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

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CSE 421 Algorithms Richard Anderson Lecture 15 Fast Fourier Transform
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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
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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)
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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
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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
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Polynomial Multiplication n-1 degree polynomials A(x) = a
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Lecture15 - CSE 421 Algorithms Richard Anderson Lecture 15...

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