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I can provide a list of key differences between non-asymptotic and asymptotic analysis in theory and illustrate these differences with equations:

1. Focus on Finite Values vs. Limiting Behavior:

• Non-asymptotic Analysis: Concerned with understanding the behavior of functions or sequences for specific, finite values or ranges of parameters.
• Asymptotic Analysis: Focuses on the behavior of functions or sequences as they approach limiting values or conditions (usually as the independent variable goes to infinity or some other limiting value).

2. Error Analysis:

• Non-asymptotic Analysis: Provides quantitative information about the error or deviation from the exact value for finite values. Involves bounding errors.
• Asymptotic Analysis: Typically does not provide information about the absolute error but focuses on relative error or how functions behave as they approach limits.

3. Rate of Convergence:

• Non-asymptotic Analysis: Emphasizes the speed at which a sequence or function converges to a limit for finite values of parameters.
• Asymptotic Analysis: Emphasizes the behavior of a sequence or function as it approaches a limiting condition, often characterized by expressions involving "big O" notation.

4. Specificity vs. Generality:

• Non-asymptotic Analysis: Often provides specific results for particular values or ranges of parameters.
• Asymptotic Analysis: Tends to provide more general results that apply as the limiting condition is approached.

5. Practical vs. Theoretical:

• Non-asymptotic Analysis: Often employed to address practical questions and problems with finite and bounded values.
• Asymptotic Analysis: Often used to gain insights into the overall behavior of functions without necessarily being tied to practical, finite cases.

6. Equations:

• Non-asymptotic Example: Consider the convergence of a finite geometric series:

  For a geometric series , the finite sum is given by:

          -------------------------------------- [3963a]

• Non-asymptotic analysis may involve studying this sum for specific values of n, a, and r.

• Asymptotic Example: Consider the asymptotic behavior of a factorial function n! as n approaches infinity:

          -------------------------------------- [3963b]

• Asymptotic analysis provides insights into how the factorial grows without specifying a particular finite value of � n.

In summary, non-asymptotic analysis is concerned with specific, finite values and provides practical insights, while asymptotic analysis focuses on the behavior of functions as they approach limits, often providing more general and theoretical insights.

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The statement that "non-asymptotics only hide the absolute constant in the bound" is a common way to emphasize the difference between asymptotic analysis and non-asymptotic analysis in mathematics and computer science. Let's break down what this statement means:

1. Asymptotic Analysis: In asymptotic analysis, the focus is on how a function behaves as its input (usually denoted as "n" or "x") approaches infinity. It disregards constant factors and lower-order terms and focuses on the dominant growth rate. Common examples include big O notation, which provides an upper bound on the growth rate of a function.

   For example, in big O notation, if you have a function f(n) = 3n^2 + 7n + 5, the asymptotic analysis would say that f(n) is O(n^2) because n^2 is the dominant term as n approaches infinity. The constant factors like 3 and 7 are ignored in this analysis.

2. Non-asymptotic Analysis: In contrast, non-asymptotic analysis considers the behavior of a function for finite values of its input. It does not focus on what happens as n approaches infinity. Instead, it takes into account all terms, including constants and lower-order terms, to provide a precise bound or estimate for the function's behavior within a given range of input values.

   Using the same example, in non-asymptotic analysis, you would consider all terms of the function f(n) = 3n^2 + 7n + 5, without simplifying or ignoring any part of it. This provides a more accurate analysis of the function's behavior for specific values of n, regardless of how large or small n is.

When someone says that "non-asymptotics only hide the absolute constant in the bound," they are highlighting that non-asymptotic analyses provide a more detailed and accurate picture of a function's behavior but may result in bounds that include constants that aren't relevant in the long run. In other words, when you perform non-asymptotic analysis, you're considering all the details, including the constant factors, which can be important for practical purposes. However, for large inputs (asymptotically), these constants become less significant, and the bound approaches the simpler asymptotic analysis.

Therefore, the statement is a reminder that non-asymptotic analyses provide a "complete" description of a function's behavior but may seem more complex due to the inclusion of constants that eventually become negligible in the asymptotic limit.