Question

Show how to apply the fast Fourier Transform to multiply three polynomials A(x) = a0 +a1x+a2x 2 +· · ·+an−1x n−1 of degree n−1, B(x) = b0 +b1x+b2x 2 +· · ·+bn−1x n−1 of degree n − 1, and C(x) = c0 + c1x + c2x 2 + · · · + cn−1x n−1 of degree n − 1 when n is a power of 2. Analyze its time complexity.

Answer 1

First multiply two polynomials using FFT, then multiply the result by the third polynomial.

The Time complexity will be nlog(n).

Use the following algorithm:

Algorithm
1. Add n higher-order zero coefficients to A(x) and B(x)
2. Evaluate A(x) and B(x) using FFT for 2n points
3. Pointwise multiplication of point-value forms
4. Interpolate C(x) using FFT to compute inverse DFT