Table Of ContentKronecker Product Properties
Kronl If A, B, C, and D are matrices, then (A® B)(C ® D) = (AC)®(BD),
assuming that the ordinary multiplications AC and BD are defined.
Kron2 If A and B are nonsingular matrices, then A® B is nonsingvlar and
T
Kron3 If A and B are matrices, then (A ® B) = A* ® B .
Kron4 If P and Q are permutations, then so is P® Q.
PXr
Kron5 Suppose n = re. If A <E , x g C", and y = (It®A)x, then yrxc — Axr*c-
cXi:
KronS Suppose n = re. If A £ <E , z £ C", and y = (A ® I r ) x , then yT*c — X r x t A .
Kron7 If A is a matrix, then Ip ® (/, ® A) = Ifq ® A.
)i >
KronS Ifn = u\andA<C * ',thennlin(Ir®A) = (AQI^ftZ^.
7
Kron9 Suppose d = abc and x 6 Cr. // y — (Ic ® H j,)z I = X(Q'a — t O'b — 1, 0:c — 1),
and y = Y(0:b — l,0:a — 1, 0:c - 1), then Y(0,a,j) = X(a,{l, 7).
KronlC Suppose d = ate and x 6 C*1. // y — (/c ® na,.k)z, a: S -Y(0:6 - l,0:o - l,0:c - 1),
andy = K(0:a - 1,0:6- l,0:c- I ) , then Y(a,0,f) = X(0,a,-f).
Kronl 1 Suppose d = abc and i 6 Cd. // y = (fl^j,,. ® /a)z, i = X(0:a - 1,0:6 - l,0:c - I),
and v = y(0:o - l,0:c - 1,0:6 - 1), then y(o,7,/?) = X(a,0, 7).
Kronl2 Suppose d = o6c and i e C^. // y = (IU.tc ® /0)i, r = A'(0:a - l,0:c - 1, 0:* - 1),
and K = y(0:a - 1, 0:6 - 1, 0:c - !), tfien y{a,j3, 7) = X ( a . ~ f . j 3 ) .
Computational Frameworks
for the Fast Fourier Transform
F R O N T I E R S IN A P P L I E D
M A T H E M A T I C S
The SIAM series on Frontiers in Applied Mathematics publishes monographs dealing
with creative work in a substantive field involving applied mathematics or scientific
computation. All works focus on emerging or rapidly developing research areas that
report on new techniques to solve mainstream problems in science or engineering.
The goal of the series is to promote, through short, inexpensive, expertly written
monographs, cutting edge research poised to have a substantial impact on the solutions
of problems that advance science and technology.The volumes encompass a broad
spectrum of topics important to the applied mathematical areas of education,
government, and industry.
EDITORIAL BOARD
H.T. Banks, Editor-in-Chief, North Carolina State University
Richard Albanese, U.S. Air Force Research Laboratory, Brooks AFB
Carlos Castillo Chavez, Cornell University
Doina Cioranescu, Universite Pierre et Marie Curie (Paris VI)
Pat Hagan, NumeriX, New York
Matthias Heinkenschloss, Rice University
Belinda King,Virginia Polytechnic Institute and State University
Jeffrey Sachs, Merck Research Laboratories, Merck and Co., Inc.
Ralph Smith, North Carolina State University
AnnaTsao, Institute for Defense Analyses, Center for Computing Sciences
B O O K S P U B L I S H E D IN F R O N T I E R S IN
A P P L I E D MATHEMATICS
Kelley, C.T., Iterative Methods for Optimization
Greenbaum.Anne, Iterative Methods for Solving Linear Systems
Kelley, C.T., Iterative Methods for Linear and Nonlinear Equations
Bank, Randolph E., PLTMG:A Software Package for Solving Elliptic Partial Differential Equations.
Users'Guide 7.0
More, Jorge J. and Wright, Stephen J., Optimization Software Guide
Rude, Ulrich, Mathematical and Computational Techniques for Multilevel Adaptive Methods
Cook, L. Pamela, Transonic Aerodynamics: Problems in Asymptotic Theory
Banks, H.T., Control and Estimation in Distributed Parameter Systems
Van Loan, Charles, Computational Frameworks for the Fast Fourier Transform
Van Huffel, Sabine andVandewalle.Joos, The Total Least Squares Problem: Computational
Aspects and Analysis
Castillo, Jose E., Mathematical Aspects of Numerical Grid Generation
Bank, R. E., PLTMG: A Software Package for Solving Elliptic Partial Differential Equations.
Users' Guide 6.0
McCormick, Stephen F, Multilevel Adaptive Methods for Partial Differential Equations
Grossman, Robert, Symbolic Computation:Applications to Scientific Computing
Coleman.Thomas F. and Van Loan, Charles, Handbook for Matrix Computations
McCormick, Stephen F, Multigrid Methods
Buckmaster.John D., The Mothemot/cs of Combustion
Ewing, Richard E., The Mathematics of Reservoir Simulation
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Computational Frameworks
for the Fast Fourier Transform
Charles Van Loan
Cornell University
Ithaca, New York
Society for Industrial and Applied Mathematics
Philadelphia
Copyright © 1992 by the Society for Industrial and Applied Mathematics
1 0 9 8 7 6 5 4 3
All rights reserved. Printed in the United States of America. No part of this book may be
reproduced, stored, or transmitted in any manner without the written permission of the
Publisher. For information, write the Society for Industrial and Applied Mathematics, 3600
University City Science Center, Philadelphia, PA 19104-2688.
No warranties, express or implied, are made by the publisher, authors, and their employers that
the programs contained in this volume are free of error. They should not be relied on as the
sole basis to solve a problem whose incorrect solutions could result in injury to person or
property. If the programs are employed in such a manner, it is at the user's own risk and the
publisher, authors, and their employers disclaim all liability for such misuse.
Trademarked names may be used in this book without the inclusion of a trademark symbol.
These names are used in an editorial context only; no infringement of trademark is intended.
Library of Congress Cataloging-in-Publication Data
Van Loan, Charles.
Computational Frameworks for the fast Fourier transform / Charles
Van Loan.
p. cm. — (Frontiers in applied mathematics : 10)
Includes bibliographical references and index.
ISBN 0-89871-285-8
I. Fourier transformations. I.Title. II. Series.
QA403.5.V35 1992
5l5'.723-dc20 92-4450
is a registered trademark.
Dedicated to
Marian,Ted, and Elizabeth
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