# Introduction to numerical analysis 3rd - Preface to the...

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Preface to the Third Edition A new edition of a text presents not only an opportunity for corrections and minor changes but also for adding new material. Thus we strived to improve the presentation of Hermite interpolation and B-splines in Chap- ter 2, and we added a new Section 2.4.6 on multi-resolution methods and B-splines, using, in particular, low order B-splines for purposes of illustra- tion. The intent is to draw attention to the role of B-splines in this area, and to familiarize the reader with, at least, the principles of multi-resolution methods, which are fundamental to modern applications in signal- and im- age processing. The chapter on di ff erential equations was enlarged, too: A new Section 7.2.18 describes solving di ff erential equations in the presence of disconti- nuities whose locations are not known at the outset. Such discontinuities occur, for instance, in optimal control problems where the character of a di ff erential equation is a ff ected by control function changes in response to switching events. Many applications, such as parameter identification, lead to di ff erential equations which depend on additional parameters. Users then would like to know how sensitively the solution reacts to small changes in these pa- rameters. Techniques for such sensitivity analyses are the subject of the new Section 7.2.19. Multiple shooting methods are among the most powerful for solving boundary value problems for ordinary di ff erential equations. We dedicated, therefore, a new Section 7.3.8 to new advanced techniques in multiple shoot- ing, which especially enhance the e ciency of these methods when applied to solve boundary value problems with discontinuities, which are typical for optimal contol problems. Among the many iterative methods for solving large sparse linear equa- tions, Krylov space methods keep growing in importance. We therefore treated these methods in Section 8.7 more systematically by adding new subsections dealing with the GMRES method (Section 8.7.2), the biorthog- onalization method of Lanczos and the (principles of the) QMR method (Section 8.7.3), and the Bi-CG and Bi-CGSTAB algorithms (Section 8.7.4). Correspondingly, the final Section 8.10 on the comparison of iterative meth- ods was updated in order to incorporate the findings for all Krylov space methods described before. The authors are greatly indebted to the many who have contributed vii
viii Preface to the Third Edition to the new edition. We thank R. Grigorie ff for many critical remarks on earlier editions, M. v. Golitschek for his recommendations concerning B- splines and their application in multi-resolution methods, and Ch. Pflaum for his comments on the chapter dealing with the iterative solution of linear equations. T. Kronseder and R. Callies helped substantially to establish the new sections 7.2.18, 7.2.19, and 7.3.8. Suggestions by Ch. Witzgall, who had helped translate a previous edition, were highly appreciated and went beyond issues of language. Our co-workers M. Preiss and M. Wenzel helped us read and correct the original german version. In particular, we appreciate the excellent work done by J. Launer and Mrs.