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Unformatted text preview: 1 CSE 421 Algorithms Richard Anderson Lecture 15 Fast Fourier Transform 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 i e i = cos + i sin 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, + i , - i , - + i , - - i where = sqrt(2) e 2 ki/n e 2 i = 1 e i = -1 n th roots of unity: e 2 ki/n for k = 0 n-1 Notation: k,n = e 2 ki/n Interesting fact: 1 + k,n + 2 k,n + 3 k,n + . . . + 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 Polynomial Multiplication n-1 degree polynomials...
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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.
- Fall '06