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Computational Frameworks for the Fast Fourier Transform

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Charles Van Loan


1992 / xiii + 273 pages / Softcover / ISBN: 978-0-898712-85-8 / List Price $66.50 / SIAM Member Price $46.55 / Order Code FR10

"This finely crafted work fills a gap in the library of books on the fast Fourier Transform (FFT). It provides a complete and elegant mathematical formulation of the family of algorithms that compute FFTs. It is written for students and professionals who already have a working knowledge of computational linear algebra. It leads the student directly into the current major fields of FFT research. . . . This work is an excellent contribution to the modern FFT literature." – J.R. Hubbard, Computing Reviews, September 1993.

"This comprehensive book contains many examples, problems, notes and hints to the literature. It will be very useful for students as well as researchers who are interested in the application of the FFT." – M. Tasche (Rostock), Zentralblatt fur Mathematik, pre757, page 37.

"The fast Fourier transform (FFT) is one of the truly great computational developments. As the author emphasizes in the preface, the central feature of the FFT is essentially the factorization of the discrete Fourier transform (DFT) matrix. This point of view will not only unify the FFT and make it more understandable to outsiders, but also give key aspects for advanced scientific computing, e.g. vectorization or parallelization. . . This is only a brief glimpse of an excellent survey on the FFT, which will be valuable to all who wish to use it." – S. Hitotumatu, Mathematical Reviews, Issue 93a.

"...the FFT has found application in almost every field of modern science, including fields as diverse as astronomy, acoustics, image processing, fluid dynamics, petroleum exploration, medicine and quantum physics, among many others. It is not an exercise in hyperbole to say that the world as we know it would be different without the FFT." – David H. Bailey, SIAM Review, March 1993.

"It is generally accepted that `Life as we know it would be very different without the FFT.' The author adds to this `Life as we know it would be considerably different if, from the 1965 Cooley-Tukey paper onwards, the FFT community had made systematic and heavy use of the matrix-vector notation.' This book contains a very readable and up-to-date presentation of FFT techniques, their theory and application. Together with many explicit computational algorithms, the extensive annotated list of references add greatly to the scientific value of this
reference text." – Short Book Reviews, August 1992, Vol. 12, No. 2 

The most comprehensive treatment of FFTs to date. Van Loan captures the interplay between mathematics and the design of effective numerical algorithms--a critical connection as more advanced machines become available. A stylized Matlab notation, which is familiar to those engaged in high-performance computing, is used.

The Fast Fourier Transform (FFT) family of algorithms has revolutionized many areas of scientific computation. The FFT is one of the most widely used algorithms in science and engineering, with applications in almost every discipline. This volume is essential for professionals interested in linear algebra as well as those working with numerical methods. The FFT is also a great vehicle for teaching key aspects of scientific computing.

Chapter 1: The Radix-2 Frameworks. Matrix Notation and Algorithms; The FFT Idea; The Cooley-Tukey Factorization; Weight and Butterfly Computations; Bit Reversal and Transposition; The Cooley-Tukey Framework; The Stockham Autosort Frameworks; The Pease Framework; Decimation in Frequency and Inverse FFTs; Chapter 2: General Radix Frameworks. The General Radix Ideas; Index Reversal and Transposition; Mixed-Radix Factorizations; Radix-4 and Radix-8 Frameworks; The Split-Radix Frameworks; Chapter 3: High Performance Frameworks. The Multiple DFT Problem; Matrix Transposition; The Large Single-Vector FFT Problem; Multi-Dimensional FFT Problem; Distributed Memory FFTs; Shared Memory FFTs; Chapter 4: Selected Topics. Prime Factor FFTs; Convolution; FFTs of Real Data; Cosine and Sine Transforms; Fast Poisson Solvers; Bibliography; Index.

ISBN: 9780898712858

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