Free ebooks ==> www.Ebook777.com www.Ebook777.com Free ebooks ==> www.Ebook777.com Texts in Applied Mathematics 43 Editors J.E Marsden L Sirovich M Golubitsky S.S Antman Advisors G.looss P Holmes D Barkley M Dellnitz P Newton Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo www.Ebook777.com Texts in Applied Mathematics I 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 SiTOvich: Introduction to Applied Mathematics Wiggins: Introduction to Applied Nonlinear Dynamical Systems and Chaos Hale/Kor;ak: Dynamics and Bifurcations Chorin/Marsden: A Mathematical Introduction to Fluid Mechanics, 3rd ed Hubbard/West: Differential Equations: A Dynamical Systems Approach: Ordinary Differential Equations Sontag: Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd ed Perko: Differential Equations and Dynamical Systems, 3rd ed 5mborn: Hypergeometric Functions and Their Applications Pipkin: A Course on Integral Equations Hoppensteadt/Peskin: Modeling and Simulation in Medicine and the Life Sciences, 2nd cd Braun: Differential Equations and Their Applications, 4th ed Stoer/Bulirsch: Introduction to Numerical Analysis, 3rd ed Renardy/Rogers: An Introduction to Partial Differential Equations Banks: Growth and Diffusion Phenomena: Mathematical Frameworks and Applications Brenner/Scott: The Mathematical Theory of Finite Element Methods, 2nd ed Van de Velde: Concurrent Scientific Computing Marsden/Ratiu: Introduction to Mechanics and Symmetry, 2nd ed Hubbard/West: Differential Equations: A Dynamical Systems Approach: Higher-Dimensional Systems Kaplan/Glass: Understanding Nonlinear Dynamics Holmes: Introduction to Perturbation Methods Curtain/Zwart: An Introduction to Infinite-Dimensional Linear Systems Theory Thomas: Numerical Partial Differential Equations: Finite Difference Methods Taylor: Partial Differential Equations: Basic Theory Merkin: Introduction to the Theory of Stability of Motion Naber: Topology, Geometry, and Gauge Fields: Foundations Polderman/Willems: Introduction to Mathematical Systems Theory: A Behavioral Approach Reddy: Introductory Functional Analysis with Applications to Boundary-Value Problems and Finite Elements Gustafson/Wilcox: Analytical and Computational Methods of Advanced Engineering Mathematics Tveito/Winther: Introduction to Partial Differential Equations: A Computational Approach Gasquet/Witomski: Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets (continued after index) Peter Deuflhard Andreas Hohmann Numerical Analysis in Modern Scientific Computing An Introduction Second Edition With 65 Illustrations , Springer Free ebooks ==> www.Ebook777.com Andreas Hohmann AMS D2 Vodafone TPAI Dusseldorf, D-40547 Germany andreas.hohmann@d2vodafone.de Peter Deuflhard Konrad-Zuse-Zentrum (ZIB) Berlin-Dahlem, D-14195 Germany deuflhard@zib.de Series Editors J.E Marsden Control and Dynamical Systems 107-S1 California Institute of Technology Pasadena, CA 91125 USA marsden@cds.caltech.edu L Sirovich M Golubitsky Department of Mathematics University of Houston Houston, TX 77204-3476 USA S.S Antman Department of Mathematics Division of Applied Mathematics Brown University Providence, RI 02912 USA chico@camelot.mssm.edu and Institute for Physical Science and Technology University of Maryland College Park, MD 20742-4015 USA ssa@math.umd.edu Mathematics Subject Classification (2000): 65-XX, 6S-XX, 65-01, 65Fxx, 6SNxx Library of Congress Cataloging-in-Publication Data Deuflhard P (Peter) Numerical analysis in modern scientific computing: an introduction / Peter Deuflhard, Andreas Hohmann.-2nd ed p cm - (Texts in applied mathematics; 43) Rev ed of: Numerical analysis 1995 Includes bibliographical references and index I Numerical analysis-Data processing (Peter) Numerische Mathematik I English QA297 D45 2003 519.4-dc21 ISBN 978-1-4419-2990-7 DOl 10.1007/978-0-387-21584-6 I Hohmann, Andreas 1964- II Deutlhard, P III Title IV Series ISBN 978-0-387-21584-6 (eBook) 2002030564 Printed on acid-free paper © 2003 Springer-Verlag New York, Inc Softcover reprint of the hardcover 1st edition 2003 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights I SPIN 10861791 www.springer-ny.com Springer-Verlag New York Berlin Heidelberg A member (If' Berte/smannSpringer Sciellce+Business Media GmbH www.Ebook777.com Series Preface Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics This renewal of interest, both in research and teaching, has led to the establishment of the series Texts in Applied Mathematics (TAM) The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathematical Sciences (AMS) series, which will focus on advanced textbooks and