1. DFT and FFT
By using the complex roots of unity, we can evaluate and interpolate a polynomial in O(n lg n)
An example, here are the solutions to 8 = 1
The principal root is
We often write these values in trigonometric form

2. Properties of Roots of Unity
The halving lemma is essential since it guarantees the recursive solutions of subproblems are only half as large

3. Discrete Fourier Transform
We wish to evaluate at the points
We can assume n to be a power of 2 by adding high-order zero coefficients as necessary
Assuming a = (a0, a1, …, an-1) we define
This calculation is O(n2), but by using the complex roots of unity, it can be done in O(n lg n)

4. Fast Fourier Transform
We separate the even and odd index coefficients
Our desired result is
So to evaluate A(x) at the complex roots of unity

5. FFT - continued
Due to the halving lemma, there are not n distinct values, rather only n/2 complex (n/2)th roots
Since the solutions of the subproblems are of the same form as the original problem, these recursive solutions are also only half in size
So we can solve an n-element DFTn computation by solving two n/2-element DFTn/2 computations
In the following recursive FFT algorithm, we compute the DFT for the vector where n is a power of 2

6. FFT Algorithm - 1 Lines 2-3 handle the base case
Lines 4,5,13 update  so that at lines we have
Lines 6-7 define the coefficient vectors for the subpolynomials

7. FFT Algorithm - 2 Lines 8-9 solve the subproblems recursively
The algorithm complexity is

8. Interpolation The DFT can be written as the matrix product y = Vna
The DFTn-1 can be computed in O(n lg n) time too