Max-linear Systems: Theory and Algorithms
Recent years have seen a significant rise of interest in max-linear theory and techniques. In addition to providing the linear-algebraic background in the field of tropical mathematics, max-algebra provides mathematical theory and techniques for solving v
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Peter Butkoviˇc
Max-linear Systems: Theory and Algorithms
Peter Butkoviˇc School of Mathematics University of Birmingham Birmingham, UK
ISSN 1439-7382 ISBN 978-1-84996-298-8 DOI 10.1007/978-1-84996-299-5
e-ISBN 978-1-84996-299-5
Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2010933523 Mathematics Subject Classification (2000): 15A80 © Springer-Verlag London Limited 2010 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To Eva, Eviˇcka and Alenka
Preface
Max-algebra provides mathematical theory and techniques for solving nonlinear problems that can be given the form of linear problems, when arithmetical addition is replaced by the operation of maximum and arithmetical multiplication is replaced by addition. Problems of this kind are sometimes of a managerial nature, arising in areas such as manufacturing, transportation, allocation of resources and information processing technology. The aim of this book is to present max-algebra as a modern modelling and solution tool. The first five chapters provide the fundamentals of max-algebra, focusing on one-sided max-linear systems, the eigenvalue-eigenvector problem and maxpolynomials. The theory is self-contained and covers both irreducible and reducible matrices. Advanced material is presented from Chap. 6 onwards. The book is intended for a wide-ranging readership, from undergraduate and postgraduate students to researchers and mathematicians working in industry, commerce or management. No prior knowledge of max-algebra is assumed. We concentrate on linear-algebraic aspects, presenting both classical and new results. Most of the theory is illustrated by numerical examples and complemented by exercises at the end of every chapter. Chapter 1 presents essential definitions, examples and basic res
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