1. Introduction to Algorithms
Polynomials and the FFT
My T. UF

2. Polynomials A polynomial in the variable x: Polynomial addition
Polynomial multiplication
where
My T. Thai

3. Representing polynomials
A coefficient representation of a polynomial d of degree bound n is a vector of coefficients
Horner’s rule to compute A(x0)
Time: O(n)
Given a = , b =
Sum: c = a + b, takes O(n) time
Product: c = (convolution of a and b), takes O(n2) time
My T. Thai

4. Representing polynomials
A point-value representation of a polynomial A(x) of degree-bound n is a set of n point-value pairs d
All of the xk are distinct
Proof:
Da = y
|D| =
=> a = D-1y
Vandermonde
matrix
My T. Thai

5. Operations in point-value representation
B :
Addition: C:
Multiplication:
Extend A, B to 2n points:
Product:
My T. Thai

6. Fast multiplication of polynomials in coefficient form
Evaluation: coefficient representation  point-value representation
Interpolation: point-value representation coefficient form of a polynomial
My T. Thai

7. Compute evaluation and interpolation
Evaluation: using Horner method takes O(n2) => not good
Interpolation: computing inversion of Vandermonde matrix takes O(n3) time => not good
How to complete evaluation and interpolation in O(n log n) time?
Choose evaluation points: complex roots of unity
Use Discrete Fourier Transform for evaluation
Use inverse Discrete Fourier Transform for interpolation
My T. Thai

8. Complex roots of unity
A complex nth root of unity is a complex number such that
There are exactly n complex nth roots of unity
Principal nth root of unity
All other complex nth roots of
unity are powers of
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9. Properties of unity’s Complex roots
Proof:
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10. Halving lemma Proof: Cancellation lemma: My T. Thai

11. Summation lemma Proof: Note: k is not divisible by n only when k
is divisible by n
My T. Thai

12. Discrete Fourier Transform
Given:
The values of A at evaluation points
Vector is discrete Fourier transform (DFT) of the coefficient vector d Denote
My T. Thai

13. Fast Fourier Transform
Divide A into two polynomials based on the even-indexed and odd-indexed coefficients:
Combine:
Evaluation points of A[0] and A[1]
are actually the n/2th roots of unity
My T. Thai

14. Fast Fourier Transform
Time:
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15. Inversion of Vandermonde matrix
Proof:
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16. Interpolation at the complex roots of unity
Compute by modifying FFT algorithm
Switch the roles of a and y
Replace
Divide each element by n
My T. Thai

17. FFT and Polynomial mutiplication
My T. Thai

18. Efficient FFT implementations
Line 10 – 12: change the loop to compute only once storing it in t ( Butterfly operation)
Butterfly operation
My T. Thai

19. Structure of RECURSIVE-FFT
Each RECURSIVE-FFT invocation makes two recursive calls
Arrange the elements of a into the order in which they appear in the leaves ( take log n for each element)
Compute bottom up
My T. Thai

20. An iterative FFT implementation
Time:
My T. Thai

21. A parallel FFT circuit
My T. Thai