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A Mathematical Framework for Parallel Computing of Discrete
A Mathematical Framework for Parallel Computing of Discrete

... Abstract: This paper presents a mathematical framework for a family of discrete-time discrete-frequency transforms in terms of matrix signal algebra. The matrix signal algebra is a mathematics environment composed of a signal space, a finite dimensional linear operators and special matrices where al ...
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Bessel Functions and Their Application to the Eigenvalues of the
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... Figure 5: This table compares the size of n and x against the amount of time it takes the downwards recurrence algorithm written for this project to compute Jn (x). Note that order of n and x is given in number of digits, so order 1 and 2 in this case refers to numbers ranging from 1 to 99, and 5 re ...
Robust Ray Intersection with Interval Arithmetic
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Chapter 12: Copying with the Limitations of Algorithm Power
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1

Cooley–Tukey FFT algorithm



The Cooley–Tukey algorithm, named after J.W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size N = N1N2 in terms of smaller DFTs of sizes N1 and N2, recursively, to reduce the computation time to O(N log N) for highly composite N (smooth numbers). Because of the algorithm's importance, specific variants and implementation styles have become known by their own names, as described below.Because the Cooley-Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT. For example, Rader's or Bluestein's algorithm can be used to handle large prime factors that cannot be decomposed by Cooley–Tukey, or the prime-factor algorithm can be exploited for greater efficiency in separating out relatively prime factors.The algorithm, along with its recursive application, was invented by Carl Friedrich Gauss. Cooley and Tukey independently rediscovered and popularized it 160 years later.
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