research-level monographs Pasadena, California Providence, Rhode Island Houston, Texas College Park, Maryland J.E Marsden L Sirovich M Golubitsky S.S Antman Preface For quite a number of years the rapid progress in the development of both computers and computing (algorithms) has stimulated a more and more detailed scientific and engineering modeling of reality New branches of science and engineering, which had been considered rather closed until recently, have freshly opened up to mathematical modeling and to simulation on the computer There is clear evidence that our present problem-solving ability does not only depend on the accessibility of the fastest computers (hardware), but even more on the availability of the most efficient algorithms (software) The construction and the mathematical understanding of numerical algorithms is the topic of the academic discipline Numerical Analysis In this introductory textbook the subject is understood as part of the larger field Scientific Computing This rather new interdisciplinary field influences smart solutions in quite a number of industrial processes, from car production to biotechnology At the same time it contributes immensely to investigations that are of general importance to our societies-such as the balanced economic and ecological use of primary energy, global climate change, or epidemiology The present book is predominantly addressed to students of mathematics, computer science, science, and engineering In addition, it intends to reach computational scientists already on the job who wish to get acquainted with established modern concepts of Numerical Analysis and Scientific Computing on an elementary level via personal studies viii Preface The field of Scientific Computing, situated at the confluence of mathematics, computer science, natural science, and engineering, has established itself in most teaching curricula, sometimes still under the traditional name Numerical Analysis However, basic changes in the contents and the presentation have taken place in recent years, and this already at the introductory level: classical topics, which had been considered important for quite a time, have just dropped out, new ones have entered the stage The guiding principle of this introductory textbook is to explain and exemplify essential concepts of modern Numerical Analysis for ordinary and partial differential equations using the simplest possible model problems Nevertheless, readers are only assumed to have basic knowledge about topics typically taught in undergraduate Linear Algebra and Calculus courses Further knowledge is definitely not required The primary aim of the book is to develop algorithmic feeling and thinking After all, the algorithmic approach has historically been one of the roots of today's mathematics It is no mere coincidence that, besides contemporary names, historical names like Gauss, Newton, and Chebyshev are found in numerous places all through the text The orientation toward algorithms, however, should by no means be misunderstood In fact, the most efficient algorithms often require a substantial amount of mathematical theory, which will be developed in the book As a rule, elementary mathematical arguments are preferred In topics like interpolation or integration we deliberately restrict ourselves to the one-dimensional case Wherever meaningful, the reasoning appeals to geometric intuition-which also explains the quite large number of graphical representations Notions like scalar product and orthogonality are used throughout-in the finite dimensional case as well as in infinite dimensions (functions) Despite the elementary presentation, the book contains a significant number of otherwise unpublished material Some of our derivations of classical results differ significantly from traditional derivations-in many cases they are simpler and nevertheless more stringent As an example we refer to our condition and error analysis, which requires only multidimensional differentiation as the main analytical prerequisite Compared to the first English edition, a polishing of the book as a whole has been performed The essential new item is Section 5.5 on stochastic eigenvalue problems-a problem class that has gained increasing importance and appeared to be well-suited for an elementary presentation within our conceptual frame As a recent follow-up, there exists an advanced textbook on numerical ordinary differential equations [22] Preface ix Of course, any selection of material expresses the scientific taste of the authors The first author founded the Zuse Institute Berlin (ZIB) as a research institute for Scientific Computing in 1986 He has given Numerical Analysis courses at the Technical University of Munich and the University of Heidelberg, and is now teaching at the Free University of Berlin Needless to say, he has presented his research results in numerous invited talks at international conferences and seminars at renowned universities and industry places all over the world The second author originally got his mathematical training in pure mathematics and switched over to computational mathematics later He is presently working in the communication industry We are confident that the combination of a senior and a junior author, of a pure and an applied mathematician, as well as a member of academia and a representative from industry has had a stimulating effect on our presentation At this point it is our pleasure to thank all those who have particularly helped us with the preparation of this book The first author remembers with gratitude his early time as an assistant of Roland Bulirsch (Technical University of Munich, retired since 2001), in whose tradition his present views on Scientific Computing have been shaped Of course, our book has significantly profited from intensive discussions with numerous colleagues, some of which we want to mention explicitly here: Ernst Hairer and Gerhard Wanner (University of Geneva) for discussions on the general concept of the book; Folkmar Bornemann (Technical University of Munich) for the formulation of the error analysis, the different condition number concepts, and the definition of the stability indicator in Chapter 2; Wolfgang Dahmen (RWTH Aachen) for Chapter 7; and Dietrich Braess (Ruhr University Bochum) for the recursive derivation of the Fast Fourier Transform in Section 7.2 The first edition of this textbook, which already contained the bulk of material presented in this text, was translated by Florian Potra and Friedmar Schulz-again many thanks to them For this, the second edition, we cordially thank Rainer Roitzsch (ZIB), without whose deep knowledge about a rich variety of fiddly TEX questions this book could never have appeared Our final thanks go to Erlinda Kornig and Sigrid Wacker for all kinds of assistance Berlin and Dusseldorf, March 2002 Peter Deufihard and Andreas Hohmann Free ebooks ==> www.Ebook777.com Outline This introductory textbook is, in the first place, addressed to students of mathematics, computer science, science, and engineering In the second place, it is also addressed to computational scientists already on the job who wish to get acquainted with modern concepts of Numerical Analysis and Scientific Computing on an elementary level via personal studies The book is divided into nine chapters, including associated exercises, a software list, a reference list, and an index The contents of the first five and of the last four chapters are each closely related In Chapter we begin with Gaussian elimination for linear systems of equations as the classical prototype of an algorithm Beyond the elementary elimination technique we discuss pivoting strategies and iterative refinement as additional issues Chapter contains the indispensable error analysis based on the fundamental ideas of J H Wilkinson The condition of a problem and the stability of an algorithm are presented in a unified framework, well separated and illustrated by simple examples The quite unpopular "E-battle" in linearized error analysis is avoided~which leads to a drastic simplification of the presentation and to an improved understanding A stability indicator arises naturally, which allows a compact classification of numerical stability On this basis we derive an algorithmic criterion to determine whether a given approximate solution of a linear system of equations is acceptable or not In Chapter we treat orthogonalization methods in the context of Gaussian linear least-squares problems and introduce the extremely useful calculus of pseudo-inverses It is immediately applied in the following Chapter 4, where we present iterative www.Ebook777.com 322 Definite Integrals Exercise 9.5 Derive the formula Tik Ti k-1 - T i - k-1 ' = Ti ,k-1 + ' ( ni) ni-k+l for the extrapolation tableau from the one of the Aitken-Neville algorithm Exercise 9.6 Every element Tik in the extrapolation tableau of the extrapolated trapezoidal rule can be considered as a quadrature formula Show that when using the Romberg sequence and polynomial extrapolation, the following results hold: (a) T22 (b) Tik, is equal to the value, which is obtained by applying the Simpson rule; T33 corresponds to the Milne