Get Computational Frameworks for the Fast Fourier Transform PDF

By Charles Van Loan

ISBN-10: 0898712858

ISBN-13: 9780898712858

The main finished therapy of FFTs so far. Van personal loan captures the interaction among arithmetic and the layout of powerful numerical algorithms--a severe connection as extra complex machines develop into on hand. A stylized Matlab notation, that's regularly occurring to these engaged in high-performance computing, is used. the short Fourier remodel (FFT) kinfolk of algorithms has revolutionized many components of medical computation. The FFT is without doubt one of the most generally used algorithms in technology and engineering, with functions in virtually each self-discipline. This quantity is key for pros drawn to linear algebra in addition to these operating with numerical equipment. The FFT can also be an outstanding motor vehicle for educating key features of clinical computing.

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In many advanced computer architectures, large power-of-two strides can severely degrade performance. Machines with interleaved memories serve as a nice case study. " For example, the 4-bank storage of a length-25 vector would be arranged as shown in Fig. 2. The figure shows that component Xj is stored in bankQ mod 4). In an interleaved memory system, the retrieval of a vector would proceed as follows. When component Xj is retrieved, bank(j mod 4) is "tied up" for four machine cycles because the path between the bank and the CPU is active and cannot be disturbed.

Stable On-Line Creation of Sines and Cosines of Successive Angles," Proc. IEEE 75, 1434-1435. C. Chu (1988). D. Thesis, Center for Applied Mathematics, Cornell University, Ithaca, NY. J. Oliver (1975). "Stable Methods for Evaluating the Points cos(tV/n)," J. Inst. Maths. Applic. 16, 247-257. Once the properties of the computed weights are known, then it is possible to address the accuracy of the FFT itself. The following papers take a statistical approach to the problem: 36 CHAPTER 1. THE RADix-2 FRAMEWORKS R.

1 Summary of roundoff error behavior behavio (ci = cos(27r/L)). 6) 3 Roundoff Roundoff in uu>{ L 0(u) 0(nj) O( ulog j) oMH + v/hp + iX) OMidi + vicip + i)10") O(u logj) large j, we can expect trouble with repeated scaling and forward recursion. For this reason we regard these methods as unstable. 6. Assume that L — 1q — 8m and that we have computed w^(Q\m). , Wjk = cos(kO) and Vk = —s'm(kd) where 0 = lirfL. (0:(L/8)), the rest of w^ can be determined without additional floating point arithmetic.

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Computational Frameworks for the Fast Fourier Transform (Frontiers in Applied Mathematics) by Charles Van Loan

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