Which Asymptotic Equation is More Accurate for Large Values of Rho?

In summary, asymptotic approximation is a mathematical technique used to estimate the behavior of a function as its input approaches a particular value, typically infinity. It allows for the simplification of complex functions and estimation without having to evaluate every point. The two most common types are Big O notation and Little O notation. It differs from exact approximation in that it focuses on the behavior near a specific point rather than finding an exact value. However, it may not be accurate for functions with finite values or those that behave differently at different points, depending on the choice of the simpler function used.
  • #1
EngWiPy
1,368
61
Hello all,

Which of the following asymptotic equations (as rho goes to infinity) correct:

[tex]1-\prod_{m=1}^M(1-\rho^{-x_m})\doteq \sum_{m=1}^M \rho^{-x_m}[/tex]

or

[tex]1-\prod_{m=1}^M(1-\rho^{-x_m})\doteq \rho^{-\underset{m}{\min}x_m}[/tex]

Thanks
 
Last edited:
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  • #2
The first expression is more accurate. The second will do as long as the min xm appears only once. If it appears more than once, you need to multiply by the number of occurrences.
 

FAQ: Which Asymptotic Equation is More Accurate for Large Values of Rho?

What is asymptotic approximation?

Asymptotic approximation is a mathematical technique used to estimate the behavior of a function as its input approaches a particular value, typically infinity. It involves finding a simpler function that behaves similarly to the original function near the point of interest.

Why is asymptotic approximation useful?

Asymptotic approximation allows us to simplify complex functions and make them more manageable to work with. It also provides a way to estimate the behavior of a function without having to evaluate it at every single point, which can be computationally expensive.

What are the common types of asymptotic approximations?

The two most common types of asymptotic approximations are the Big O notation and the Little O notation. Big O notation provides an upper bound on the behavior of a function, while Little O notation provides a stricter bound that is closer to the actual behavior of the function.

How is asymptotic approximation different from exact approximation?

Exact approximation aims to find an exact value for a function at a particular point, while asymptotic approximation focuses on the behavior of the function near a particular point. Asymptotic approximation is a more general and flexible approach, as it can be used to estimate the function at multiple points.

What are the limitations of asymptotic approximation?

Asymptotic approximation is based on the assumption that the input to the function approaches a particular value, typically infinity. Therefore, it may not be accurate for functions with finite values or those that behave differently at different points. Additionally, the quality of the approximation may depend on the choice of the simpler function used to approximate the original function.

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