rule i > k is obtained by 2i - k -fold application of the quadrature formula, which belongs to to suitably chosen subintervals Tkk (c) For every Tik, the weights of the corresponding quadrature formula are positive Hint: By using (b), show that the weights which corresponds to Tkk, satisfies max Ai n i' :::; of the quadrature formula, Ai,n 4k Ai i n Exercise 9.7 Implement the Romberg algorithm by only using one single vector of length n (note that only one intermediate value of the table needs to be extra stored) Exercise 9.8 Experiment with an adaptive Romberg quadrature program, test it with the "needle function" I(n):= 2-n -1 4- n +t dt, for n = 1,2, and determine the n, for which your program yields the value zero for a given precision of eps = 10- Exercise 9.9 Consider the computation of the integrals In = 12 (In x)ndx, n = 1,2, (a) Show that the In satisfy the recurrence relation In = 2(ln2)n - nIn - , n ~ (R) (b) Note that h = 0.3863 and h = 0.0124 Investigate the increase of the input error in the computation of (1) (2) h from h by means of (R) h from h by means of (R) (forward recursion), (backward recursion) Exercises 323 Assume an accuracy of four decimal places and neglect any rounding errors (c) Use (R) as a backward recursion for the computation of In from In+k with starting value In+k = How is k to be chosen in order to compute 17 accurately up to digits by this method? Exercise 9.10 Consider integrals of the following form: In(a) := 11 t 2n +a sin(7rt)dt where a > -1 and n = 0, 1,2, (a) For In, derive the following inhomogeneous two-term recurrence relation I ()=~_(2n+a)(2n+a-1)I () n a 7r 7r n-l a (b) Show that lim In(a) n ->oo =0 and 0::; In+l(a) ::; In(a) for n 2: (c) Give an informal algorithm for the computation of Io(a) (compare Chapter 6.2-3) Write a program to compute Io(a) for a given relative precision Exercise 9.11 A definite integral over [-1, + 1] is to be computed Based on the idea of the Gauss-Christoffel quadrature, derive a quadrature formula 11 +1 n-l f(t)dt::::; f-Lof( -1) + f-Lnf(l) + ~ f-Ld(t i ) with fixed nodes -1 and + and variable nodes to be determined such that the order is as high as possible (Gauss-Lobatto quadrature) References [1] ABDULLE, A., AND WANNER, G 200 years of least squares method Elemente der Mathematik (2002) [2] ABRAMOWITZ, M., AND STEGUN, A Pocketbook of Mathematical Functions Verlag Harri Deutsch, Thun, Frankfurt/Main, 1984 [3] AIGNER, M Diskrete Mathematik, ed Vieweg, Braunschweig, Wiesbaden, 200l [4] ANDERSON, E., BAI, Z., BISCHOF, C., DEMMEL, J., DONGARRA, J., DUCROZ, J., GREENBAUM, A., HAMMARLING, S., McKENNEY, A., OSTRUCHOV, S., AND SORENSEN, D LAPACK Users' Guide SIAM, Philadelphia, 1999 [5] ARNOLDI, W E The principle of minimized iterations in the solution of the matrix eigenvalue problem Quart Appl Math (1951), 17-29 [6] BABUSKA, 1., AND RHEINBOLDT, W C Error estimates for adaptive finite element computations SIAM J Numer Anal 15 (1978), 736-754 [7] BJ0RCK, A Iterative refinement of linear least squares solutions I BIT (1967), 257-278 [8] BOCK, H G Randwertproblemmethoden zur Parameteridentijizierung in Systemen nichtlinearer Differentialgleichungen PhD thesis, Universitiit zu Bonn, 1985 [9] BORNEMANN, F A An Adaptive Multilevel Approach to Parabolic Equations in two Dimensions PhD thesis, Freie Universitiit Berlin, 1991 [10] BRENT, R P Algorithms for Minimization Without Derivatives Prentice Hall, Englewood Cliffs, N.J., 1973 [11] BULIRSCH, R Bemerkungen zur Romberg-Integration Numer Math (1964),6-16 326 References [12] BUSINGER, P., AND GOLUB, G H Linear least squares solutions by Householder transformations Numer Math 'l (1965), 269-276 [13] CULLUM, J., AND WILLOUGHBY, R Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol I, II Birkhiiuser, Boston, 1985 [14] DE BOOR, C An algorithm for numerical quadrature In Mathematical Software, J Rice, Ed Academic Press, London, 1971 [15] DE BOOR, C A Practical Guide to Splines, reprint ed Springer-Verlag, Berlin, Heidelberg, New York, 1994 [16] DEUFLHARD, P On algorithms for the summation of certain special functions Computing 17 (1976), 37-48 [17] DEUFLHARD, P A summation technique for minimal solutions of linear homogeneous difference equations Computing 18 (1977), 1-13 [18] DEUFLHARD, P A stepsize control for continuation methods and its special application to multiple shooting techniques Numer Math 33 (1979), 115146 [19] DEUFLHARD, P Order and stepsize control in extrapolation methods Numer Math 41 (1983), 399-422 [20] DEUFLHARD, P Newton Methods for Nonlinear Problems Affine Invariance and Adaptive Algorithms Springer