The Fast Fourier Transform on Hypercube Parallel Computers

The Fast Fourier Transform on Hypercube Parallel Computers
Author: Cornell University. Dept. of Computer Science
Publisher:
Total Pages: 217
Release: 1987
Genre: Fourier series
ISBN:

The Fast Fourier Transform appears frequently in scientific computing. Therefore it is desirable to implement it efficiently on parallel computers. In this thesis, we investigate several different aspects of parallel Fast Fourier Transform implementation techniques for distributed-memory message-passing systems such as hypercube multiprocessors. We describe various Fast Fourier Transform algorithms using a matrix notation. An error analysis is presented that considers the effect of different methods used in the computation of the Fourier Transform coefficients as well as accumulated roundoff. New implementations of one and two-dimensional Fast Fourier Transforms are presented along with comparisons with existing methods. New algorithms for symmetric transforms are also developed and the results show excellent speedup when implemented on the Intel iPSC hypercube.

Inside the FFT Black Box

Inside the FFT Black Box
Author: Eleanor Chu
Publisher: CRC Press
Total Pages: 346
Release: 1999-11-11
Genre: Mathematics
ISBN: 9781420049961

Are some areas of fast Fourier transforms still unclear to you? Do the notation and vocabulary seem inconsistent? Does your knowledge of their algorithmic aspects feel incomplete? The fast Fourier transform represents one of the most important advancements in scientific and engineering computing. Until now, however, treatments have been either brief, cryptic, intimidating, or not published in the open literature. Inside the FFT Black Box brings the numerous and varied ideas together in a common notational framework, clarifying vague FFT concepts. Examples and diagrams explain algorithms completely, with consistent notation. This approach connects the algorithms explicitly to the underlying mathematics. Reviews and explanations of FFT ideas taken from engineering, mathematics, and computer science journals teach the computational techniques relevant to FFT. Two appendices familiarize readers with the design and analysis of computer algorithms, as well. This volume employs a unified and systematic approach to FFT. It closes the gap between brief textbook introductions and intimidating treatments in the FFT literature. Inside the FFT Black Box provides an up-to-date, self-contained guide for learning the FFT and the multitude of ideas and computing techniques it employs.

Computational Frameworks for the Fast Fourier Transform

Computational Frameworks for the Fast Fourier Transform
Author: Charles Van Loan
Publisher: SIAM
Total Pages: 286
Release: 1992-01-01
Genre: Mathematics
ISBN: 9781611970999

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.

Ordered Fast Fourier Transforms on a Massively Parallel Hypercube Multiprocessor

Ordered Fast Fourier Transforms on a Massively Parallel Hypercube Multiprocessor
Author: National Aeronautics and Space Administration (NASA)
Publisher: Createspace Independent Publishing Platform
Total Pages: 34
Release: 2018-07-09
Genre:
ISBN: 9781722467104

Design alternatives for ordered Fast Fourier Transformation (FFT) algorithms were examined on massively parallel hypercube multiprocessors such as the Connection Machine. Particular emphasis is placed on reducing communication which is known to dominate the overall computing time. To this end, the order and computational phases of the FFT were combined, and the sequence to processor maps that reduce communication were used. The class of ordered transforms is expanded to include any FFT in which the order of the transform is the same as that of the input sequence. Two such orderings are examined, namely, standard-order and A-order which can be implemented with equal ease on the Connection Machine where orderings are determined by geometries and priorities. If the sequence has N = 2 exp r elements and the hypercube has P = 2 exp d processors, then a standard-order FFT can be implemented with d + r/2 + 1 parallel transmissions. An A-order sequence can be transformed with 2d - r/2 parallel transmissions which is r - d + 1 fewer than the standard order. A parallel method for computing the trigonometric coefficients is presented that does not use trigonometric functions or interprocessor communication. A performance of 0.9 GFLOPS was obtained for an A-order transform on the Connection Machine. Tong, Charles and Swarztrauber, Paul N. Unspecified Center NCC2-387...

