Mathematical Analysis and the Mathematics of Computation PDF
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Werner Römisch · Thomas Zeugmann Mathematical Analysis and the Mathematics of Computation Mathematical Analysis and the Mathematics of Computation ö Werner R misch Thomas Zeugmann (cid:129) Mathematical Analysis and the Mathematics of Computation 123 WernerRömisch ThomasZeugmann Institut für Mathematik Division of Computer Science Humboldt-Universität zuBerlin Hokkaido University Berlin Sapporo Germany Japan ISBN978-3-319-42753-9 ISBN978-3-319-42755-3 (eBook) DOI 10.1007/978-3-319-42755-3 LibraryofCongressControlNumber:2016952899 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart 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 orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. 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 authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To our wives, Ute and Yasuyo Preface Thisbookaimstoprovideacomprehensiveintroductiontothefieldof(clas- sical) mathematical analysis and mathematics of computation. While many students may have a perception of what mathematical analysis might be, which is often formed in school and college, there is, in our experience, a rather large surprise when studying mathematical analysis at the university level. Concerningmathematicsofcomputationthesituationmaybeevenworse. This field is rich and has seen an enormous development during recent decades. It comprises numerical analysis, computational discrete mathemat- ics, including algebra and combinatorics, number theory, and stochastic nu- mericalmethodsaswellascertainaspectsrelatedtomathematicaloptimiza- tion. In addition there are many interdisciplinary applications of all these researchsubjects,e.g.,supportvectormachines,kernel-basedlearningmeth- ods, pattern recognition, statistical learning theory, computer graphics, ap- proximation theory, and many more. So having a solid understanding of the foundations of mathematical anal- ysis and of its relations to the mathematics of computation is indispensable. However, these subjects are conventionally taught in separate courses which may considerably differ in the level of abstraction involved, and it may be difficult to relate the material taught to one another. Expressed a bit differ- ently,wearefacedwiththesituationthatstudentswhohavestudiedmodern expositions of mathematical analysis are very good at understanding mod- ern ideas but often have serious difficulties if it comes to really computing something. On the other hand, students who focused on more elementary presentationsaregoodatcalculatingbuttoooftenhaveseriousdifficultiesto understand the modern lines of thought and their benefits, e.g., why, when, and where we need a Banach space. Another aspect is that students who grew up with modern computing equipment no longer have a serious feeling for what it means to calculate function values, e.g., root functions, the sine function, or logarithm func- tions.Tohaveanotherexample,studentsmaybewellawareofapowerseries VII VIII Preface representation of the sine function or the cosine function, but may fail to explainwhythisrepresentationcoincideswiththesinefunctionorthecosine function, respectively, which they learned in school. On a higher level, when using computer programs to solve more difficult problemssuchasintegralequationsorordinarydifferentialequations,thenit is often not clear what the original problem formulated as an operator equa- tion in an infinite-dimensional space has to do with the computed solutions obtained by solving (linear) equations in a finite-dimensional space. Therefore, our goal is to present the whole material in one book and to carefully elaborate all these points. That is, we shall aim to develop the whole theory starting from a fairly simple axiom system for the real num- bers. Then we lay the foundations to a certain extent, i.e., we develop the theory, and then we exemplify where the theory developed so far is applica- ble. This in turn provides motivation for why a further development of the theory explained so far is necessary. In this way we go from sets, structures, and numbers to metric spaces, continuous functions in metric spaces, and then on to linear normed spaces and linear mappings. Subsequently, we turn our attention to the differential calculus and its applications, the integral calculus, the Gamma function, and linear integral operators. Then we study important aspects of approximation theory including numerical integration. The remaining parts of the book are devoted to ordinary differential equa- tions, the discretization of operator equations, and numerical solutions of ordinary differential equations. The intended audience ranges from undergraduate students in mathemat- ics, computer science, and related fields to all graduate students who are interested in studying the foundations of mathematical analysis and its wide range of applications. Moreover, the book may be useful as a reference and compendiumfordoctoralandotherstudentswhowishtogetadeeperunder- standing of the methodology, the techniques, and the groundwork of several applications they are trying to pursue. In general, it is intended as a four semester course comprising 15 lectures per semester, provided some choices are made. Thebookalsocontainsnumerousexercisesofvaryingdegreesofdifficulty, and at the end of each chapter additional problems are provided. The only differencebetweenexercisesandproblemsisthattheformershouldbesolved by the reader and/or as homework assignments, since they can be solved by having just studied the material up to the point where they appear. On the other hand, the problems are to