International, 2002 [21] DEUFLHARD, P., AND BAUER, H J A note on Romberg quadrature Preprint 169, Universitiit Heidelberg, 1982 [22] DEUFLHARD, P., AND BORNEMANN, F Scientific Computing with Ordinary Differential Equations Springer, New York, 2002 [23] DEUFLHARD, P., FIEDLER, B., AND KUNKEL, P Efficient numerical pathfollowing beyond critical points SIAM J Numer Anal 18 (1987), 949-987 [24] DEUFLHARD, P., HUISINGA, W., FISCHER, A., AND SCHUTTE, C Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains Lin Alg Appl 315 (2000), 39-59 [25] DEUFLHARD, P., LEINEN, P., AND YSERENTANT, H Concept of an adaptive hierarchical finite element code Impact of Computing in Science and Engineering 1, (1989), 3-35 [26] DEUFLHARD, P., AND POTRA, F A A refined Gauss-Newton-Mysovskii theorem ZIB Report SC 91-4, ZIB, Berlin, 1991 [27] DEUFLHARD, P., AND POTRA, F A Asymptotic mesh independence for Newton-Galerkin methods via a refined Mysovskii theorem SIAM J Numer Anal 29,5 (1992), 1395-1412 [28] DEUFLHARD, P., AND SAUTTER, W On rank-deficient pseudoinverses Lin Alg Appl 29 (1980),91-111 [29] ERICSSON, T., AND RUHE, A The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems Math Compo 35 (1980), 1251-1268 [30] FARIN, G Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide Academic Press, New York, 1988 References 327 [31J FLETCHER, R Conjugate gradient methods In Pmc Dundee Biennial Conference on Numerical Analysis Springer Verlag, New York, 1975 [32J FORSYTHE, G E., AND MOLER, C Computer Solution of Linear Algebra Systems Prentice Hall, Englewood Cliffs, N.J., 1967 [33J FRANCIS, J G F The QR-transformation A unitary analogue to the LRtransformation - Part and Compo J (1961/62),265-271 and 332-344 [34J GATERMANN, K., AND HOHMANN, A Symbolic exploitation of symmetry in numerical pathfollowing Impact Compo Sci Eng 3,4 (1991), 330-365 [35J GAUSS, C F Theoria Motus Corporum Coelestium Vol Perthes et Besser, Hamburgi, 1809 [36J GAUTSCHI, W Computational aspects of three-term recurrence relations SIAM Rev (1967), 24-82 [37J GENTLEMAN, W M Least squares computations by Givens transformations without square roots J Inst Math Appl 12 (1973), 189-197 [38J GEORG, K On tracing an implicitly defined curve by quasi-Newton steps and calculating bifurcation by local perturbations SIAM J Sci Stat Comput 2, (1981), 35-50 [39J GEORGE, A., AND LIU, J W Computer Solution of Large Sparse Positive Definite Systems Prentice Hall, Englewood Cliffs, N.J., 1981 [40J GOERTZEL, G An algorithm for the evaluation of finite trigonometric series Amer Math Monthly 65 (1958), 34-35 [41J GOLUB, G H., AND VAN LOAN, C F Matrix Computations, second ed The Johns Hopkins University Press, Baltimore, MD, 1989 [42J GOLUB, G H., AND WELSCH, J H Calculation of Gauss quadrature rules Math Compo 23 (1969), 221-230 [43J GRADSHTEYN, S., AND RYZHlK, W Table of Integral Series and Products, sixth ed Academic Press, New York, San Francisco, London, 2000 [44J GRIEWANK, A., AND CORLISS, G F Automatic Differentiation of Algorithms: Theory, Implementation, and Application SIAM Publications, Philadelphia, PA, 1991 [45J HACKBUSCH, W Multi-Grid Methods and Applications Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1985 [46J HAGEMAN, L A., AND YOUNG, D M Applied Iterative Methods Academic Press, Orlando, San Diego, New York, 1981 [47J HAIRER, E., N0RSETT, S P., AND WANNER, G Solving Ordinary Differential Equations I, Nonstiff Problems Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1987 [48J HALL, C A., AND MEYER, W W Optimal error bounds for cubic spline interpolation J Appmx Theory 16 (1976), 105-122 [49J HAMMARLING, S A note on modifications to the Givens plane rotations J [nst Math Appl 13 (1974), 215-218 [50J HESTENES, M R., AND STIEFEL, E Methods of conjugate gradients for solving linear systems J Res Nat Bur Stand 49 (1952), 409-436 328 References [51J HIGHAM, N J How accurate is Gaussian elimination? In Numerical Analysis, Pmc 13th Biennial Conf., Dundee / UK 1989 Pitman Res Notes Math Ser 228, 1990, pp 137-154 [52J HOUSEHOLDER, A S The Theory of Matrices in Numerical Analysis Blaisdell, New York, 1964 [53J IpSEN, I C F A history of inverse iteration In Helmut Wielandt, Mathematische Werke, Mathematical Works, B Huppert and H Schneider, Eds., vol II: Matrix Theory and Analysis Walter de Gruyter, New York, 1996, pp.464-72 [54J KATO, T Perturbation Theory for Linear Operators, reprint ed Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1995 [55J KNOPP, K Theorie und Anwendung der unendlichen Reihen Springer Verlag, Berlin, Heidelberg, New York, (5 Aufiage) 1964 [56J KUBLANOVSKAYA, V