Numerical Analysis (Second Edition)

1 Walter GautschiNumerical AnalysisSecond EditionWalter GautschiDepartment of Computer SciencesPurdue University250 N. University StreetWest Lafayette, IN 978-0-8176-8258-3e-ISBN 978-0-8176-8259-0 DOI New York Dordrecht Heidelberg LondonLibrary of Congress Control Number: 2011941359 Mathematics Subject Classification (2010): 65-01, 65D05, 65D07, 65D10, 65D25, 65D30, 65D32,65H04, 65H05, 65H10, 65L04, 65L05, 65L06, 65L10c Springer Science+Business Media, LLC 1997, 2012 All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer ScienceCBusiness Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly Analysis .

2 Use inconnection with any form of information storage and retrieval, electronic adaptation, computer software,or by similar or dissimilar methodology nowknown or hereafter developed is use in this publication of trade names, trademarks, service marks, and similar terms, even if they arenot identified as such, is not to be taken as an expression of opinion as to whether or not they are subjectto proprietary on acid-free to the Second EditionIn this second edition, the outline of chapters and sections has been preserved. Thesubtitle An Introduction , as suggested by several reviewers, has been deleted. Thecontent, however, is brought up to date, both in the text and in the notes.

3 Manypassages in the text have been either corrected or improved. Some biographicalnotes have been added as well as a few exercises and computer assignments. Thetypographical appearance has also been improved by printing vectors and matricesconsistently in boldface regard to computer language in illustrations and exercises, we now adoptuniformly Matlab. For readers not familiar with Matlab, there are a number ofintroductory texts available, some, like Moler [2004], Otto and Denier [2005],Stanoyevitch [2005] that combine Matlab with Numerical computing, others, likeKnight [2000], Higham and Higham [2005], Hunt, Lipsman and Rosenberg [2006],and Driscoll [2009], more exclusively focused on major novelty, however, is a complete set of detailed solutions to all exercisesand machine assignments.

4 The solution manual is available to instructors uponrequest at the publisher s Selected solutions are also included in the text to give students an idea ofwhat is expected. The bibliography has been expanded to reflect technical advancesin the field and to include references to new books and expository accounts. As aresult, the text has undergone an expansion in size of about 20%.West Lafayette, IndianaWalter GautschiNovember 2011viiPreface to the First EditionThe book is designed for use in a graduate program in Numerical Analysis thatis structured so as to include a basic introductory course and subsequent morespecialized courses.

5 The latter are envisaged to cover such topics as numericallinear algebra, the Numerical solution of ordinary and partial differential equations,and perhaps additional topics related to complex Analysis , to multidimensionalanalysis, in particular optimization, and to functional Analysis and related functionalequations. Viewed in this context, the first four chapters of our book could serve asa text for the basic introductory course, and the remaining three chapters (whichindeed are at a distinctly higher level) could provide a text for an advanced courseon the Numerical solution of ordinary differential equations. In a sense, therefore,the book breaks with tradition in that itdoes no longer attempt to deal with allmajor topics of Numerical mathematics.

6 It is felt by the author that some of thecurrent subdisciplines, particularly those dealing with linear algebra and partialdifferential equations, have developed into major fields of study that have attaineda degree of autonomy and identity that justifies their treatment in separate booksand separate courses on the graduate level. The term Numerical Analysis asused in this book, therefore, is to be taken in the narrow sense of the numericalanalogue of Mathematical Analysis , comprising such topics as machine arithmetic,the approximation of functions, approximate differentiation and integration, and theapproximate solution of nonlinear equations and of ordinary differential is being covered, on the other hand, is done so with a view towardstressing basic principles and maintaining simplicity and student-friendliness as faras possible.

7 In this sense, the book is An Introduction . Topics that, even thoughimportant and of current interest, require a level of technicality that transcends thebounds of simplicity striven for, are referenced in detailed bibliographic notes at theend of each chapter. It is hoped, in this way,to place the material treated in propercontext and to help, indeed encourage, the reader to pursue advanced modern topicsin more significant feature of the book is the large collection of exercises thatare designed to help the student develop problem-solving skills and to provideinteresting extensions of topics treated in the text.

8 Particular attention is given toixxPreface to the First Editionmachine assignments, where the studentis encouraged to implement numericaltechniques on the computer and to make use of modern software author has taught the basic introductory course and the advanced course onordinary differential equations regularly at Purdue University for the last 30 yearsor so. The former, typically, was offered both in the fall and spring semesters, to amixed audience consisting of graduate (and some good undergraduate) students inmathematics, computer science, and engineering, while the latter was taught only inthe fall, to a smaller but also mixed audience.

9 Written notes began to materialize inthe 1970s, when the author taught the basic course repeatedly in summer courses onMathematics held in Perugia, Italy. Indeed, for some time, these notes existed onlyin the Italian language. Over the years, they were progressively expanded, updated,and transposed into English, and along with that, notes for the advanced course weredeveloped. This, briefly, is how the present book long gestation period such as this, of course, is not without dangers, themost notable one being a tendency for the material to become dated. The authortried to counteract this by constantly updating and revising the notes, adding newerdevelopments when deemed appropriate.

10 There are, however, benefits as well: overtime, one develops a sense for what is likely to stand the test of time and whatmay only be of temporary interest, and one selects and deletes accordingly. Anotherbenefit is the steady accumulation of exercises and the opportunity to have themtested on a large and diverse student purpose of academic teaching, in the author s view, is twofold: to transmitknowledge, and, perhaps more important, to kindle interest and even enthusiasmin the student. Accordingly, the author did not strive for comprehensiveness even within the boundaries delineated but rather tried to concentrate on what isessential, interesting and intellectuallypleasing, and teachable.