Table Of ContentWerner 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
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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