N On some algorithms for the solution of the complete eigenvalue problem USSR Compo Math Phys (1961),637-657 [57J LANCZOS, C An iteration method for the solution of the eigenvalue problem of linear differential and integral operators J Res Nat Bur Stand 45 (1950), 255-282 [58J MANTEUFFEL, T A The Tchebychev iteration for nonsymmetric linear systems Numer Math 28 (1977), 307-327 [59J MEIJERINK, J., AND VAN DER VORST, H An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix Math Compo 31 (1977), 148-162 [60J MEIXNER, J R., AND SCHAFFKE, W Mathieusche Funktionen und Sphiiroidfunktionen Springer Verlag, Berlin, Gi:ittingen, Heidelberg, 1954 [61J MEYER, C D Matrix Analysis and Applied Linear Algebra SIAM Publications, Philadelphia, PA, 2000 [62J MILLER, J C P Bessel Functions, Part II (Math Tables X) Cambridge University Press, Cambridge, UK, 1952 [63J NASHED, M Z Generalized Inverses and Applications Academic Press, New York, 1976 [64J NIKIFOROV, A F., AND UVAROV, V B Special Functions of Mathematical Physics Birkhiiuser, Basel, Boston, 1988 [65J PERRON, O tiber Matrizen Math Annalen 64 (1907),248-263 [66J POINCARE, H Les Methodes Nouvelles de la Mecanique Celeste GauthierVillars, Paris, 1892 [67J POPPE, C., PELLICIARI, C., AND BACHMANN, K Computer analysis of Feulgen hydrolysis kinetics Histochemistry 60 (1979), 53-60 [68J PRAGER, W., AND OETTLI, W Compatibility of approximate solutions of linear equations with given error bounds for coefficients and right hand sides Numer Math (1964), 405-409 [69J PRIGOGINE, I., AND LEFEVER, R Symmetry breaking instabilities in dissipative systems II J Chem Phys 48 (1968), 1695-170l [70J REINSCH, C A note on trigonometric interpolation Manuscript, 1967 References 329 [71] RIGAL, J L., AND GACHES, J On the compatibility of a given solution with the data of a linear system J Assoc Comput Mach 14 (1967), 543-548 [72] ROMBERG, W Vereinfachte Numerische Integration Det Kongelige Norske Videnskabers Selskabs Forhandlinger Bind 28, (1955) [73] SAUER, R., AND SZABO, Mathematische Hilfsmittel des Ingenieurs Springer Verlag, Berlin, Heidelberg, New York, 1968 [74] SAUTTER, W Fehlerfortpfianzung und Rundungsfehler bei der verallgemeinerten Inversion von Matrizen PhD thesis, TU Miinchen, Fakultiit fiir Allgemeine Wissenschaften, 1971 [75] SHANNON, C E The Mathematical Theory of Communication The University of Illinois Press, Urbana, Chicago, London, 1949 [76] SKEEL, R D Scaling for numerical stability in Gaussian elimination J ACM 26, (1979), 494-526 [77] SKEEL, R D Iterative refinement implies numerical stability for Gaussian elimination Math Compo 35, 151 (1980), 817-832 [78] SONNEVELD, P A fast Lanczos-type solver for nonsymmetric linear systems SIAM J Sci Stat Comput 10 (1989), 36-52 [79] STEWART, G W Introduction to Matrix Computations Academic Press, New York, San Francisco, London, 1973 [80] STEWART, G W On the structure of nearly uncoupled Markov chains In Mathematical Computer Performance and Reliability, G Iazeolla, P J Courtois, and A Hordijk, Eds Elsevier, New York, 1984 [81] STOER, J Solution of large systems of linear equations by conjugate gradient type methods In Mathematical Programming, the State of the Art, A Bachem, M Grotschel, and B Korte, Eds Springer Verlag, Berlin, Heidelberg, New York, 1983 [82] SZEGO, G Orthogonal Polynomials, fourth ed AMS, Providence, RI, 1975 [83] TRAUB, J., AND WOZNIAKOWSKI, H General Theory of Optimal Algorithms Academic Press, Orlando, San Diego, San Francisco, 1980 [84] TREFETHEN, L N., AND SCHREIBER, R S Average-case stability of gaussian elimination SIAM J Matrix Anal Appl 11,3 (1990), 335-360 [85] TUKEY, J W., AND COOLEY, J W An algorithm for the machine calculation of complex Fourier series Math Comp 19 (1965), 197-30l [86] VARGA, J Matrix Iterative Analysis Prentice Hall, Englewood Cliffs, N.J., 1962 [87] WILKINSON, J H Rounding Errors in Algebraic Processes Her Majesty's Stationary Office, London, 1963 [88] WILKINSON, J H The Algebraic Eigenvalue Problem Oxford University Press, Oxford, UK, 1965 [89] WILKINSON, J H., AND REINSCH, C Handbook for Automatic Computation, Volume II, Linear Algebra Springer Verlag, New York, Heidelberg, Berlin, 1971 [90] WITTUM, G Mehrgitterverfahren Spektrum der Wissenschajt (April 1990), 78-90 330 [91] References WULKOW, M Numerical treatment of countable systems of ordinary differential equations ZIB Report TR 90-8, ZIB, Berlin, 1990 [92] Xu, J Theory of Multilevel Methods PhD thesis, Penn State University, 1989 [93] H On the multi-level splitting of finite element spaces Numer Math 49 (1986), 379-4l2 YSERENTANT, Software 