Hierarchical Matrices: Algorithms and Analysis
This self-contained monograph presents matrix algorithms and their analysis. The new technique enables not only the solution of linear systems but also the approximation of matrix functions, e.g., the matrix exponential. Other applications include the sol
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Wolfgang Hackbusch
Hierarchical Matrices: Algorithms and Analysis
Springer Series in Computational Mathematics Volume 49
Editorial Board R.E. Bank R.L. Graham W. Hackbusch J. Stoer R.S. Varga H. Yserentant
More information about this series at http://www.springer.com/series/797
Wolfgang Hackbusch
Hierarchical Matrices: Algorithms and Analysis
2123
Wolfgang Hackbusch MPI für Mathematik in den Naturwissenschaften Leipzig, Germany
ISSN 0179-3632 ISSN 2198-3712 (electronic) Springer Series in Computational Mathematics ISBN 978-3-662-47323-8 ISBN 978-3-662-47324-5 (eBook) DOI 10.1007/978-3-662-47324-5 Library of Congress Control Number: 2015954369
Mathematics Subject Classification (2010): 65Fxx, 65F05, 65F08, 65F10, 65F15, 65F30, 65F60, 65F99, 65H99, 65Nxx, 65N22, 65N38, 65N99, 65Rxx, 65R99, 15Axx, 39B42 Springer Heidelberg New York Dordrecht London © Springer-Verlag Berlin Heidelberg 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer-Verlag GmbH Berlin Heidelberg is part of Springer Science+Business Media (www.springer.com)
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Preface
Usually one avoids numerical algorithms involving operations with large, fully populated matrices. Instead one tries to reduce all algorithms to matrix-vector multiplications involving only sparse matrices. The reason is the large number of floating point operations; e.g., O(n3 ) for multiplying two general n × n matrices. Starting with Strassen’s algorithm [236], one has tried to reduce the work to O(nγ ) with γ < 3. However, these attempts cannot be satisfactory since γ ≥ 2 is a lower bound, and even quadratic work is unacceptable for large-scale matrices. The hierarchical matrix (H-matrix) technique provides tools for performing matrix operations in almost linear work O(n log∗ n). This is no contradiction to the previously mentioned lower bound, since the former statement holds for exact computations, whereas H-matrix operations yield approximations. The approximation errors are nev
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