Parallel Algorithms for Linear Models Numerical Methods and Estimati
Parallel Algorithms for Linear Models provides a complete and detailed account of the design, analysis and implementation of parallel algorithms for solving large-scale linear models. It investigates and presents efficient, numerically stable algorithms f
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Advances in Computational Economics VOLUME 15
SERIES EDITORS Hans Amman, University ofAmsterdam, Amsterdam, The Netherlands Anna Nagurney, University of Massachusetts at Amherst, USA
EDITORIAL BOARD Anantha K. Duraiappah, European University Institute John Geweke, University of Minnesota Manfred Gilli, University of Geneva Kenneth L. Judd, Stanford University David Kendrick, University of Texas at Austin Daniel McFadden, University of California at Berkeley Ellen McGrattan, Duke University Reinhard Neck, University of Klagenfurt Adrian R. Pagan, Australian National University John Rust, University of Wisconsin Berc Rustem, University of London Hal R. Varian, University ofMichigan
The titles published in this series are listed at the end of this volume.
Parallel Algorithms for Linear Models Numerical Methods and Estimation Problems by
Erricos John Kontoghiorghes Universite de Neuchâtel, Switzerland
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Springer Science+Business Media, LLC
Library of Congress Cataloging-in-Publication Data Kontoghiorghes, Erricos John. Parallel algorithms for linear models : numerical methods and estimation problems / by Erricos John Kontoghiorghes. p. cm. -- (Advances in computational economics; v. 15) lncludes bibliographical references and indexes. ISBN 978-1-4613-7064-2 ISBN 978-1-4615-4571-2 (eBook) DOI 10.1007/978-1-4615-4571-2 1. Linear models (Statistics)--Data processing. 2. Parallel algorithms. 1. Title. II. Series. QA276 .K645 2000 519.5'35--dc21
99-056040
Copyright © 2000 by Springer Science+Business Media New York Originally published by Kluwer Academic Publishers, New York in 1992 Softcover reprint ofthe hardcover lst edition 1992 AII rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photo-copying, recording, or otherwise, without the prior written permission of the publisher, Springer Science + Business Media, LLC Printed on acid-free paper.
To Laurence and Louisa
Contents
List of Figures List of Tables List of Algorithms Preface
ix xi xiii xv
1. LINEAR MODELS AND QR DECOMPOSmON 1 Introduction 2 Linear model specification 2.1 The ordinary linear model 2.2 The general linear model 3 Forming the QR decomposition 3.1 The Householder method 3.2 The Givens rotation method 3.3 The Gram-Schmidt orthogonalization method 4 Data parallel algorithms for computing the QR decomposition 4.1 Data: parallelism and the MasPar SIMD system 4.2 The Householder method 4.3 The Gram-Schmidt method 4.4 The Givens rotation method 4.5 Computational results 5 QRD of large and skinny matrices 5.1 The CPP GAMMA SIMD system 5.2 The Householder QRD algorithm 5.3 QRD of skinny matrices 6 QRD of a set of matrices 6.1 Equal size matrices 6.2 Mattices with different number of columns
1 1 1 2 7 10 11 13 16 17 17 19 21 22 23 23 24 25 27 29 29 34
2. OLM Nor OF FULL RANK 1 Introduction 2 The QLD of the coefficient matrix 2.1 SIMD implementation 3 Triangularizing the lower trapezoid
39 39 40 41 43
viii
PARALLEL ALGORITHMS FOR UNEAR
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