## Computational Frameworks for the Fast Fourier TransformThe 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. |

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### Contents

1 | 17 |

5 | 36 |

8 | 60 |

JL General Radix Frameworks | 76 |

O HighPerformance Frameworks | 121 |

Selected Topics | 188 |

259 | |

### Common terms and phrases

Acoust applied approach arithmetic assume autosort bit reversal bit-reversal permutation block matrix C.S. Burrus cache column complete conjugate-even convolution Cooley-Tukey framework Cosine Transform defined derivation develop DFT matrix DFT problem diagonal Discrete Cosine Transform Discrete Fourier Transform Discrete Sine Transform discussed DST-II end end end equation example external memory Fast Fourier Transform FFT Algorithm FFT framework flops are required following algorithm computes following algorithm overwrites Gentleman-Sande IEEE IEEE Trans implementation in-place in-place algorithm index-reversal integer inverse involves Kronecker product Lemma length loop m.id main memory Mixed Radix mixed-radix multiple DFT multirow DFT notation Notes and References obtain the following Parallel Comput Pease perfect shuffle Pn(p prime factor Proc Proc(/i Proc(l procedure processor Proof Radix radix-2 radix-p splitting recursive References for Section roundoff shared-memory Speech Signal Process split-radix Suppose Temperton Theorem transposed Stockham two-dimensional unit stride update workspace x(kL