Discrete Fourier Transform
7 hours ago
- #Algorithm Optimization
- #FFT
- #Polynomial Multiplication
- Polynomial multiplication using coefficient representation requires O(N^2) operations.
- Value representation allows polynomial multiplication in O(N) operations by sampling points and interpolating.
- Fast Fourier Transform (FFT) enables efficient conversion between coefficient and value representations in O(N log N) time.
- FFT leverages roots of unity and recursive decomposition to reduce complexity.
- Interpolation, the inverse of evaluation, can be performed using the inverse DFT matrix.
- The implementation of FFT and its inverse (IFFT) can be concise, with IFFT requiring division by n.
- Testing the implementation with sample polynomials confirms the correctness of the approach.