331 Software For most of the algorithms described in this book there exists rather sophisticated software, which is public domain Of central importance is the netlib, a library of mathematical software, data, documents, etc Its address IS http://www.netlib.org/ Linear algebra (LAPACK): http://www.netlib.org/lapack Especially linear eigenvalue problems (EISPACK): http://www.netlib.org/eispack Please study the therein given hints carefully (e.g., README, etc.) to make sure that you download all necessary material Sometimes a bit of additional browsing in the neighborhood is needed The commercial program package MATLAB also offers a variety of methods associated with topics of this book In addition, the book presents a series of algorithms as informal algorithms which can be easily programmed from this description-such as the fast summation of spherical harmonics Numerous further programs (not only by the authors) can be downloaded from the electronic library Elib by ZIB, either via the ftp-oriented address http://elib.zib.de/pub/elib/codelib/ or via the web-oriented address http://www.zib.de/SciSoft/CodeLib/ All of the there available programs are free as long as they are exclusively used for research or teaching purposes Index A-orthogonal, 250 Abel's theorem, 126 Aigner, M., 306 Aitken's L~.2-method, 114 Aitken-Neville algorithm, 184 algorithm invariance, 12 reliability, speed, almost singular, 45, 73 Arnoldi method, 254 Arrhenius law, 79 asymptotic expansion, 292 automatic differentiation, 92 B-spline basis property, 224 recurrence relation, 221 Bezier curve, 208 points, 208 Babuska, 1., 315 backward substitution, Bernoulli numbers, 288 Bessel functions, 159, 177 maze, 159 bi-cg-method, 255 bifurcation point, 100 Bjorck, A., 66, 78 Bock, H G., 96 Bornemann, F A., 261 Brent, R P., 85 Brusselator, 112 Bulirsch sequence, 297 Bulirsch, R., 293 Businger, P., 72 cancellation, 27 cascadic principle, 267 Casorati determinant, 157, 177 cg-method, 252 preconditioned, 257 termination criterion, 252, 260 Chebyshev abscissas, 195 approximation problem, 60 iteration, 247 nodes, 184, 196 polynomials, 193, 248 min-max property, 193, 246, 253 Cholesky decomposition rational, 15 Christoffel-Darboux formula, 285 334 Index complexity of problems, condition intersection point, 24 condition number absolute, 26 componentwise, 32 of addition, 27 of multiplication, 32 of scalar product, 33 relative, 26 Skeel's, 33 conjugate gradients, 252 continuation method, 92 classical, 102 order, 104 tangent, 103, 108 convergence linear, 85 model, 309 monitor, 307 quadratic, 85 super linear, 85 Cooley, W., 202 cost QR-factorization, 69, 72 Cholesky decomposition, 16 Gaussian elimination, QR method for singular values, 137 QR-algorithm, 132 Cramer's rule, Cullum, J., 266 cylinder functions, 159 de Boor algorithm, 235 de Boor, C., 204, 309 de Casteljau algorithm, 213 detailed balance, 143 Deuflhard, P., 73, 90, 261, 308 eigenvalue derivative, 120 Perron, 140 elementary operation, 23 Ericsson, T., 266 error absolute, 25 analysis backward, 36 forward, 35 equidistribution, 316 linearised theory, 26 relative, 25 extrapolation algorithm, 295 local, 316 methods, 291, 295 sub diagonal error criterion, 304 tableau, 292 Farin, G., 204 FFT,203 fixed-point Banach theorem, 84 equation, 82 iteration, 82, 239 method symmetrizable, 242 Fletcher, R., 255 floating point number, 22 forward substitution, Fourier series, 152, 200 transform, 197 fast, 201 Francis, J G F., 127 Frobenius, F G., 140 Gaches, J., 50 Gauss Jordan decomposition, Newton method, 109 Seidel method:, 240 Gauss, C F., 4, 57 Gautschi, W., 164 generalized inverse, 76 Gentleman, W M., 70 Givens fast, 70 rational, 70 rotations, 68 Givens, W., 68 Goertzel algorithm, 171 Goertzel, G., 171 Golub, G H., 47, 72, 119 graph, 140 irreducible, 140 Index strongly connected, 140 greedy algorithm, 306 Green's function discrete, 157, 177 Griewank, A., 92 Lebesgue constant, 183 Leibniz formula, 191 Leinen, P., 261 Levenberg-Marquardt method, 98, 117, 149 Hackbusch, W., 244 Hagemann, L A., 244 Hall, C A., 230 Hammarling, S., 70 Hermite interpolation cubic, 186 Hestenes, M R., 252 Higham, N J., 46 homotopy, 111 method,l11 Horner algorithm, 169 generalized, 170 Householder reflections, 70 Householder, A S., 70 Manteuffel, T A., 249 Markov chain, 137 nearly uncoupled, 147, 150 reversible, 144 uncoupled, 145 process, 137 Markov, A A., 137 Marsden identity, 223 matrix bidiagonal, 134 determinant, 1, 12, 30 Hessenberg, 132, 254 incidence, 141 irreducible, 140 norms, 53 numerical range, 17 permutation, primitive, 142 Spd-, 14 stochastic, 137 