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Polynomials and Fast Fourier Transform Chapter 30, pp.823-848 new editionPowerPoint Presentation

Polynomials and Fast Fourier Transform Chapter 30, pp.823-848 new edition

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### Polynomials and Fast Fourier TransformChapter 30, pp.823-848 new edition

CS8550

Polynomials

- Polynomial in coefficient representation
A(x) = a0 + a1x+a2 x2+…+ an-2 xn-2+ an-1 xn-1

- Operations over polynomials:
- polynomial degree n = highest nonzero coeff
- addition = O(n)
- multiplication = O(n2)!!! Bad -- too slow!
- evaluation (finding the value in a point) = O(n) !!! Good --
Horner’s rule = stack-based

A(x0 ) = a0 + x0 (a1 + x0 (a2 + …+ x0 (an-2 + x0 (an-1))…)

- Point-value representation
- (x0 ,y0 ), (x1 ,y1 ), …, (xn-1 ,yn-1 ) - n point-values are sufficient

CS8550

Point-value representation

- Interpolation = getting polynomial coefficients from point-value representation
- Theorem (unique interpolation):
for any set of distinct n point-value pairs,

! polynomial of degree less than n

- Operations over polynomials:
- addition = O(n)
- multiplication = O(n)!!! Good --
- how?

- evaluation (finding the value in a point) = O(n2)!!! Bad -- too slow!

- IDEA: Use both representations!!!

CS8550

Simple Transformations

- Coefficient => point-value (evaluation)
- just O(n) per point
- O(n2)!!! Bad -- too slow!

- Point-value => coefficient (interpolation)
- Lagrange’s formula
n-1

yk (x-xj) / (xk-xj)

k=0 jk jk

- O(n2)!!! Bad -- too slow!

- Lagrange’s formula
- GOAL
- both transformations in O(n log n)

CS8550

O(n log n) Multiplication

- Double-degree bound
- 2n point-value pairs
- O(n)

- Evaluate
- Compute point point-value representations using FFT
- in (2n)-roots of unity
- O(n log n)

- Point-wise multiply
- Multiply the values for each of 2n points
- O(n)

- Interpolate
- Compute coefficient representation of product using FFT
- O(n log n)

CS8550

Complex Roots of Unity

- Point-value representation in complex roots of 1
= DFT = discrete Fourier Transform

- Complex n-th root of 1: wn = 1
- Complex numbers: i = -1
- The principal n-th root is
wn1= e2 i /n = cos(2/n)+ i sin (2/n)

- n roots of n-th power :
wn0 = 1, wn1 = principal , wn2 = wnwn, …, wnn-1

- Properties:
- Cancellation: wdndk= wnk
- Halving: if n is even then the squares of nth roots are n/2-roots
- Summation: n-1
(wnk ) j= 0

j=0

CS8550

Fast Fourier Transform

- Problem:
- Given polynomial
A(x) = a0 + a1x+a2 x2+…+ an-2 xn-2+ an-1 xn-1

- Find values in roots of unity

- Given polynomial
- FFT = divide and conquer:
A[0](x)= a0 + a1x+a2 x2+…+ an-2 xn/2-1

A[1](x) = a1 + a3x+a5 x2+…+ an-1 xn/2-1

- A(x) = A[0](x2)+ xA[1](x2)

- Recursive procedure:
- Evaluate A[0](x) and A[1](x) in points (wn0 )2 , (wn1 )2 ,…, (wnn-1 )2
- n/2 roots each

- Combine the results

- Evaluate A[0](x) and A[1](x) in points (wn0 )2 , (wn1 )2 ,…, (wnn-1 )2
- T(n) = 2T(n/2) +(n) = (n log n) (master theorem)

CS8550

Inverse FFT Interpolation

y = (y0 y1 y2...yn-1), a = (a0 a1 a2...an-1), x = (x0 x1 x2…xn-1)

Vn = Vn (x) Vandermonde matrix

y = a Vn (x)

Replace x = (x0 x1 x2…xn-1) => wn = (wn0w n1 w n2 …wnn-1)

DFT = y = a Vn (wn)

a = y Vn-1 (wn)

CS8550

Efficient FFT

CS8550

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