Fast Fourier Transform Algorithms for Parallel Computers

Fast Fourier Transform Algorithms for Parallel Computers
Author: Daisuke Takahashi
Publisher: Springer Nature
Total Pages: 120
Release: 2019-10-05
Genre: Computers
ISBN: 9811399654

Following an introduction to the basis of the fast Fourier transform (FFT), this book focuses on the implementation details on FFT for parallel computers. FFT is an efficient implementation of the discrete Fourier transform (DFT), and is widely used for many applications in engineering, science, and mathematics. Presenting many algorithms in pseudo-code and a complexity analysis, this book offers a valuable reference guide for graduate students, engineers, and scientists in the field who wish to apply FFT to large-scale problems. Parallel computation is becoming indispensable in solving the large-scale problems increasingly arising in a wide range of applications. The performance of parallel supercomputers is steadily improving, and it is expected that a massively parallel system with hundreds of thousands of compute nodes equipped with multi-core processors and accelerators will be available in the near future. Accordingly, the book also provides up-to-date computational techniques relevant to the FFT in state-of-the-art parallel computers. Following the introductory chapter, Chapter 2 introduces readers to the DFT and the basic idea of the FFT. Chapter 3 explains mixed-radix FFT algorithms, while Chapter 4 describes split-radix FFT algorithms. Chapter 5 explains multi-dimensional FFT algorithms, Chapter 6 presents high-performance FFT algorithms, and Chapter 7 addresses parallel FFT algorithms for shared-memory parallel computers. In closing, Chapter 8 describes parallel FFT algorithms for distributed-memory parallel computers.

Parallel Computers 2

Parallel Computers 2
Author: R.W Hockney
Publisher: CRC Press
Total Pages: 642
Release: 2019-08-16
Genre: Mathematics
ISBN: 1000716112

Since the publication of the first edition, parallel computing technology has gained considerable momentum. A large proportion of this has come from the improvement in VLSI techniques, offering one to two orders of magnitude more devices than previously possible. A second contributing factor in the fast development of the subject is commercialization. The supercomputer is no longer restricted to a few well-established research institutions and large companies. A new computer breed combining the architectural advantages of the supercomputer with the advance of VLSI technology is now available at very attractive prices. A pioneering device in this development is the transputer, a VLSI processor specifically designed to operate in large concurrent systems. Parallel Computers 2: Architecture, Programming and Algorithms reflects the shift in emphasis of parallel computing and tracks the development of supercomputers in the years since the first edition was published. It looks at large-scale parallelism as found in transputer ensembles. This extensively rewritten second edition includes major new sections on the transputer and the OCCAM language. The book contains specific information on the various types of machines available, details of computer architecture and technologies, and descriptions of programming languages and algorithms. Aimed at an advanced undergraduate and postgraduate level, this handbook is also useful for research workers, machine designers, and programmers concerned with parallel computers. In addition, it will serve as a guide for potential parallel computer users, especially in disciplines where large amounts of computer time are regularly used.

Mathematics of Multidimensional Fourier Transform Algorithms

Mathematics of Multidimensional Fourier Transform Algorithms
Author: Richard Tolimieri
Publisher: Springer Science & Business Media
Total Pages: 241
Release: 2012-12-06
Genre: Technology & Engineering
ISBN: 1468402056

The main emphasis of this book is the development of algorithms for processing multi-dimensional digital signals, and particularly algorithms for multi-dimensional Fourier transforms, in a form that is convenient for writing highly efficient code on a variety of vector and parallel computers.

Massively Parallel Hypercube FFTs

Massively Parallel Hypercube FFTs
Author:
Publisher:
Total Pages: 20
Release: 1991
Genre:
ISBN:

On parallel computers, the way the data elements are mapped to the processors may have a large effect on the timing performance of a given algorithm. In our previous paper, we have examined a few mapping strategies for the ordered radix-2 DIF (decimation-in-frequency) Fast Fourier Transform. In particular, we have shown how reduction of communication can be achieved by combining the order and computational phases through the use of i-cycles. A parallel methods was also presented for computing the trigonometric factors which requires neither trigonometric function evaluation nor interprocessor communication. This paper first reviews some of the experimental results on the Connection Machine to demonstrate the importance of reducing communication in a parallel algorithm. The emphasis of this paper, however, is on analyzing the numerical stability of the proposed method for generating the trigonometric factors and showing how the error can be improved. 16 refs., 12 tabs.