# Lecture15 - CSE 421 Algorithms Richard Anderson Lecture 15...

This preview shows pages 1–7. Sign up to view the full content.

CSE 421 Algorithms Richard Anderson Lecture 15 Fast Fourier Transform

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
FFT, Convolution and Polynomial Multiplication FFT: O(n log n) algorithm Evaluate a polynomial of degree n at n points in O(n log n) time Polynomial Multiplication: O(n log n) time
Complex Analysis Polar coordinates: re θ i e θ i = cos θ + i sin θ a is an n th root of unity if a n = 1 Square roots of unity: +1, -1 Fourth roots of unity: +1, -1, i, -i Eighth roots of unity: +1, -1, i, -i, β + i β , β - i β , - β + i β , - β - i β where β = sqrt(2)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
e 2 π ki/n e 2 π i = 1 e π i = -1 n th roots of unity: e 2 π ki/n for k = 0 …n-1 Notation: ϖ k,n = e 2 π ki/n Interesting fact: 1 + ϖ k,n + ϖ 2 k,n + ϖ 3 k,n + . . . + ϖ n-1 k,n = 0 for k != 0
FFT Overview Polynomial interpolation – Given n+1 points (x i ,y i ), there is a unique polynomial P of degree at most n which satisfies P(x i ) = y i

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Polynomial Multiplication n-1 degree polynomials A(x) = a
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 02/25/2012 for the course CSE 421 taught by Professor Richardanderson during the Fall '06 term at University of Washington.

### Page1 / 19

Lecture15 - CSE 421 Algorithms Richard Anderson Lecture 15...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online