Algorithms by Jeff Erickson PDF

By Jeff Erickson

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N/2 − 1]) ) + i sin( 2π ) ωn ← cos( 2π n n ω←1 for j ← 0 to n/2 − 1 P ∗ [ j] ← U ∗ [ j] + ω · V ∗ [ j] ∗ P [ j + n/2] ← U ∗ [ j] − ω · V ∗ [ j] ω ← ω · ωn return P ∗ [0 .. 6 Inverting the FFT We also need to recover the coefficients of the product from the new sample values. Recall that the transformation from coefficients to sample values is linear; the sample vector is the product of a Vandermonde matrix V and the coefficient vector. For the discrete Fourier transform, each entry in V is an nth root of unity; specifically, v jk = ωnjk for all integers j and k.

Jk − jk In other words, if W = V −1 then w jk = v jk /n = ωn /n = ωn /n. What this means for us computer scientists is that any algorithm for computing the discrete Fourier transform can be easily modified to compute the inverse transform as well. INVERSEFFT(P ∗ [0 .. n − 1]): if n = 1 return P for j ← 0 to n/2 − 1 U ∗ [ j] ← P ∗ [2 j] V ∗ [ j] ← P ∗ [2 j + 1] U ← INVERSEFFT(U[0 .. n/2 − 1]) V ← INVERSEFFT(V [0 .. n/2 − 1]) ωn ← cos( 2π ) − i sin( 2π ) n n ω←1 for j ← 0 to n/2 − 1 P[ j] ← 2(U[ j] + ω · V [ j]) P[ j + n/2] ← 2(U[ j] − ω · V [ j]) ω ← ω · ωn return P[0 ..

585 ) time. (f) What are the running times of FACTORIAL(n) and FACTORIAL2(n, n) if we use the modified Karatsuba multiplication from part (e)? c Copyright 2011 Jeff Erickson. 0/). Free distribution is strongly encouraged; commercial distribution is expressly forbidden. edu/~jeffe/teaching/algorithms/ for the most recent revision. 18 Lecture 2: Fast Fourier Transforms [Fa’10] Algorithms Calvin: Here’s another math problem I can’t figure out. What’s 9+4? Hobbes: Ooh, that’s a tricky one. You have to use calculus and imaginary numbers for this.

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Algorithms by Jeff Erickson

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