triangular, Vandermonde, 181 maximum likelihood method, 59 measurement tolerance, 59 Meixner, J., 162 Meyer, C D., 119 Meyer, W W., 230 Miller algorithm, 167 Miller, J C P., 166, 168 monotonicity test natural, 90, 106 standard, 90 multigrid methods, 244, 313, 320 incidence matrix, 141 information theory Shannon, 308 initial value problem, 270 interpolation Hermite, 185 nodes, 179 iterative refinement for linear equations, 13 for linear least-squares problems, 66,78 Jacobi method, 240 Kato, T., 122 Krylov spaces, 250 Kublanovskaja, V N., 127 Lagrange polynomials, 181 representation, 182 Lagrange, J L., Lanczos method spectral, 266 Lanczos, C., 262 Landau symbol, 26 LAPACK,13 Nashed, M Z., 76 needle impulse, 298, 309, 319 Neumann functions, 159, 177 series, 29 Neville scheme, 185 Newton correction, 88 335 336 Index simplified correction, 91 Newton method affine invariance, 88, 90 complex, 115 for square root, 86 nodes Gauss-Christoffel, 282 of a quadrature formula, 273 nonlinear least-squares problem almost compatible, 96 compatible, 93 norm £1_,271 energy, 249 Frobenius, 53 matrix, 53 spectral, 53 vector, 53 normal distribution, 59 numerical rank, 73 numerically singular, 45 Oettli, W., 50, 55 Ohm's law, 58 pcg-method, 257, 267 Penrose axioms, 76 Perron cluster, 147 analysis, 143 Perron, 0., 138 pivot element, row, pivoting column, conditional, 238 partial, total, polynomials Bernstein, 205 Chebyshev, 154, 176, 193, 248, 285 Hermite, 186, 285 Laguerre, 285 Legendre, 164, 176, 285 orthogonal, 153, 279, 285 trigonometric, 197 power method direct, 124 inverse, 125 Prager, W., 50, 55 preconditioning diagonal, 260 incomplete Cholesky, 260 pseudo-inverse, 75, 93, 109 QR-factorization, 76 singular value decomposition, 133 QR decomposition column permutation, 72 QR-algorithm shift strategy, 131 quadratic equation, 28, 81 quadrature condition of problem, 271 error, 283 estimator, 301 formula, 273 Gauss-Christoffel, 282 Newton-Cotes, 275 Gauss-Chebyshev, 285 Gauss-Christoffel, 285 Gauss-Hermite, 285 Gauss-Laguerre, 285 Gauss-Legendre, 285 numerical, 270 parameter-dependent, 312 rank decision, 73 determination, 96 Rayleigh quotient, 262 generalized, 265 refinement global, 317 local, 317 Reinsch, C., 41, 132, 171 residual, 49 Rheinboldt, W C., 315 Richardson method, 240 relaxed, 243 Rigal, J L., 50 Ritz-Galerkin approximation, 249 Romberg quadrature adaptive algorithm, 306 Romberg sequence, 296 Ruhe, A., 266 Rutishauser, H., 127 Index Sautter, W., 47, 73 scaling, 12 column, 12 row, 12 Schiiffke, W., 162 Schreiber, R S., 49 Schur normal form, 132 Schur, I., 20 Shannon, C E., 308 shift strategy, 130 Skeel, R D., 14, 33, 51, 56 Sonneveld, P., 255 sparse solvers, 238, 266 sparsing, 92 special functions, 151 spectral equivalence, 259 spherical harmonics algorithm, 163, 166 fast summation, 171 splines complete, 232 minimization property, 229 natural, 232 stability indicator, 37, 42 statistical model inadequate, 96 steepest descent method, 255 step size, 103 basic, 295, 299 internal, 295 step-size control, 300 Stewart, G W., 147 Stiefel, E., 252 stochastic process, 137 Stoer, J., 255 Sturm sequence, 178 subcondition, 73 substitution backward, forward, Taylor interpolation, 186 three-term recurrence relation adjoint, 170 condition, 161 dominant solution, 162 homogeneous, 156 inhomogeneous, 156, 158 337 minimal solution, 162 symmetric, 156 trigonometric, 40, 162, 170 Traub, J., Trefethen, L N., 49 Tukey, J W., 202 turning point, 100 van Loan, C., 47, 119 van Veldhuizen, R., 321 von Mises, R., 124 weight function, 279 weights Gauss-Christoffel, 282, 285 Newton-Cotes, 275 of a quadrature formula, 273 Wielandt, H., 125, 143 Wilkinson pathological example, 48, 56 Wilkinson, J H., 36, 46, 47, 123, 129, 131, 132 Willoughby, R., 266 Wittum, G., 244 work per unit step, 306 Wozniakowski, H., Wronski determinant, 158 Xu, J., 261 Young, D M., 244 Yserentant, H., 261 Free ebooks ==> www.Ebook777.com Texts in Applied Mathematics (continued from 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(Peter) Numerical analysis in modern scientific computing: an introduction / Peter Deuflhard, Andreas Hohmann. -2nd ed p cm - (Texts in applied mathematics; 43) Rev ed of: Numerical analysis 1995 Includes... Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets (continued after index) Peter Deuflhard Andreas Hohmann Numerical Analysis in Modern Scientific Computing An Introduction. .. Systems Kaplan/Glass: Understanding Nonlinear Dynamics Holmes: Introduction to Perturbation Methods Curtain/Zwart: An Introduction to Infinite-Dimensional Linear Systems Theory Thomas: Numerical