Journal ArticleDOI
Note on a Lower Bound on the Linear Complexity of the Fast Fourier Transform
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A lower bound for the number of additions necessary to compute a family of linear functions by a linear algorithm is given when an upper bound c can be assigned to the modulus of the complex numbers involved in the computation.Abstract:
A lower bound for the number of additions necessary to compute a family of linear functions by a linear algorithm is given when an upper bound c can be assigned to the modulus of the complex numbers involved in the computation. In the case of the fast Fourier transform, the lower bound is (n/2) log2n when c = 1.read more
Citations
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Book
The Design and Analysis of Computer Algorithms
Alfred V. Aho,John E. Hopcroft +1 more
TL;DR: This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.
Journal ArticleDOI
Computing Fourier Transforms and Convolutions on the 2-Sphere
James R. Driscoll,D. M. Healy +1 more
TL;DR: Convolution theorems generalizing well known and useful results from the abelian case are used to develop a sampling theorem on the sphere, which reduces the calculation of Fourier transforms and convolutions of band-limited functions to discrete computations.
Algebraic Complexity Theory.
TL;DR: Algebraic complexity theory as mentioned in this paper is a project of lower bounds and optimality, which unifies two quite different traditions: mathematical logic and the theory of recursive functions, and numerical algebra.
Book
Arithmetic Circuits: A Survey of Recent Results and Open Questions
Amir Shpilka,Amir Yehudayoff +1 more
TL;DR: The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what it finds to be the most interesting and accessible research directions, with an emphasis on works from the last two decades.
Journal ArticleDOI
A Modified Split-Radix FFT With Fewer Arithmetic Operations
Steven G. Johnson,Matteo Frigo +1 more
TL;DR: A simple recursive modification of the split-radix algorithm is presented that computes the DFT with asymptotically about 6% fewer operations than Yavne, matching the count achieved by Van Buskirk's program-generation framework.
References
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Journal ArticleDOI
An algorithm for the machine calculation of complex Fourier series
J.W. Cooley,John W. Tukey +1 more
TL;DR: Good generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series, applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices.
Book ChapterDOI
On linear algorithms
TL;DR: In this article, the authors focus on linear algorithms and explain how to bound the number of additions necessary to compute by a linear algorithm a given set of linear functions of several variables.