About the Project
This project involves modeling the runtime of an algorithm based on input size using power series. The runtime can be expressed as a Taylor series.
Key Steps
- Determine the Runtime Function: Assume the runtime has the form ( T(n) = a_0 + a_1 n + a_2 n^2 + a_3 n^3 + \ldots ).
- Determine Coefficients via Derivatives: Coefficients are obtained from the derivatives of ( T(n) ) evaluated at ( n = 0 ).
- First Term: ( T(0) = 2 ) ms.
- Second Term: ( T’(0) = 3 ).
- Third Term: ( T”(0) = 2 ).
- Fourth Term: ( T'''(0) = 1.5 ).
Power Series Expansion
Substitute the coefficients into the Taylor series: [ T(n) = 2 + 3n + \frac{1}{2} n^2 + \frac{1}{6} n^3 ]
Approximate Runtime for ( n = 5 )
[ T(5) = 2 + 3(5) + \frac{1}{2}(5^2) + \frac{1}{6}(5^3) ] [ T(5) \approx 39.92 \text{ ms} ]
Conclusion
Using power series, we can approximate the runtime of an algorithm based on input size. This method provides an effective approximation for analyzing and optimizing algorithms in computer engineering.
Contributions
Contributions are welcome. If you have any ideas to improve the application or find any bugs, please open an issue or send a pull request in the GitHub repository.
Thank you for exploring this project! If you have any questions or suggestions, feel free to open an issue or contact directly.