a certain extent intended to shed additional lightonmanyinterestingfeaturesandoftenrequireadeeperunderstandingof theunderlyingconcepts.Sotheymaybebettersuitedforclassroomseminars or study groups. The material presented in this book goes to a large extent back to lec- tures, seminars, and compositions read, taught, and made by the authors at Humboldt University for students in mathematics and computer science at different stages of our own careers. Preface IX Wearegreatlyindebtedtotheinspiringlectures,seminars,anddiscussions at Humboldt University in Berlin and elsewhere which deeply influenced our view and passion for mathematical analysis and the mathematics of com- putation. In particular, our colleagues and teachers Roswita M¨arz, Konrad Gr¨oger,ArnoLangenbach,UdoPirl,WolfgangTutschke,andHelmutWolter carefully guided us through all the stages necessary to get acquainted with mathematical analysis and the mathematics of computation. WealsogratefullyacknowledgethesupportprovidedbyHeinzW.Englat Johannes Kepler University Linz, who shared with the first author his own lecture notes on related subjects. The second author would like to express his sincere gratitude to Norihiro YamadaandCharlesJordanfortheircarefulreadingofapreliminaryversion ofthisbookandforthemanyenlighteningdiscussionswehadonthematerial presented in these notes. Of course, all possible errors you may find in this book are ours. Finally,weheartilythankSpringer-Verlagforprofessionalsupportandad- vice. In particular, we gratefully acknowledge the encouragement, guidance, patience, and excellent cooperation with Ronan Nugent of Springer. Berlin, Sapporo Werner R¨omisch August 2016 Thomas Zeugmann Contents 1 Sets, Structures, Numbers 1 1.1 Sets and Algebraic Structures ......................... 1 1.2 The Real Numbers.................................. 6 1.3 Natural Numbers, Rational Numbers, and Real Numbers.... 10 1.4 Roots ............................................ 18 1.5 Representations of the Real Numbers ................... 20 1.6 Mappings and Numerosity of Sets ...................... 23 1.7 Linear Spaces ...................................... 34 1.8 Complex Numbers .................................. 38 Problems for Chapter 1 .................................. 43 2 Metric Spaces 47 2.1 Introducing Metric Spaces ............................ 47 2.2 Open and Closed Sets ............................... 50 2.3 Convergent Sequences ............................... 53 2.4 Banach’s Fixed Point Theorem ........................ 58 2.5 Compactness ...................................... 62 2.6 Connectedness ..................................... 69 2.7 Product Metric Spaces............................... 71 2.8 Sequences in R ..................................... 75 2.9 Sequences in the Euclidean Space Rm ................... 84 2.10 Infinite Series ...................................... 85 2.10.1 Rearrangements .............................. 95 2.11 Power Series and Elementary Functions ................. 102 2.11.1 Power Series ................................. 102 2.11.2 Elementary Functions.......................... 104 Problems for Chapter 2 .................................. 116 XI XII Contents 3 Continuous Functions in Metric Spaces 119 3.1 Introducing Continuous Mappings...................... 119 3.2 Properties of Continuous Functions..................... 126 3.3 Semicontinuous Functions ............................ 132 3.4 Variations of Continuity.............................. 135 3.5 Continuous Continuations ............................ 138 3.6 Continuous Functions over R.......................... 142 3.7 Functional Equations ................................ 148 Problems for Chapter 3 .................................. 155 4 Linear Normed Spaces, Linear Operators 157 4.1 Linear Normed Spaces ............................... 157 4.2 Spaces of Continuous Functions........................ 167 4.3 The Arzel`a–Ascoli Theorem........................... 176 4.4 Linear Bounded Operators............................ 179 4.5 The Space L(X ,X )................................. 182 1 2 4.6 The Banach–Steinhaus Theorem ....................... 187 4.7 Invertible Linear Operators ........................... 190 4.8 Compact Operators ................................. 196 Problems for Chapter 4 .................................. 198 5 The Differential Calculus 201 5.1 Real-Valued Functions of a Single Real Variable ........... 201 5.1.1 Mean Value Theorems ......................... 208 5.1.2 Derivatives of Power Series...................... 213 5.1.3 The Graph of the Sine Function and of the Cosine Function .................................... 226 5.1.4 Taylor’s Theorem ............................. 233 5.2 The Fr´echet Derivative and Partial Derivatives............ 239 5.2.1 Directional Derivatives, Partial Derivatives, and Fr´echet Derivatives............................ 242 5.2.2 Criterions ................................... 247 5.2.3 Higher-Order Partial Derivatives ................. 250 5.2.4 The Chain Rule .............................. 253 5.2.5 Generalized Mean Value Theorems ............... 258 5.2.6 Taylor’s Theorem Generalized ................... 263 5.2.7 A Linear Unbounded Operator................... 266 Problems for Chapter 5 .................................. 267 6 Applications of the Differential Calculus 269 6.1 Numerical Solutions of Nonlinear Systems of Equations ..... 269 6.1.1 Newton-Like Methods.......................... 271 6.1.2 Solving Systems of Linear Equations .............. 284 6.1.3 Quasi-Newton Methods ........................ 293 6.2 Solving Extremal Problems ........................... 296
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