but just don't quite understand why the earlier rule pertaining to basic multiplication and division no longer applies. What is the reason for a discrepancy between the two methods of error propagation? Can't exponentiation (using positive integers) be considered as just an extension of multiplication?
 
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  • #2
Notice that in Q = (a)(b)(c) the terms a, b and c are considered to be independent in the sense that the error of one of them is independent from the error to another. But if a = b = c then the "three" terms are not independent.
 
  • #3
One suggestion is to consider a function like S=(A+B)^2 where A and B are two variables that can each take on the values of +1 and -1. If A and B are uncorrelated, you will have less spread in the S distribution than if A=B. In computing experimental uncertainties, often the "delta" is just an estimate and it can be difficult to account for correlations in the variables that are often considered to have random uncertainties.
 

1. How do you calculate the error propagation of exponentials?

The error propagation of exponentials can be calculated using the formula: Δf = |f'(x)| * Δx, where Δf is the error in the output, f'(x) is the derivative of the exponential function, and Δx is the error in the input.

2. What is the purpose of calculating error propagation for exponentials?

Calculating error propagation for exponentials helps to determine the uncertainty or error in the final result when there are uncertainties in the input values. This allows for more accurate and reliable data analysis in scientific experiments.

3. Can error propagation of exponentials be applied to all exponential functions?

Yes, the error propagation formula for exponentials can be applied to all exponential functions, as long as the function is differentiable and the error in the input is known.

4. How does the error in the input affect the error in the output for exponentials?

The error in the input directly affects the error in the output for exponentials. The larger the error in the input, the larger the error in the output will be.

5. Is there a way to minimize error propagation in exponential calculations?

Yes, there are methods to minimize error propagation in exponential calculations, such as using more precise measurement tools, reducing experimental uncertainties, and using more accurate mathematical approximations.

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